singularities and integrability of birational...
TRANSCRIPT
Simion Stoilow Institute of Mathematics Romanian Academy
HABILITATION THESIS
SINGULARITIES AND INTEGRABILITY OF BIRATIONALDYNAMICAL SYSTEMS ON PROJECTIVE PLANE
ADRIAN STEFAN CARSTEA
Specialisation: Mathematical Physics
Bucharest, 2013
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Contents
1 Abstract 4
2 Rezumat 6
3 Overview 83.1 Role of singularities . . . . . . . . . . . . . . . . . . . . . . . . 143.2 Integrable discrete systems . . . . . . . . . . . . . . . . . . . . 17
3.2.1 Singularity Confinement . . . . . . . . . . . . . . . . . 183.2.2 Complexity growth and algebraic entropy . . . . . . . . 20
3.3 Deautonomisation . . . . . . . . . . . . . . . . . . . . . . . . . 22
4 QRT mapping 244.1 The A1 matrices for various QRT mappings . . . . . . . . . . 28
5 Rational surfaces and elliptic fibrations 315.1 Discrete mappings and surfaces . . . . . . . . . . . . . . . . . 315.2 Preliminaries on rational elliptic surfaces . . . . . . . . . . . . 325.3 Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
6 Examples 376.1 Case ii-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386.2 Case i-2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446.3 Case ii-2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
7 Q4 mapping 527.1 Resolution of singularities and symmetry group . . . . . . . . 54
8 Minimization of elliptic surfaces from birational dynamics 588.1 Blowing down structure . . . . . . . . . . . . . . . . . . . . . 618.2 A simple example which needs blowing down . . . . . . . . . . 648.3 Discrete Nahm equations with tetrahedral symmetry . . . . . 668.4 Discrete Nahm equations with octahedral symmetry: . . . . . 678.5 Discrete Nahm equations with icosahedral symmetry . . . . . 69
9 Linearizable mappings 739.1 A non-autonomous linearizable mapping . . . . . . . . . . . . 739.2 Discrete Suslov system . . . . . . . . . . . . . . . . . . . . . . 74
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9.3 Other new linearisable systems . . . . . . . . . . . . . . . . . 769.4 Linearisable mappings of Q4 family . . . . . . . . . . . . . . . 80
10 Ultradiscrete (tropical) mappings 8310.1 Ultradiscrete singularities and their confinement . . . . . . . . 8510.2 Nonintegrable systems with confined singularities and inte-
grable systems with unconfined singularities . . . . . . . . . . 8910.3 A family of integrable mappings and their ultradiscrete coun-
terparts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9110.4 Complexity growth of ultradiscrete systems . . . . . . . . . . . 9410.5 Linearisable ultradiscrete dynamics: example from a biological
model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
11 General conclusions 103
12 Future research directions 104
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1 Abstract
The main topic of this thesis is the singularity analysis and integrability oftwo dimensional discrete dynamical systems (mappings). It is based essen-tially on the papers [1, 2, 3, 4, 5, 6, 7, 8] which posed the problem in thecontext of autonomous dynamical systems. The main tools are based onsingularity confinement, algebraic entropy and their rigurous formulation us-ing algebraic geometry of rational elliptic surfaces. It is an outcome of theprogress accomplished in the domain of discrete integrability which startedin 1990 with the introduction of singularity confinement and culminatig in2001 with the definitive classification of discrete Painleve equations [57] usinggeneralised Halphen surfaces and affine Weyl groups.
Gradually it was realised that the methods o algebraic geometry (ap-pearred for the first time in the pioneering work of Okamoto [54]) can be ex-tended to analyse the integrability and symmetries/invariants of two dimen-sional mappings. The paradigmatic example is the so called QRT (Quispel-Roberts-Thomson) mapping which is a very general birational dynamicalsystem possesing an invariant expressed by a ratio of biquadratic polyno-mials and which can be parametrised by Jacobi elliptic functions. Later onit served as the fundamental skeleton on which the whole discrete Painlevehierarchy has been erected.
In this thesis we shall be concerned with mappings different from QRT.There are physical and mathematical motivation for this aspect. The physi-cal motivation is that birational nonlinearity appears very often in biochem-ical and molecular biological models so the integrability and construction ofinvariants are extremely important because they give a kind of global under-standing of the phenomenon (unlike numerical simulations which give onlysolution to an initial value problem). Mathematically the algebraic geometryof discrete dynamical systems is done mainly for chaotic case (focusing onconstruction of invariant measures, entropy etc. [11], [10]). We consider thatintegrable discrete dynamical systems deserves the same treatment and webelieve that algebraic geometric methods will be extremely fruitfull.
In the chapter 3, we shall present an overview of the main instancesof integrability. The presentation is a physicist oriented one focusing onexamples and intuitive explanations. Various concepts and tools are brieflydiscussed for both ODE’s and PDE’s. Then the case of discrete systemsis presented in particular underlining the role played by the confining ofsingularities and algebraic entropy.
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In the chapter 4 we present the original result [1] of integrating the generalasymmetric QRT mapping and chapter 5 (which is the integral reproducingof [2])deals with the rigurous formulation of singularity confinement based onrational elliptic surfaces. We present a classification o mappings that preservean elliptic fibration and exchange or not singular fibers during evolution. Thisclassification is illustrated with various examples in chapter 6.
In the chapter 7 we present the original result about Q4 lattice equation[3]. Since it is a master equation for all soliton partial discrete equationconsistent around the cube we were interested to see how its travelling wavereduction fits in our classification.
Chapter 8 is devoted to the original results about systems which can belifted to automorphisms of non-minimal rational elliptic surfaces [4]. By us-ing blowing down structure we minimize them and show how this mappingspossesses invariants of higher order. Integrable discretisations of Nahm equa-tions are among the analysed cases.
Chapter 9 presents our original results on linearisable systems [7], [8].Here the power of algebraic geometry is limited since they have an infinitenumber o singularities and accordingly an infinite number o blowing ups isneeded. However for some examples the blow down structure can be used andlinearisation is found. Other examples (including degenerations of Q4) areanalysed experimentally showing some peculiar aspects at deautonomisation(the general presence of a free function).
Chapter 10 deals with ultradiscrete (tropical) mappings. It is shown thatpractically here there is no efective integrability detector. The tropicalisationo singularity confinement is critically analysed on various examples [5]. A niceexample [6] taken from a biological model is ending the chapter. Conclusions(ch. 11) and possible new research directions (ch.12) are the last chapters ofthis thesis
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2 Rezumat
Subiectul principal al acestei teze este analiza de singularitati si integra-bilitatea sistemelor dinamice discrete bidimensionale ( mappings ). Ea sebazeaza pe lucrarile [1, 2, 3, 4, 5, 6, 7, 8] in care s-au analizat in special sis-teme dinamice autonome (se discuta si deautonomizarea celor liniarizabile).Principalele instrumente sunt metoda confinarii singularitatilor, metoda en-tropiei algebrice si formularea lor riguroasa, folosind geometria algebrica asuprafetelor rational eliptice. Domeniul teoriei integrabilitatii discrete a in-ceput practic in 1990 (desi studii de ecuatii integrabile discrete au aparutodata cu nasterea teoriei solitonilor in anii 70) prin introducerea metodeiconfinarii singularitatilor si a culminat in 2001 cu clasificarea definitiva aecuatiilor Painleve discrete [57] folosind suprafetele Halphen generalizate sigrupurile Weyl afine .
Treptat, s-a observat ca metodele de geometria algebrica (aparute pentruprima data n lucrarea de pionierat a Okamoto [54]) pot fi extinse pentrua analiza integrabilitatea si simetriile/ invariantii sistemelor dinamice dis-crete bidimensionale. Exemplul paradigmatic este asa-numita ecuatie QRT(Quispel - Roberts - Thomson), care este un sistem dinamic birational foartegeneral, ce poseda un invariant exprimat printr-un raport de polinoame bi-quadratice si care poate fi parametrizat de functii eliptice Jacobi . Mai tarziuacest exemplu a reprezentat scheletul fundamental pe care toata ierarhiaPainleve discreta a fost construita.
n aceasta teza vom discuta ecuatii diferite de QRT . Exista atit motivatiefizica cit si matematica pentru aceasta. Motivatia fizica vine din faptul caneliniaritatea birationala apare foarte des n modele biologice, biochimice simoleculare, astfel incit integrabilitatea si constructia invariantilor sunt ex-trem de importante, deoarece dau un fel de ntelegere globala a fenomenului(spre deosebire de simularile numerice care dau doar o solutie la o problemaparticulara de conditie initiala/de frontiera) . Cea matematica este legatade faptul ca geometria algebrica se aplica cu succes in domeniul sistemelordinamice discrete haotice (cu precadere pentru calculul entropiei topologice[11], [10] etc.). Noi consideram ca si sistemele dinamice discrete integrabilemerita acelasi tratament iar apliactiile geometriei algebrice s-au dovedit pinaacum fructuoase.
n capitolul 3, vom prezenta o imagine de ansamblu a principalelor con-cepte de integrabilitate. Prezentarea este una folosita in general de fizicieniconcentrandu-se pe exemple si explicatii intuitive. Diferite concepte si instru-
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mente sunt discutate pe scurt atat pentru ecuatii diferentiale cit si pentrucele cu derivate partiale . Apoi, cazul sistemelor discrete este prezentat maiin detaliu subliniind rolul jucat de singularitati si entropiea algebrica .
n capitolul 4 vom prezenta rezultatul original [1] legat de integrarea di-recta a sistemului QRT asimetric; capitolul 5 ( care este reproducerea in-tegrala a lucrarii [2] ) se ocupa cu formularea riguroasa a confinarii de sin-gularitati bazata pe teoria suprafetelor eliptice rationale . Vom prezenta oclasificare a transformarilor care invariaza o fibrare eliptica si schimba saunu fibrele singulare in timpul evolutiei. Aceasta clasificare este ilustrata cuexemple variate in capitolul 6.
n capitolul 7 vom prezenta rezultatele originale despre ecuatia discretaQ4 [3] . Avnd n vedere ca este o ecuatie master pentru toate ecuatiile soli-tonice partial discrete cu consistenta cubica am fost interesati sa vedem cumreducerea de tip “unda progresiva” se incadreaza in clasificarea noastra .
Capitolul 8 este dedicat rezultatelor originale cu privire la sistemele carepot fi ridicate la automorfisme ale suprafetelor eliptice relativ non-minimale[4]. Prin utilizarea structurii de “blow-down” (eclatare inversa sau contractie)suprafata se minimalizeaza lucru care conduce in mod remarcabil la invariantide ordin superior . Discretizarile integrabile dale ecuatiilor reduse Nahm dinteoria cimpurilor de etalonare sunt printre cazurile analizate .
Capitolul 9 prezinta rezultatele noastre originale cu privire la sistemelelinearisable [7],[8]. Aici metodele de geometrie algebrica nu mai merg, deoareceaceste sisteme au un numar infinit o singularitati si in consecinta este nece-sar un numar infinit de eclatari (blow ups). Cu toate acestea, pentru uneleexemple daca se contarcta anumite clase de divizori exceptionali gasiti directprocedeul de liniarizare rezulta imediat. Alte exemple (inclusiv degenerariale Q4) sunt analizate experimental precum si o serie de aspecte particularecare apar la deautonomisare (prezenta generala a unei functii arbitrare indefinirea coeficientilor) .
Capitolul 10 se refera la sistemele ultradiscrete (tropicale) . Se arataca practic, aici nu exista nici un detector de integrabilitate. Tropicalizareametode confinarii singularitatatilor este analizata critic pe diverse exemple[5]. Capitolul se incheie cu un exemplu interesant de sistem partial ultra-discret liniarizabil venit dintr-un model biologic [6]. Concluzii (cap.11) siposibile noi directii de cercetare (cap. 12) sunt ultimele capitole ale acesteiteze
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3 Overview
The nonlinear science is coping today with a very deep problem; can onesingle out and describe to a certain degree of accuracy the complexity andself-organization exhibited by a nonlinear dynamical system? This fact im-poses the next question related to existence or non existence of some amountof hidden symmetry which would help in decribing that. Today these prob-lems are still open, despite many deep results obtained so far. In this context,the integrable nonlinear dynamical systems play a special role. First of all,the fact that they are integrable might give the impression that they arenot very important since they are very rare. Indeed! The great majorityof dynamical systems emerging from models (in physics, biology, economyetc) are not integrable and chaotic. But on the other hand, integrability ex-hibits a huge amount of hidden symmetry in various ways. This in turn givesrich structure which can be described in a clear and accurate way. It turnsout that many problems with unexpected structure and self-organization arerelated in some way to integrable systems. Roughly speaking, nonlinear dy-namical systems are “rules of evolution” for given quantities subjected toself-interaction (otherwise the dynamics would have been linear and not in-teresting). These rules can be put in a form of a nonlinear differential/discreteequations (ODE/O∆E), partial differential/discrete (PDE/P∆E) equationsor cellular automata (CA) (which are also discrete equations with the depen-dent variables having values in a denumerable or finite field).
Now, what is integrability? Given a nonlinear finite or infinite dimen-sional dinamical system (continuous or discrete) when can we say that it iscompletely integrable? This is a question with no clear cut answer. And thisis because there are many characterisation of integrability. First one whichseems to be quite intuitive is related to the possibility of finding a solutionwith enough number of free constants such that this solution could be con-sidered general. However this approach can somehow be misleading. Letstake the example of the famous logistic map which exhibits chaotic behaviourthrough period doubling in cascade birfucations [12]
xn+1 = 4xn(1− xn) (3.1)
The coefficient 4 in front of the RHS of the equation places the dynamicsin the fully chaotic region. However the equation can be solved analytically,
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namely the solution is given by:
xn =1
2(1− cos(2nc0))
which depends on an arbitrary constant c0 (fixed by the initial condition). Soone can say that the system has a general solution but still the system is in thechaotic region. This can be easily seen from the “buterfly effect” (exponentialgrowth related to the small variation of initial condition) namely:
dxndc0
= 2n−1 cos(2nc0)
Accordingly, the definition of integrability must be somehow posed ina different setting. A more appropriate way to characterize the integrabil-ity would be not relying on the exact solutions but rather on the globalinformation given by the integrals (or invariants or conservation laws) andsymmetries (we shall define later what means the symmetry). The first defi-nition of integral has been given by Darboux and Goursat [13]. They say that“an integral is general and useful if provides all the arbitrary data needed forexpressing the solution (whose existence must be proved or it is guaranteed bythe Cauchy theorem)”. In the same spirit, according to Poincare, to integratea differential equation is to find, for the general solution, a finite expression,possible multivalued, in terms of a finite number of functions. This definitionhas a very important connection with the concept of singularity as we shallsee in the next part.
In order to clarify the integrability concept we are going to give a verybrief description of main types of integrability used for ODE’s/PDE’s andthen to discrete systems. This review is based on the informations found in[65, 66]
The first type of integrability is the so called “integrability by quadra-tures”. For instance, if we take the nonlinear ordinary differential equation:
x = ax+ bx2 + cx3
the integral is given by:
I =x2
2− ax2
2− bx3
3− cx4
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and the integration is realised by the quadrature
t− t0 =∫
dx√2(I + ax2/2 + bx3/3 + cx4/4)
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The integral can be easily computed using Jacobi elliptic functions. Since Iis known by the initial condition the solution is readily obtained by inversion.This type of integrability is rather restrictive. For instance the equation
x+ x2 + t = 0
cannot be integrated by quadratures since it is non-autonomous. Howeverit is integrable by direct linearisation since we use the substitution x(t) =˙y(t)/y(t) then we get
y(t) + ty(t) = 0
which is solvable by Airy functions (this is somehow tautological since theAiry functions are defined by such a liniar ODE
The concept of integral is well known and it is applied widely speciallyin the hamiltonian mechanics (the above example is a particular one di-mensional hamiltonian system). For an N dimensional hamiltonian systemthe concept of integrability is clear. Because of the symplectic structure ofphase space the identification of N invariants of motion (integrals) allowssimplification to a set of trivial integrations (equations of motion in the socalled action-angle variables). This type of integrability is called Liouvilleintegrability.
The existence of integrals is not limited only to hamiltonian (conservative)systems. Even strongly dissipative systems can be integrated using the socalled time-dependent integrals. For instance, the famous Lorentz system[14]
x′ = σ(y − x)y′ = ρx− y − xzz′ = xy − bz
The following values of the parameters make the system integrable [15]:i)σ = 0ii)σ = 1/2, b = 1, ρ = 0iii)σ = 1, b = 2, ρ = 1/9iv)σ = 1/3, b = 0, ρ = freeIn the case:i) the system is linear
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ii) it has two time dependent integrals namely [16]:
I1 = (y2 + z2)e2t, I2 = et(x2 − z)
which reduces the system to a quadrature and the solutions can be expressedin terms of elliptic functions.iii) we do have one time dependent integral I3 = e2t(x2− 2z) which turn thesystem, after a change of variables, into the Painleve II equation.iv) can be combined in a third order equation which can be integrated togive the following time dependent integral:
I4 = e4t/3(xx− x2 + x4/4)
Changing variable to X = xet/3 and T = e−t/3 transform the above equationin the Painleve III equation
X ′′ =X ′2
X− X ′
T+X3 − 1
X(3.2)
where the new variables are defined through:
x(t) =2ic
3e−t/3X(T ), T = ce−t/3, c4 − (3K)4 = 0
The most powerfull type of integrability is the so called integrability byLax pair or by spectral methods. The idea is to write the nonlinear ODEunder consideration as a compatibility condition of two linear operators andto move the whole analysis in a different space where everything is linear(in the case of ODE in the space of monodromy data). Then, by inversespectral transform one can obtain the solution using some singular integrallinear equations (which cannot be solved in a closed form but still providesa lot of informations about solutions and their asymptotology)[17]
In the domain of PDE’s the situation is somehow similar. We do havehere direct linearisation, for instance in the case of two dimensional Navier-Stokes equation (Burgers equation) ut + uux + uxx = 0. The Cole-Hopftransform u = ∂x logF will turn it into the heat equation Ft + Fxx = 0.Finding integrals for a given PDE is a rather complicated procedure sinceany PDE is an infinite dimensional dynamical system. Accordingly, completeintegrability would require infinite number of integrals. It turns out that thisis the case for the so called soliton equations which can be cast in the so
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called Lax representation. For example, we take the famous Korteweg deVries equation which is a paradigmatic soliton equation:
ut + 6uux + uxxx = 0
It can be written in the following form [18]:
ut + 6uux + uxxx =∂L
∂t− [L,B] = 0
where the operators L = ∂2x + u(x, t) and B = −4∂3x − 3∂x, u are calledLax pairs. There is no algorithm for finding such operators and moreoverthey are not unique, making problem very hard. Since B is an antisymmetricoperator, the spectrum of L is invariant in time. So the problem of initialcondition will fix the spectral data and then, using the inverse method inthe scattering theory associated to L, the solution can be computed at anymoment of time by means of a linear integral equation. In addition, theintegrals of motion can be computed by traces of powers of operator L andevery integral of motion can be considered as a hamiltonian generating aflow. The family of such flows (equations) forms the so called KdV-hierarchy.In this way the Liouville integrability is intimately related to the existenceof Lax operators [19]. Related to this method is another vey interestingaspect specific to integrable PDE’s namely the bihamiltonian structure. Forexample, KdV equation it can be written easily in the hamiltonian form:
ut = J0δ
δu
∫
R
dx
(u3 − u2x
2
)≡ J0δuH1
where the symplectic operator is J0 = ddx. In the late seventies Magri dis-
covered a remarkable fact that it can be written in a different way using adifferent symplectic form
ut =
(d
dx3+ 2u
d
dx+ 2
d
dxu
)δ
δu
∫
R
dx
(u2
2
)≡ J1δuH0
The main consequence of this aspect is the possibility of generating the wholeKdV hierarchy using the recursion operator R = J−1
0 J1 through:
utn = J0δuHn+1 = J1δuHn
There is also another type of integrability which is specific to PDE namelyHirota integrability or integrability by Hirota bilinear method. This method
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has been introduced by Hirota in 1971 [20] and it says that if a quasilinearhyperbolic equation has a general N -soliton solution for any N then theequation is completely integrable. Physically speaking N -soliton solutionmeans that multiple collisions of arbitrary solitons are allowed. The mainadvantage of this method is that it is a direct one and can be applied to anyequation (continuous, discrete or semidiscrete). Again we take the KdV casefor illustration:
ut + 6uux + uxxx = 0 (3.3)
and we put u(x, t) = 2∂2x logF (x, t). Then our KdV will transform into amore complicated equation but bilinear and quadratic:
FFxt − FxFt + FFxxxx − 4FxFxxx + 3F 2xx = 0
which can be written as:
(DtDx +D4x)F · F = 0 (3.4)
The bilinear antisymmetric operator D is given by:
Dnxa(x) · b(x) := (∂x − ∂y)na(x)b(y)|x=y (3.5)
So in this way the nonlinearity of the original KdV equation has been swal-lowed and the bilinear equation shows practically the dispersion relationof the linear part of the KdV (this can be easily seen if we formally putDt → ω,Dx → k). We have to emphasize that this operator appeared alsolong time ago in the papers of Borel and Chazy [21], [22] where they showedthat equations written with this operator have solution which are complexentire functions.
Hirota proved that the N -soliton solution of the KdV equation can bewritten in terms of exponentials for the function F and has the followingexpression:
F (x, t) =∑
µ1,...,µN∈0,1
exp
(N∑
i=1
µi(kix− ωit) +∑
i<j
µiµjAij
)(3.6)
where ωi = −k3i is the dispersion relation of the linearised equation andAij = ((ki − kj)/(ki + kj))
2 is the interaction term between the soliton i andsoliton j.
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The method can be applied to any equation and it was observed that veryfew equations possesses N -soliton solution. The great majority have one andmaximum 2-soliton solution (namely N = 2 in the above formula (3.6)). Onthe other hand it was also observed that once 3-soliton solution is allowedthen automatically N -soliton solution is as well (this is still a conjecture) andthese equations are precisely those which are completely integrable. This facthas been studied thoroughly in the middle of eighties in the papers of Jimboand Miwa who showed the deep algebraic meaning of Hirota integrability[23](bilinear hierarchies are related to vertex operators representations ofaffine Lie algebras).
The importance of Hirota integrability relies on the fact that it appliesequatlly to discrete and differential-discrete equations and, moreover makesconnection with the role of singularities in the working definition of integra-bility. In the next chapter we shall discuss the role of singularities:
3.1 Role of singularities
We have seen that in the definition of Poincare solution means a finite ex-pression possible multivalued in terms of a finite number of functions. Whenwe discussed integrability by linearisation we encounter the special functions(Airy function in the example). However any special function is defined by alinear differential equation which is better studied when we extend the analy-sis to complex domain. The importance of the analysis in the complex spacehas been given in [24] where the analysis of a purely real power spectrum ofa signal has been done. It was shown that the high frequency behaviour ofthe Fourier transform depends on the location and nature of singularities inthe complex time plane.
Also, thanks to modern analytic theory in the complex domain, the globalinformation for an ODE can be obtained by an analytic continuation of lo-cally defined solutions. Now, we have to fix the statements here; A singularpoint is a point which breaks the analyticity of a solution for an ODE. Ifthere is multivaluedness in its neighbourhood, then the singularity is critical,or sometimes is called branched singularity. If one wish to define a functionthere is a requirement to treat singular points such that to restore the single-valuedness. This can be done by the so called uniformisation (by introducingcontours, Riemann surfaces etc.). This procedure can be always done in thecase of solutions of linear ODE, and this is possible because the location ofsingularities is fixed - namely are completely determined by the coefficients.
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So, according to Poincare definition every solution of a linear ODE definesa function and any linear ODE is an integrable system.
The nonlinear ODE lose this property because the location of singularitiesdepends on the initial conditions (or equivalently the integration constants).So defining a function as a solution of a nonlinear ODE becomes a hard prob-lem. Based on the results of Kowalevskaya [25], Fuchs and Painleve definedthe so called Painleve property and moreover they were able to constructthe most general order two nonlinear ODE’s which define new special func-tions beyond the elliptic ones, namely Painleve transcendents [26]. In a fewwords, the Painleve property imposes that the movable singularities (mean-ing that they depend on the initial conditions) of a given nonlinear ODE inthe complex plane, be at most poles. This fact places the dynamics of theconsidered ODE, on the Riemann sphere which is a compact and “regular”object and, accordingly, it is considered to be compatible with a smooth, pre-dictable dynamics (integrability). On the other hand, presence of branchingand essential singularities would proliferate the number of Riemann sheetsand the evolution is no longer “integrable”. We must stress on importantpoint here. It is considered that Painleve property is not just a predictor ofintegrability but practically a definition of integrability. As such it becomesrather a tautology than a criterion. This is the case because in the eightiesit was discovered that practically all integrable soliton PDE’s when reducedthey become equations which obey Painleve property. But on the other handit is crucial to make distinction between Painleve property and various al-gorithms for investigation (like Painleve test for instance which search formovable branch points subject to certain assumptions). There is no algo-rithm so far that guarantee Painleve property. However the application ofthese algorithms (mainly Painleve test) gave many interesting results evento chaotic ODE’s and PDE’s [27] . Still there are systems which are solvable(by quadratures and cascade linearization) and are not related to singularitystructure. Accordingly the Painleve property is not always equivalent withsolvability.
The key steps in application of the Painleve test are the following: Sup-pose we start with a system of nonlinear ODEs:
wi = Φ(w1, w2, ..., wn; t) (3.7)
Then the main idea is to see the asimptotology of a solution around a sin-gularity. For instance if t0 is a generic point one tries to see the dominant
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behaviour of the solution in the form:
wi = ai(t− t0)pi
where some reals,parts of pi are negative. Substituting in the sistem andrelying on the maximals balance principle [49] one can find possible dominantbehaviours. If one of the pi is noninteger then we are in the situation thatt0 is a movable branch point which is incompatible with Painleve propertyso in this case our system of ODE is likely to be not integrable (furtherrefinements can be done in the terms of the so called weak Painleve propertybut we shall not dwell on this). If all pi are integers then for each of themthen the leading behaviour can be seen as the dominant term of a Laurentseries around a movable pole:
wi = (t− t0)pi∞∑
0
a(m)i (t− t0)m (3.8)
where a(0)i = ai and the location of t0 is the first integration constant. The rest
of n− 1 constants are among the coefficients a(m)i and if their corerspondent
powers m are integers as well then the system is free of any branching andfrom the existence of n constants of integration, it is likely to be a completelyintegrable one. However this is just a necesary condition and moreover it doesnot capture the presence of essential movable singularities. Further investi-gation are necessary to establish the sufficiency by constructing integrals orLax pairs.
The bad role of branching of singularities can be grasped by the followingvery simple ODE [27]:
dx(t)
dt=
A
t− a +B
t− b +C
t− c (3.9)
Its integration by quadratures gives:
I = x(t)− A log(t− a)− B log(t− b)− C log(t− c) (3.10)
It well known that in the complex plane the logarithm is defined up to aninteger multiple of 2iπ so the integral (3.10) is determined up to the term2iπ(kA + mB + nC) with k,m, n ∈ Z. Now if one or two of the A,B,Care zero one can construct a two or one dimensional lattice and define I in
16
a unique way. But if ABC is not zero and A,B,C are linearly independentover the integers then we have a big indeterminacy in constructing I beacuseits value can fill densely the whole plane. So the integral is not useful if wehave dense multivaluedness and accordingly the above ODE is not integrablein this case.
Anyway one can argue that practically any dynamical system has anintegral, namely the initial condition. For instance if we consider the sistemof ODE:
xi = Fi(t, x1, ..., xn), i = 1, ..., n (3.11)
with initial condition xi(t0) = ci. The general solution of the system is:
xi(t) = fi(t, c1, ..., cn)
and by inverting it we get ci = Ii(t, x1, ..., xn) which we can consider to bethe integrals. However this inversion is not at all guaranteed to be singlevalued.
As we have said Painleve analysis has been thoroughly applied to nonlin-ear ODE and PDE’s and we are not going to insists here. We shall concen-trate mainly on discrete systems.
3.2 Integrable discrete systems
In the case of discrete systems the problem is completely different. Sincenow we have practically a recurence relation and everything is not local it isimpossible to apply the instruments of complex analysis (expansions aroundsingularities since now there are not neighbourhoods at all). In addition, it isimpossible to “discretise” the results from continuous systems because thereare many (in fact infinity) ways to discretise a continuum system and also adiscrete system can have many continuum limits. Many of the properties ofa continuous system are not at all preserved by the discretisation procedure.For instance, the Riccati equation
x′ = ax2 + bx+ c (3.12)
can be discretised either as writing the derivative as finite difference
xn+1 = ax2n + (b+ 1)xn + c (3.13)
17
or adding also some factor to the nonlinear term
xn+1 − xn = axnxn+1 + bxn + c⇐⇒ xn+1 =(b+ 1)xn + c
1− axn(3.14)
There is a huge difference between (3.13) and (3.14). The first one is a lo-gistic type mapping which is fully chaotic and the second is a homographicmapping which is integrable by a Cole-Hopf transform. So only the seconddiscretisation preserve the properties of the initial continuum Riccati equa-tion (3.12).
Because of these many ways to discretize a dynamical system one needs atool to detect at least the necessary conditions for integrability. By integra-bility in the discrete case we understand the same as in the continuos one,namely for a k-dimensional discrete nonlinear system we need [65]:
• existence of a sufficient number of integrals or conservation laws ex-pressed as rational expression Fk(xn, xn+1, ...xn+k) invariant under theaction of the mapping (however defining a hamiltonian structure is notguaranteed)
• possible linearisability by some transformations of dependent variables(like the above Cole-Hopf)
• existence of a Lax pair
• existence of general multisoliton solution in the infinite dimensionalcase
3.2.1 Singularity Confinement
A very efficient tool in detecting possible candidates for integrability is theso called singularity confinement test discovered in 1991 by A. Ramani, B.Grammaticos, V. Papageorgiou [28]. The idea has roots in the Painleveanalysis for continuos systems. As we have seen in the integrable case thesingularities are just poles. In the nonintegrable case sigularities may accu-mulate in fractal boundaries (so a natural boundary appears). Here in thediscrete setting the analysis is not based on Laurent expansion but ratheron the behaviour of iterations in some movable singular points. More pre-cisely, if the mapping leads to a singularity (depending on initial conditions)then after a finite number of steps (iterations) the singularity must dissapear
18
(confinement) without loss of information of initial condition. Thus the con-finement is reminiscent to absence of natural boundaries (where singularitiesaccumulate) in integrable continuous systems. On the other hand, preservingof information of initial condition is in contrast with chaotic dynamics wherestrange/fractal attractors absorb initial information. In order to implementpractically the criterion let us see how it works on a given example:
xn+1 + xn + xn−1 =a
xn+ b (3.15)
and suppose that starting with a given initial condition namely x−1 = f(where f is an arbitrary complex number) we hit at the next iteration x0 = 0.Now let us see what happens further on:
• x−1 = f
• x0 = 0
• x1 = −0− f + a/0 + b =∞
• x2 = −∞− 0 + a/∞+ b = −∞
• x3 =∞−∞− a/∞+ b =?
So one can see that the emergent infinities (which are just apparent singu-larities since they can be treated as nonsingular in the projective space) leadin the expression of x3 to a real singularity given by the ambiguity of∞−∞.To cope with this situation we use the argument of continuity with respectto initial conditions and consider x−1 = f and x0 = ǫ and then expand inpower series of ǫ. We get:
• x−1 = f
• x0 = ǫ
• x1 = aǫ−1 + b− f − ǫ
• x2 = −aǫ−1 + f + ǫ+ f−baǫ2 +O(ǫ3)
• x3 = aǫ−1−f−ǫ+ f−baǫ2−aǫ−1−b+f+ǫ+ a
−aǫ−1+f+O(ǫ)+b = ǫ+O(ǫ2)
• x4 = f +O(ǫ)
19
so the ambiguity is resolved and the initial information is recovered. Accord-ingly, the mapping is a possible candidate for an integrable one and indeedthe mapping can be integrated in terms of elliptic functions being in fact anautonomous limit of a discrete Painleve equation. This integrability detectorcan be applied also to partial discrete equations and what is really interest-ing it has a closed connection with Hirota bilinear formalism and existenceof multisoliton solution [29].
In our case we have seen that the singularity pattern (the sequence ofvalues of xn from initial condition up to its recovery) is (f, 0,∞,∞, 0, f) soit suggests that we can express xn using an entire function through:
xn =Fn−1Fn+2
FnFn+1
(3.16)
Now it is convenient to work with the discrete derivative of (3.15) namely:
xn+2+xn+xn+1−a
xn+1
−b−xn+1−xn−xn−1+a
xn+b = xn+2−xn−1−
a
xn+1
+a
xn= 0
Introducing (3.16) we get
Fn−1(Fn+4Fn − aF 2n+2) = Fn+3(Fn+2Fn−2 − aF 2
n)
which gives immediately the following Hirota bilinear form
Fn+2Fn−2 − aF 2n − Fn+1Fn−1 := (exp 2Dn − expDn − a)F · F = 0
where we have used the Hirota bilinear operators introduced in (3.5). Thesolution of the bilinear equation is an entire function and it can be shownthat is given by Riemann theta function (in fact the bilinear equation isnothing but a particular case of the famous Fay identitiy for Riemann thetafunctions)
3.2.2 Complexity growth and algebraic entropy
Unfortunately the singularity confinement test is just a necesary condition.This can be seen by the famous counterexample the so called Hietarinta-Viallet mapping [30]:
xn+1 + xn−1 = xn +1
x2n(3.17)
20
It has the following singularity pattern (f, 0,∞2,∞2, 0, f) so we do have con-fining. However a pathology can be seen in the numerical simulation of theequation which shows fully developed chaos. So a new stronger criterion isneeded. Here we give the most powerfull integrability criterion namely thealgebraic entropy. It is based on the notion of complextity introduced byArnold [31] which is the number of intersection points of a fixed curve withthe image of a second curve obtained under the iteration of the mapping.This idea has been extended by Viallet and colaborators who introduced theidea of algebraic entropy which encodes globally the complexity by meansof degrees of iterates. More precisely, if a birational mapping starts witha polynomial degree (of numerator or denominator) d then the n − th iter-ate will have the degree dn. When the mapping is integrable, some strongsimplifications occur and the degree growth is polynomial in n instead ofexponential. Let us illustrate on the example given by the mapping (3.15).Suppose x0 = p and x1 = q/r (which practically means iteration of numbersin the projective space). Then the following sequence of polynomial degreeof the denominator we have [32]:
1, 2, 4, 8, 13, 20, 28, 38, 49, 62, 76...
which can be fitted by the formula,
dn =1
8(9 + 6n2 − (−1)n)
where n is the iteration. So clearly the growth is polynomial in accord withthe integrability. On the other hand the Hietarinta-Viallet mapping which isconfining but chaotic has the following sequence of degrees:
0, 1, 3, 8, 23, 61, 162, 425...
and the degree growth obeys the recursion relation dn+4 = 3(dn+3−dn−1)+dn.This gives the expression of algebraic entropy
S = limn→∞
log dn/n
which in this case is S = (3 +√5)/2. A nonzero algebraic entropy is the
sign of chaos. The zero algebraic entropy (or equivalently the polynomialdegree growth) is the detector of integrability. The singularity confinementand algebraic entropy were proved to be instrumental in the majority ofanalysis of discrete systems. The discovery and properties of discrete Painleveequations relies on them.
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3.3 Deautonomisation
Another important aspect of singularity confinement and complexity growthis the procedure of deautonomisation [29]. It means that for a mappingone can put coefficients to depend on the independent variable and still themapping to be integrable. The method is quite simple namely to impose thesame singularity pattern (or complexity growth) for both autonomous andnonautonomous mapping. This will result in a constraint on coefficients. Letus illustrate on the same mapping (3.15) but having a = a(n)
xn+1 + xn + xn−1 =anxn
+ b (3.18)
So the singularity confinement is:
• xn−1 = f
• xn = ǫ
• xn+1 = anǫ−1 + b− f − ǫ
• xn+2 = −anǫ−1 + f + ǫ+ an+1/an(f−b)an
ǫ2 +O(ǫ3)
• xn+3 = −an+2+an+1+anan+2
ǫ+ (an+1
anb− an+1+an+2
anf)/anǫ
2 +O(ǫ3)
• xn+4 = −an+3−an+2−an+1+anan+an+1+an+2
anǫ+O(ǫ)
And indeed for generic an xn+4 is still divergent. But if we impose thatxn+4 to have the same expression as in the autonomous limit then we get thefollowing discrete linear equation for an namely:
an+3 − an+2 − an+1 + an = 0 =⇒ an = αn+ β + γ(−1)n
and our mapping becomes:
xn+1 + xn + xn−1 =αn+ β + γ(−1)n
xn+ b
with α, β, γ are free constants. The resulting mapping is nothing is nothingbut the discrete Painleve I or II equation. The name comes from the con-tinuous limit; namely if γ = 0 then the continuous limit can be computed asfollows: Consider t = ǫn and xn = w0(t) + ǫw1(t) + ǫ2w2(t) + O(ǫ3) wherewi(t) are unknown functions. For the shifted variable xn+1 the functions
22
wi(t) appear with shifted argument as well but we expand them in Taylorseries namely:
wi(t+ ǫ) = wi(t) + ǫw′i(t) +
ǫ2
2w′′
i (t) + ...
Also we need b = b0 + ǫb1 + ... and an = a0(t) + ǫa1(t) + ... The difficult partis that we do not know which of the ai are constant or not and which of thewi(t) is the dependent variable. This requires a lot of “intuition” so that iswhy an equation can have many continuous limits. In our case if we takexn = 1 +2 w(t), an = −3− ǫ4t, b = 6 then in the limit ǫ→ 0 we get:
w′′(t) + 3w2(t) + t = 0
which is the Painleve I equation.For γ not zero the equation has bigger freedom and can be written as an
asymmetric system. Namely if Xm = x2m, Ym = x2m+1 then we have:
Ym +Xm + Ym−1 =2αm + β + γ
Xm
+ b
Xm+1 + Ym +Xm =2αm+ α + β − γ
Ym+ b
Now if
Xm = 1 + ǫw + ǫ2u, Ym = 1− ǫw + ǫ2u, 2αm+ β = 1− ǫ3m, γ = −ǫ3c/4
then we get
u =1
4(w2 − w′ + t)
leading tow′′ − 2w3 − 2tw − c = 0
which is the Painleve II equation. The symmetric and asymmetric notion ofthe mapping will be clarified in the next section when we discuss QRT map-pings. The deautonomisation procedure was deeply investigated in connnec-tion with the theory of discrete Painleve equations and their properties. Weare not going to discuss this topic since it is too vast. Rather we shallfocus on the rigurous aspects of singularity confinement using tools fromalgebraic geometry which in turn will help not only to establish the inte-grable/nonintegrable character but also to integrate effectively any mappingby computing invariants.
23
4 QRT mapping
The basic object in the study of integrability of two dimensional mappingsis the so called QRT system. It was introduced in the beginning of ninetiesby Quispel, Roberts and Thomson [33] and by now is considered to be theparadigm of discrete integrability (in 2010 a whole book appeared dedicatedto QRT mappings [33]). The importance o this system relies on the act thatit gives a rather general discrete equation with a solution written in terms oelliptic function and possessing a biquadratic invariant.
There exist two families of QRT mappings [29], [33], [34], [35] whichare dubbed respectively symmetric and asymmetric for reasons which willbecome obvious below. One starts by introducing two 3×3 matrices, A0 andA1 of the form
Ai =
αi βi γiδi ǫi ζiκi λi µi
(4.1)
If both these matrices are symmetric the mapping is called symmetric. Oth-
erwise it is called asymmetric. Next one introduces the vector ~X =
x2
x1
and constructs the two vectors ~F ≡
f1f2f3
and ~G ≡
g1g2g3
through
~F = (A0~X)× (A1
~X) (4.2)
~G = (A0~X)× (A1
~X)
where the tilde denotes the transpose of the matrix. The components fi, giof the vectors ~F , ~G are, in general, quartic polynomial of x. Given the fi, githe mapping assumes the form:
xn+1 =f1(yn)− xnf2(yn)f2(yn)− xnf3(yn)
(4.3)
yn+1 =g1(xn+1)− yng2(xn+1)
g2(xn+1)− yng3(xn+1)(4.4)
In the symmetric case we have gi = fi and (4.3), (4.4) reduces to a singleequation
xm+1 =f1(xm)− xm−1f2(xm)
f2(xm)− xm−1f3(xm)(4.5)
24
with the identification xn → x2n, yn → x2n+1.Since ~F and ~G are obtained as vector products it is clear that the re-
sult will be the same if one replaces the matrices A0 and A1 by the linearcombinations ρ0A0 + σ0A1 and ρ1A0 + σ1A1 where ρ0, σ0, ρ1, σ1 are four freeparameters (with the only constraint ρ0σ1 6= ρ1σ0). This transformation canbe used in order to reduce the effective number of the parameters of the sys-tem to 14 in the asymmetric case and to 8 in the symmetric one. Howeverthis is still not the number of the effective parameters since we have the fullfreedom of a homographic transformation, which amounts to three param-eters, separately for x and y in the asymmetric case and just for x in thesymmetric one. Thus the final number of genuine parameters in this systemis 8 for the asymmetric mapping and 5 for the symmetric one.
The QRT mapping possesses an invariant which is biquadratic in x andy:
(α0 +Kα1)x2ny
2n + (β0 +Kβ1)x
2nyn + (γ0 +Kγ1)x
2n + (δ0 +Kδ1)xny
2n
+(ǫ0+Kǫ1)xnyn+(ζ0+Kζ1)xn+(κ0+Kκ1)y2n+(λ0+Kλ1)yn+(µ0+Kµ1) = 0
(4.6)
where K plays the role of the integration constant. In the symmetric casethe invariant becomes just:
(α0 +Kα1)x2n+1x
2n + (β0 +Kβ1)xn+1xn(xn+1 + xn) + (γ0 +Kγ1)(x
2n+1 + x2n)
+(ǫ0 +Kǫ1)xn+1xn + (ζ0 +Kζ1)(xn+1 + xn) + (µ0 +Kµ1) = 0
(4.7)
Viewed as a relation between xn and yn equation (4.6) is a 2-2 correspon-dence (and similarly for (4.7). While the generic biquadratic correspondenceis not in general integrable [36], leading to an exponential growth of thenumber of images and preimages of a given point, this is not the case for(4.6). (The symmetric case (4.7) is a well-known exception to this, beingindeed integrable). As a matter of fact it was argued in [29], due to thespecific structure of the mapping the correspondence (4.6) leads to just a lin-ear growth of the number of images of a given point. Further results whichstrengthen the integrability argument of (4.6) are the analyses presented in
25
[37] and [38]. As we have shown in [37] both symmetric and asymmetricQRT mappings pass the singularity confinement test. Moreover in the caseof the symmetric mapping we were able to show that, within a given genericsingularity pattern, the QRT mapping was the only one to satisfy the sin-gularity confinement criterion. The degree growth of the iterates of someinitial condition was studied in [38], using algrebraic entropy techniques. Wehave shown there that both symmetric and asymmetric mappings have a zeroalgebraic entropy and in fact lead to quadratic degree growth. We turn nowto the explicit integration of the QRT mapping. In the symmetric case theintegration (which, according to Veselov [39], is due to Euler) is presentedin a pedagogical way by Baxter [40]. Still we present below the details ofthe calculation since they will help understanding the asymmetric case. So,we start with the symmetric case and work with the integrated form. Tobegin with, we drop the explicit reference to the parameters of the A0 andA1 matrices and to the integration constant K and rewrite (4.7) as
αx2y2 + βxy(x+ y) + γ(x2 + y2) + ǫxy + ζ(x+ y) + µ = 0 (4.8)
(We shall, of course, return to the explicit consideration of the A0, A1 pa-rameters and K). We introduce a homographic transformation x = (aX +b)/(cX+d) (and the same for y which, in the symmetric case is just x shiftedby one step). Moreover we take d = 1, since we are not looking for a lineartransformation (neither in x nor in 1/x), and put a = sc. We demand thatthe coefficient of the XY (X +Y ) and of (X +Y ) terms vanish and also thatthe coefficient α of the X2Y 2 term be equal to the constant term µ. Fromthe latter we obtain:
c4 =αb4 + 2βb3 + (2γ + ǫ)b2 + 2ζb+ µ
αs4 + 2βs3 + (2γ + ǫ)s2 + 2ζs+ µ(4.9)
From the vanishing of the coefficient of the (x+ y) term we find:
s = −βb3 + (2γ + ǫ)b2 + 3ζb+ 2µ
2αb3 + 3βb2 + (2γ + ǫ)b+ ζ(4.10)
Requiring as a last constraint that the coefficient of the XY (X + Y ) termto vanish we obtain an equation for b which factorizes into a quartic factorwhich is unacceptable, since it would lead to a = c = 0, and an equation of
26
degree six:
b6(2α2ζ−αβǫ−2αβγ+β3)+b5(4α2µ+2αβζ−αǫ2−4αǫγ−4αγ2+β2ǫ+2β2γ)
+5b4(2αβµ−αǫζ−2αγζ+β2ζ)+10b3(−αζ2+β2µ)+5b2(−2αµζ+βǫµ+2βγµ−βζ2)+b(−4αµ2−2βµζ+ǫ2µ+4ǫγµ−ǫζ2+4γ2µ−2γζ2)−2βµ2+ǫµζ+2γµζ−ζ3 = 0
(4.11)
We can now, by a simple division, take α = µ = 1. (Here we are treating thegeneric, αµ 6= 0, case). Thus the biquadratic relation (3.1) is reduced to:
X2Y 2 + γ(X2 + Y 2) + ǫXY + 1 = 0 (4.12)
(where we indicated by a tilde the parameters of the equation resulting fromthe homographic transformation.). The parametrisation of (4.12) can begiven in terms of elliptic functions We introduce the ansatz X = A sn(z),Y = A sn(z + q) where sn(z) denotes an elliptic sine of argument z andmodulus k. Substituting this ansatz into the (4.12) we find A2 = k andmoreover k satisfies the second-degree equation:
k2 +(γ +
1
γ− ǫ2
4γ
)k + 1 = 0 (4.13)
Having obtained k from this equation we can compute q through k sn2(q) +1 = 0. Thus the biquadratic relation (4.8) can indeed be parametrised interms of elliptic functions.
We turn now to the asymmetric case. The invariant, with the sameconventions as for (4.8), is now:
αx2y2 + βx2y + γx2 + δxy2 + ǫxy + ζx+ κy2 + λy + µ = 0 (4.14)
We introduce two distinct homographic transformations x = (aX+b)/(cX+d) and y = (eY + f)/(gY + h). As in the symmetric case we can taked = h = 1 and, in order to better organise the calculations we put a = sc,e = tg. We choose the parameters c and g so as to put κ = γ and α = µ.We find the relations:
c2
g2=
αb2t2 + βb2t+ γb2 + δbt2 + ǫbt+ ζb+ κt2 + λt+ µ
αf 2s2 + βfs2 + γs2 + δf 2s+ ǫfs+ ζs+ κf 2 + λf + µ(4.15)
27
c2g2 =αb2f 2 + βb2f + γb2 + δbf 2 + ǫbf + ζb+ κf 2 + λf + µ
αt2s2 + βts2 + γs2 + δt2s+ ǫts+ ζs+ κt2 + λt+ µ(4.16)
The parameters b, f, s, t are chosen so as to put to zero the coefficientsβ, δ, ζ , λ. We find
s = − δbf2 + ǫbf + ζb+ 2κf 2 + 2λf + 2µ
2αbf 2 + 2βbf + 2γb+ δf 2 + ǫf + ζ(4.17)
t = −βb2f + 2γb2 + ǫbf + 2ζb+ λf + 2µ
2αb2f + βb2 + 2δbf + ǫb+ 2κf + λ(4.18)
There remain two equations for f and b. Taking the resultant for f , say,we obtain for b an equation of degree 20. However it turns out that thepolynomial of degree 20 factorizes into two quartic ones and the square of apolynomial of degree 6. These expressions, obtained with the help of com-puter algebra, are prohibitively long for a display here. Still, we were ableto show that the roots of the two quartic polynomials were unacceptable:they lead to the vanishing or divergence of c and g, in which cases the wholecalculation collapses. Thus, as in the symmetric case, the condition is givenin the form of an equation of degree 6 (for which, generically, no problemarises). Once b is obtained, f can be computed from the solution of a quarticequation.
Thus in the end, after all the simplifications have been implemented,(4.14) is reduced to precisely (4.12). Thus the solution of the full “asymmet-ric” biquadratic relation is again given in terms of elliptic functions. However,since the two homographic transformations which take us back from the el-liptic sines to the x, y that parametrise (4.14) are not the same for x and y,the solutions in the asymmetric case are not simply related as in the sym-metric case where one is the ‘upshift’ of the other. Note however, that onlythe homography is different for x and y. The step q of the argument of theelliptic function is the same at each iteration.
4.1 The A1 matrices for various QRT mappings
In this section we are going to show some examples of mappings which can beanalysed with the tools developed above. It turns out that one can choosethe A1 matrix to depend only on the ‘family’ of the equation and put allthe details into the A0 matrix. For a given equation, once the A1 matrix isknown, the construction of the corresponding A0 is elementary. The utility of
28
the mappings we give below resides alos in the fact that they are autonomousform of some discrete Painleve transcendents. In what follows we present theresults without their derivation: once the form of the matrix is given one canverify the results in a straightforward way. We give the general form of theequation and the corresponding A1 matrix
(I) xn+1 + xn−1 = f(xn) A1 =
0 0 00 0 00 0 1
(II) xn+1xn−1 = f(xn) A1 =
0 0 00 1 00 0 0
(III) (xn+1 + xn)(xn + xn−1) = f(xn) A1 =
0 0 00 0 10 1 0
(IV) (xn+1xn − 1)(xnxn−1 − 1) = f(xn) A1 =
0 0 00 1 00 0 −1
(V) (xn+1+xn+2z)(xn+xn−1+2z)(xn+1+xn)(xn+xn−1)
= f(xn) A1 =
0 0 10 2 2z1 2z 0
(VI) (xn+1xn−z2)(xnxn−1−z2)(xn+1xn−1)(xnxn−1−1)
= f(xn) A1 =
1 0 00 −z2 − 1 00 0 z2
(VII) (xn+1−xn−z2)(xn−1−xn−z2)+xnz2
xn+1−2xn+xn−1−2z2= f(xn) A1 =
0 0 10 −2 −2z21 −2z2 z4
(VIII) (xn+1z2−xn)(xn−1z2−xn)−(z4−1)2
(xn+1z−2−xn)(xn−1z−2−xn)−(z−4−1)2= f(xn) A1 =
0 0 z4
0 −z2(z4 + 1) 0z4 0 (z4 − 1)2
29
The forms presented above correspond to symmetric mappings but they canbe extended to asymmetric ones directly, the A1 matrix being the same. Tothese cases one must add the explicitly asymmetric one
(IX) xn+1 + xn = f(yn), ynyn−1 = g(xn) A1 =
0 0 00 0 10 0 0
Now once the A0, A1 matrices are obtained, one can proceed to the ex-plicit integration of the mapping. To do this, one uses the invariant and theinitial conditions in order to compute the integration constant K. Then oneconstructs the corresponding α, β... of (4.8), (4.14) through α = α0 +Kα1,β = β0+Kβ1 etc. The important remark is that these equations depend ex-plicitly on the integration constant. Thus the homographic transformationsand the details of the elliptic functions (modulus k and step q) are differentfor every initial condition. This explains why the brute force computationsof the solutions of a given QRT mapping are particularly hard.
Remark 4.1. In 2004 Tsuda [41] somehow solved the miracle of integrabilityof the QRT mapping. Practically he proved that the general QRT mapping isnothing but the famous group law on an elliptic curve. More precisely if P1+P2 = P3 is the addition on the elliptic curve in the group structure then onecan coordonatize the points Pi through P1 = (xn, yn−1), P2 = (xn+1, y), P3 =(xn+1, yn+1) from (2.3a) and (2.3b)
30
5 Rational surfaces and elliptic fibrations
Starting from this chapter we are goingt to study the singularity confine-ment in a rigurous way. Practically we shall show that a rigurous singularityanalysis can be used to integrate effectively the mapping (in this case by con-structing the invariants). In this spirit we are going to use some elementarytools from algebraic geometry o rational elliptic surfaces and a s a first out-come both singularity confinement and algebraic entropy will aquire rigurousformulation. In addition we shall see how singularity analysis will give alsosymmetries and the method of linearisation for linearisable systems (herethe problem is more complicated since in the case of linearisable systems thenumber of singularities is infinite). Various examples will be given and in theend we shall focus on tropical (ultradiscrete) mappings.
5.1 Discrete mappings and surfaces
In order to see how we go from mappings to surfaces we start from the sameexample (3.15)
xn+1 + xn−1 + xn = a/xn
It is an order two equation which can be written as a system defined on C2
(or P2 if we include infinities):
φ :
xn+1 = ynyn+1 = −xn − yn + a
yn
. (5.1)
It can be seen also as a chain of birational mappings ... → (x, y) →(x, y)→ (x, y)→ ... where x = xn−1, x = xn, x = xn+1 and so on.
Each step is an automorphism of the field of rational functions C(x, y).Now singularity confinement means:
(f, 0)︸ ︷︷ ︸(x0,y0)
→ (0,∞)︸ ︷︷ ︸(x1,y1)
→ (∞,∞)︸ ︷︷ ︸(x2,y2)
→ (∞, 0)︸ ︷︷ ︸(x3,y3)
→ (0, f)︸ ︷︷ ︸(x4,y4)
The secret is the follwing: if (x0, y0) = (f, ǫ) then the foolowing products arefinite
x1y1 = a+O(ǫ),x2y2
= −1 +O(ǫ), x3y3 = −a+O(ǫ)
31
Now let us construct a surface by glueing
C2 ∪C2 =
(x1,
1
x1y1
)∪(x1y1,
1
y1
)
But this is nothing but blow up of the affine space SpecC[x, Y ] with thecenter (x, Y ) = (0, 0) which gives the surface (Y = 1/y):
X1 = (x, Y, [z0 : z1]) ∈ SpecC[x, Y ]× P1 |xz0 = Y z1 =
= SpecC[x, 1/xy] ∪ SpecC[xy, 1/y]
So by blowing up C2 in the points (x1, y1) = (0,∞), (x2, y2) = (∞,∞), (x3, y3) =
(∞, 0) the equation then make sense on this new surface given by glueing suchaffine schemes. Accordingly we do analize any discrete order two nonlinearequation by identifying the singularities and blow them up.
5.2 Preliminaries on rational elliptic surfaces
We begin by the following definition: A complex surfaceX is called a rationalelliptic surface if there exists a fibration given by the morphism: π : X → P
1
such that:
• for all but finitely many points k ∈ P1 the fibre π−1(k) is an elliptic
curve
• π is not birational to the projection : E × P1 → P
1 for any curve E
• no fibers contains exceptional curves of first kind.
Blowing up: Let X be a smooth projective surface and let p be a point on X.There exist a smooth projective surface X ′ and a morphism π : X ′ → Xsuch that π−1(p) ∼= P
1 and π represents a biholomorphic mapping fromX ′ − π−1(p) → X − (p). The morphism is called blow-down and the cor-respondence π−1 is called blow-up of X at p as a rational mapping. Forexample if X is the space C
2 and p is a point of coordinate (x0, y0) then wedenote blow-up of X in p
X ′ = (x− x0, y − y0; ζ0 : ζ1) ∈ C2×P
1 |(x− x0)ζ0 = (y − y0)ζ1
by (we use the coordinates notation rather than glueing affine schemes)
π : (x, y)←− (x− x0, (y − y0)/(x− x0)) ∪ ((x− x0)/(y − y0), y − y0)
32
Space of initial conditions: Let Yi be smooth projective surfaces and letφi : Yi → Yi+1 be a sequence of dominant rational mappings. A sequence ofrational surfaces Xi is called the space of initial conditions for the sequenceφi if each φi is lifted by blowing ups to the mappings φ′
i : Xi → Xi+1 suchthat the set of indeterminate points of φ′
i is empty.Also we denote the group of divizors of a variety X by Div(X). The
Picard group of X is the group of isomorphism classes of invertible sheaveson X and it is isomorphic to the group of linear equivalence classes of divisorson X. We denote it by Pic(X).
Total transform and proper transform: Let π−1 : X → Y be the blow upat the point p and D be a divisor on X. The bundle mapping π∗(D) on Yis called total transform of D and for any analytic subvariety V on X theclosure of π−1(V − p) in Y is called the proper tranform of V.
Let X be a surface obtained by N times blowing up of P1×P1. Then the
Picard group Pic(X) is isomorphic to a Z module (the Neron-Severi lattice)with the form:
ZHx + ZHy +N∑
i=1
ZEi
where Hx, Hy are the proper transforms of lines x = const., y = const. andEi is the total transform of the i− th blow up. In addition the intersectionnumbers of two divisors on X are given by the following basic formulas (validfor any i, j = 1...N):
Hx ·Hy = 1, Ei ·Ei = −1, Ei ·Ej = Ei ·Hx = Ei ·Hy = Hx ·Hx = Hy ·Hy = 0
A rational surface X is called a generalized Halphen surface if the anti-canonical divisor class −KX is uniquely decomposed into effective divisors as[−KX ] = D =
∑miDi(mi ≥ 1) such that Di ·KX = 0 Generalized Halphen
surfaces can be obtained from P2 by succesive 9 blow-ups. They can be clas-
sified by D in elliptic, multiplicative and additive type. A rational surface Xis called a Halphen surface of index m if the dimension of the linear system| − kKX | = 0, k = 1,m− 1 and | − kKX | = 1, k = m. A Halphen surfaceof index m is also referred to be a rational elliptic surface of index m. Thelinear system | − kKX | is the set of curves in P
2 (resp. P1×P
1) of degree3k (resp. 4k) passing through each point of blow-up with multiplicity k.It is known that any Halphen surface of index m contains a unique cubiccurve with multiplicity m It is known that if m ≥ 2 a Halphen pencil ofindex m contains a unique cubic curve C with multiplicity m, i.e. C is the
33
unique element of | −KX |. It is well known that if X is a Halphen surfaceof index m and C is nonsingular, then k(P1 + · · ·+ P9 − 3P0) is not zero fork = 1, . . . ,m−1 and zero for k = m (here + is the group law on C, P1, . . . , P9
are base points of blow-ups and 3P0 is equal by the group law to 3 crossingpoints with a generic line in P
2). Conversely, for a nonsingular cubic curveC in P
2, if k(P1 + · · · + P9 − 3P0) is not zero for k = 1, . . . ,m − 1 and zerofor k = m, then there exists a family of curves of degree 3m passing throughP1, . . . , P9 with multiplicity m, which constitutes a Halphen pencil of indexm(see chap. 5 §6 of [62] for more details).
It is known that a rational elliptic surface can be obtained by 9 blow-upsfrom P
2 and that the generic fiber of X can be put into a Weierstrass form:
f(x, y, k) = y2 + a1xy + a3y − x3 − a2x2 − a4x− a6,
where all the coefficients ai depend on k. Singular fibers can be computedeasily by the vanishing of the discriminant:
∆ ≡ −b22b8 − 8b34 − 27b26 + 9b2b4b6,
where b2 = a21+4a2, b4 = 2a4+a1a3, b6 = a23+4a6, b8 = a21a6+4a2a6−a1a3a4+a2a
23−a24. The discriminant has degree 12 which gives the number of singular
fibers together with their multiplicities. The singularities have been classifiedby Kodaira according to the irreducible components of singular fibers.
Now for any nonlinear birational discrete equation of the form:
xn+1 = f(xn, yn)
yn+1 = g(xn, yn)
In [57], Sakai showed that every discrete Painlve equation can be obtainedas a translational component of an affine Weyl group which acts on a familyof generalized Halphen surfaces, i.e. a rational surface with special divisorsobtained by 9-blow-ups from P
2. From this viewpoint the Quispel-Roberts-Thomson (QRT) mappings [33] are obtained by specializations of the surfacesso that they admit elliptic fibrations.
In autonomous setting, Diller and Favre [63] showed that if a Kahlersurface S admits an automorphism ϕ of infinite order, then (i) ϕ is ”lineariz-able”, i.e. it preserves the fibrations of a ruled surface [72]; (ii) ϕ preservesan elliptic fibration of S; or (iii) the algebraic (or topological) entropy of ϕ is
34
positive. The typical example of the second case is so called the QRT map-pings [33], while mappings not belonging to the QRT family are discoveredby several authors [70, 73, 74].
In this part, we classify these types of mappings by their relation withrational elliptic surfaces. For this purpose, we consider not only rationalelliptic surfaces but also generalized Halphen surfaces. In next section, wepropose a classification of autonomous rational mappings preserving ellipticfibrations. We also show an equivalent condition when a generalized Halphensurface becomes a Halphen surface of index m. Although our classificationis rather simple, existence of (simple) examples is nontrivial. In the nextsection, we extend this result into the case where C is singular by using ”theperiod map” for generalized Halphen surfaces.
5.3 Classification
Let X be a rational elliptic surface obtained by 9 blow-ups from P2. The
main result is the following classification.
Classification Let m be a positive integer, ϕ an automorphism of X whichpreserves the elliptic fibration αf0(x, y, z) + βg0(x, y, z) = 0. Such cases areclassified as follows.i-m) ϕ preserves α : β and the degree of fibers is 3m;ii-m) ϕ does not preserve α : β and the degree of fibers is 3m.
Remark 5.1.
• The QRT mappings belong to Case i-1) [41].
• In case ii-m), elliptic fibrations admit exchange of fibers.
• The integer m corresponds to the index m of X as a Halphen surface.
• It is well known (for example van Hoeji’s gave an algorithm [69] and [60]used it) that there exists a birational transformation on P
2 which mapsan (possibly singular) elliptic curve in P
2, αf0(x, y, z)+βg0(x, y, z) = 0,into the Wierstrass normal form. Since in general the coefficientsof this transformation are algebraic on a rational function α/β =g0(x, y, z)/f0(x, y, z), there exists a bialgebraic transformation from aHalphen surface of index m to that of index one that preserves the
35
elliptic fibrations. On the other hand, Proposition 11.9.1 of [64] showsnon-existence of a birational transformation from the Halphen surfaceof index m to that of index m′ (m 6= m′) that preserves the ellipticfibrations. (Precisely saying, Proposition 11.9.1 of [64] claims that ifan example of a mapping of the case i-2 is infinite order, then it is notbirationally conjugate to a mapping of the class i-1. But its proof isstill effective for the above assertion.) If two infinite order mappingspreserving rational elliptic fibrations are conjugate with each other bya birational mapping ψ, the mapping ψ preserves the elliptic fibrations.Thus, two infinite order mappings belonging to different classes of theabove classification are not birationally conjugate with each other.
In the rest of this section, we characterize Halphen surfaces as generalizedHalphen surfaces.
Let X be a generalized Halphen surface and Q the root lattice defined asthe orthogonal complement of D with respect to the intersection form andω a meromorphic 2-form on X with Div(ω) = −Dred, where Dred =
∑si Di.
Then, the 2-form ω determines the period mapping χ from Q to C by
χ(α) =
∫
α
ω
in modulo∑
γ Zχ(γ), where the summation is taken for all the cycles onDred (see examples in the next section and [57] for more details). Note thatif X is not a Halphen surface of index one, then the divisor D and thus ω(modulo a nonzero constant factor) are unique. The divisor D (or X itselfif X is not a Halphen surface of index one) is called elliptic, multiplicative,or additive type if the rank of the first homology group of Dred is 2, 1, or 0respectively.
Theorem 5.2.(ell) If a member of | −KX | is of elliptic type, then X is a Halphen pencil ofindex m iff χ(−kKX) 6= 0 for k = 1, . . . ,m− 1 and χ(−mKX) = 0.(mult) If a member of |−KX | is of multiplicative type, then the same assertionholds as in the elliptic case.(add) If a member of | −KX | is of additive type, then X is a Halphen pencilof index 1 iff χ(−KX) = 0, and never a Halphen pencil of index m ≥ 2.
Proof. Case (ell) is a classical result (see Remark 5.6.1 in [62] or referencestherein). Case (mult) and case (add) of index 1 are Proposition 23 in [57].
36
Similar to that proof, we can vary D and χ continuously to nonsingular case.Indeed, let P1, . . . , P9 be the points of blow-ups (possibly infinitely near, weassume P9 is the point for the last blow-up) and f0 be the cubic polynomialdefining D. There exists a pencil of cubic curves Cλ : fλ = f0 + λf1 = 0λ ∈ P
1 passing through the 8 points P1, . . . , P8. For small λ, the cubic curveCλ is close to D, and the meromorphic 2-form ωλ for Cλ is also close to ω.Let P ′
9 be a point close to P9 on Cλ such that
limλ→0
χλ(−mKX′) = limλ→0
∫
−mKX′
ω′ =
∫
−mKX
ω′ = χ(−mKX)
holds,y where X ′ is the surface obtained by blow-ups at P1, . . . , P8 and P ′9
instead of P9. Thus, χλ(−mKX′) 6= 0 holds if χ(−mKX) 6= 0 for smallλ, and therefore X does not have a pencil of degree 3m. Conversely, ifχ(−mKX) = 0, then χ′(−mKX′) is close to zero, and there exists P ′′
9 closeto P ′
9 on C ′ such that χλ(−mKX′′) = 0. Thus, we have
limλ→0
χλ(−mKX′′) = χ(−mKX).
Since X ′′ has (at least) a pencil of curves of degree 3m passing throughthe 9 points with multiplicity m and this condition is closed in the space ofcoefficients of polynomials defining curves, X also has the same property.
Remark 5.3. In Painleve context, for multiplicative case, χ is normalizedso that χ(γ) = 2πi for a simply connected cycle γ on some Di, and theparameter “q” is defined as q = expχ(−KX), i.e. the condition χ(−mKX) =0 corresponds to qm = 1. We must point out here that in [67] and [68] similarstudy has been done on q-Painleve equations, and it is reported that Eq. (3.1)of [67] with q =
√−1 preserves degree (4,4) pencil, which seems contradict
to the above theorem, but there the definition of q is different from ours (itssquare root is our q).
6 Examples
In this section, we are going to give examples for case i-2, ii-1 and ii-2. Atypical example of Case i-1 is the QRT mappings. There is some literatureon their relation to rational elliptic surfaces [41, 64], and we are not going todiscuss it here. In the first subsection, we investigate the action on the space
37
of initial conditions of some mapping of Case ii-1, which was proposed in [74].In the second subsection, we show that one of the HKY mappings belongsto Case i-2. Theoretically, from Theorem 5.2, we can construct mappings ofthe type i-m for any integers. Actually, let φ(q) be some q-discrete Painleveequation and q a primitive m-th root of unity, then φm(q) is autonomousand preserves the Halphen fibration of index m. However, the degrees ofmappings obtained in this way are very high. The HKY mapping is muchsimpler example. In the third subsection, we construct some example forCase ii-2, which we believe as the first example for this case.
6.1 Case ii-1
We start with a mapping [7, 74, 67] which preserves elliptic fibration of degree(2, 2) but exchanges the fibers:
xn+1 = −xn−1(xn − a)(xn − 1/a)
(xn + a)(xn + 1/a). (6.1)
In this subsection, studying space of initial conditions (values), we computethe conserved quantity, the parameter “q” and all singular fibers. We alsoclarify the relation with the q-discrete Painleve VI equation (qP (A
(1)3 ) in
Sakai’s notation) by deautonomizing the mapping (6.1), where the label A(1)3
corresponds to the type of space of initial conditions.First of all, in order to compactify the space of dependent variables, we
write the equations in projective space as a two component system:
φ : P1×P1 → P
1×P1, φ(x, y) = (x, y),
x = y
y = −x(y − a)(y − 1/a)
(y + a)(y + 1/a). (6.2)
We use P1×P
1 instead of P2 just because the parameters of blowing-up
points become easy to write. The projective space P1×P
1 is generated bythe following coordinate system (X = 1/x, Y = 1/y):
P1×P
1 = (x, y) ∪ (X, y) ∪ (x, Y ) ∪ (X, Y ).
38
The indeterminate points for the mappings φ and φ−1 are
P1 : (x, y) = (0,−a), P2 : (x, y) = (0,−1/a),P3 : (X, y) = (0, a), P4 : (X, y) = (0, 1/a),
P5 : (x, y) = (a, 0), P6 : (x, y) = (1/a, 0),
P7 : (x, Y ) = (−a, 0), P8 : (x, Y ) = (−1/a, 0).
Let X be the surface obtained by blowing up these points. Then, φ is liftedto an automorphism of X. Such a surface X is called the space of initialconditions. More generally, if a sequence of mappings φn is lifted to asequence of isomorphisms from a surface Xn to a surface Xn+1, each surfaceXn is called the space of initial conditions.
The Picard group of X is a Z-module:
Pic(X) = ZHx ⊕ ZHy ⊕8⊕
i=1
ZEi,
where Hx, Hy are the total transforms of the lines x = const., y = const. andEi are the total transforms of the eight points of blow-ups. The intersectionform of divisors is given by Hz ·Hw = 1− δzw, Ei ·Ej = −δij , Hz ·Ek = 0for z, w = x, y. Also the anti-canonical divisor of X is
−KX = 2Hx + 2Hy −8∑
i=1
Ei.
Let us denote an element of the Picard lattices by A = h0Hx + h1Hy +∑8i=1 eiEi (hi, ej ∈ Z), then the induced bundle mapping is acting on it as
φ∗(h0, h1, e1, ..., e8)
=(h0, h1, e1, ..., e8)
2 1 0 0 0 0 −1 −1 −1 −11 0 0 0 0 0 0 0 0 01 0 0 0 0 0 0 0 −1 01 0 0 0 0 0 0 0 0 −11 0 0 0 0 0 −1 0 0 01 0 0 0 0 0 0 −1 0 00 0 1 0 0 0 0 0 0 00 0 0 1 0 0 0 0 0 00 0 0 0 1 0 0 0 0 00 0 0 0 0 1 0 0 0 0
.
39
It preserves the decomposition of −KX =∑3
i=0Di:
D0 = Hx − E1 − E2, D1 = Hy − E5 − E6
D2 = Hx − E3 − E4, D3 = Hy − E7 − E8, (6.3)
which constitute the A(1)3 type singular fiber: xy = 1.
One can see that the elliptic curves
F ≡αxy − β((x2 + 1)(y2 + 1) + (a+ 1/a)(y − x)(xy + 1)) = 0
⇔ kxy − ((x2 + 1)(y2 + 1) + (a+ 1/a)(y − x)(xy + 1)) = 0
correspond to the anti-canonical class (these curves pass through all Ei’s forany α : β). This family of curves defines a rational elliptic surface. One cansee that even though the anti-canonical class is preserved by the mapping,the each fiber is not. More precisely, the action changes k in −k.
So, as a conclusion, the dimension of the linear system corresponding tothe anti-canonical divisor is 1. It can be written as αf1(x, y) + βf2(x, y) =0⇔ kf1(x, y)+f2(x, y) = 0 for α : β ∈ P(C) and deg f = deg g = (2, 2). Thiselliptic fibration is preserved by the action of the dynamical system but nottrivially in the sense that the fibers are exchanged. The conserved quantitybecomes higher degree as (f/g)ν for some ν > 1. In our case ν = 2 and theinvariant is exactly the same as the result of [74].
Remark 6.1. In order to have a Weierstrass model, we perform some homo-graphic transformations according to the algorithm of Schwartz [71]. Then,after long but straightforward calculations, we can compute the roots of theelliptic discriminant ∆(k) as
k1 = 0, multiplicity = 2
k2,3 = ±4(1 + a2)/a, multiplicity = 1
k4,5 = ±(1− a2)2/a2, multiplicity = 2
k6 =∞, multiplicity = 4.
We have A(1)1 singular fiber for k1 and k4,5, A
(1)0 fiber for k2,3, and A
(1)3 fiber
for k6. The mapping acts on these singular fibers as an exchange as (k1 →k1, k2 → k3 → k2, k4 → k5 → k4, k6 → k6).
Remark 6.2. If a surface is a generalized Halphen surface but not a Halphensurface of index 1, then the anti-canonical divisor −KX is uniquely decom-posed to a sum of effective divisors as −KX =
∑miDi and we can char-
acterize the surface by the type the decomposition. However, if the surface
40
is Halphen of index 1, it may have several types of singular fibers as thisexample.
Next, we consider deautonomization of the mapping φ. For that, we willuse the decomposition (6.3) of −KX which is preserved by the mapping,though the decomposition and hence the deautonomization are not unique(the fiber corresponding to k = 0 is also preserved).
The affine Weyl group symmetries are related to the orthogonal comple-ment ofDred = D1, . . . , D4. In order to see this, we note that rank Pic(X) =rank 〈Hx, Hy, E1, ...E8〉Z = 10. The orthogonal complement of Dred:
〈D〉⊥ = α ∈ Pic(X)|α ·Di = 0, i = 0, 3
has 6-generators:
〈D〉⊥ = 〈α0, α1, ..., α5〉Zα0 = E1 − E2, α1 = E3 − E4, α2 = Hy − E1 − E3
α3 = Hx − E5 − E7, α4 = E5 − E6, α5 = E7 − E8.
Figure 1: Singular fiber and orthogonal complement.
Related to them, we define elementary reflections:
wi : Pic(x)→ Pic(X), wi(αj) = αj − cijαi,
41
where cji = 2(αj · αi)/(αi · αi). One can easily see that cij is a Cartan
matrix of D(1)5 -type for the root lattice Q =
⊕5i=0 Zαi. We also introduce
permutations of roots:
σ10 : (α0, α1, α2, α3, α4, α5) 7→ (α1, α0, α2, α3, α4, α5)
σtot : (α0, α1, α2, α3, α4, α5) 7→ (α5, α4, α3, α2, α1, α0).
The group generated by reflections and permutations becomes an extendedaffine Weyl group:
W (D(1)5 ) = 〈w0, w1, ..., w5, σ10, σtot〉.
This extended affine Weyl group can be realized as an automorphisms ofa family of generalized Halphen surfaces which are obtained by allowing thepoints of blow-ups to move so that they preserve the decomposition of −KX
as
P1 : (x, y) = (0, a1), P2 : (x, y) = (0, a2),
P3 : (X, y) = (0, a3), P4 : (X, y) = (0, a4),
P5 : (x, y) = (a5, 0), P6 : (x, y) = (a6, 0),
P7 : (x, Y ) = (a7, 0), P8 : (x, Y ) = (a8, 0),
which can be normalized as a1a2a3a4 = a5a6a7a8 = 1. Accordingly, ourmapping lives in an extended affine Weyl group W (D
(1)5 ) and deautonomized
as
φ :
x = a1a2y
y = −x(y − a3)(y − a4)(y − a1)(y − a2)
with
(a1, a2, a3, a4, a5, a6, a7, a8, q)
7→ (− 1√qa6
,− 1√qa5
,−√q
a8,−√q
a7, a3, a4, a1, a2, q),
where
q =a1a2a7a8a3a4a5a6
. (6.4)
42
This mapping can be decomposed by elementary reflections as
φ∗ =σ10 σtot σ10 σtot w2 w1 w0 w2 w1 w0
and acts on the root lattice as
(α0, α1, α2, α3, α4, α5)
7→ (−α5,−α4,−α3, α2 + 2α3 + α4 + α5, α0, α1).
Hence, φ4 is a translational element of the extended affine Weyl group, andtherefore one of the q-Painleve VI equations (qP (A
(1)3 )) in Sakai’s sense, while
the original q-Painleve VI studied in [57] was
qPVI :
x = − y
a1a2
y = − (y − a1)(y − a2)x(y − a3)(y − a4)
with
(a1, a2, a3, a4, a5, a6, a7, a8, q)
7→ (−√qa5,−√qa6,−
a7√q,− a8√
q,−a1,−a2,−a3,−a4, q),
which is decomposed by elementary reflections as
qPVI = σ10 w1 w0 w2 w1 w0 w2 w1 w0
and acts on the root lattice as
(α0, α1, α2, α3, α4, α5)
7→ (−α4,−α5,−α3,−α2 + δ,−α0,−α1)
(δ = α0 + α1 + 2α2 + 2α3 + α4 + α5).
At the last of this subsection, we define the period map χ : Q → C andcompute q (6.4) by using
ω =1
2πi
dx ∧ dyxy
. (6.5)
43
For example, χ(α0) is computed as follows. The exceptional divisors E1 andE2 intersect with D0 at (x, y) = (0, a1) and (0, a2), and χ(α0) is computed as
χ(α0) =
∫
|x|=ε, y=a2∼a1
1
2πi
dx ∧ dyxy
=−∫ a1
a2
dy
y
= loga2a1,
where y = a2 ∼ a1 denotes a path from y = a2 to y = a1 in D1. Accordingto the ambiguity of paths, the result should be considered in modulo 2πiZ.Similarly, we obtain
χ(α0) = loga2a1, χ(α1) = log
a3a4, χ(α2) = log
a1a3,
χ(α3) = loga7a5, χ(α4) = log
a5a6, χ(α5) = log
a8a7,
and therefore we have
χ(−KX) = loga1a2a7a8a3a4a5a6
and q as (6.4) (see Remark 5.3). For the mapping φ, we have q = 1.
6.2 Case i-2
We consider the following HKY mapping which is a symmetric reduction ofqPV for q = −1 [70] (also [66] pg. 311).
x =(x− t)(x+ t)
y(x− 1)
y = x (6.6)
We define the space of initial conditions as a rational surface obtained byblow-ups from P
1×P1 at 8 points:
P1 : (x, y) = (a1, 0) = (t, 0), P2 : (x, y) = (a2, 0) = (−t, 0)P3 : (x, y) = (0, a3) = (0, t), P4 : (x, y) = (0, a4) = (0,−t)
P5 : (x, y) = (1,∞), P6 : (x, y) = (∞, 1)P7 : (x, y) = (∞,∞), P8 : (x, x/y) = (∞, a5) = (∞, 1),
44
where ai’s wii be used for deautonomization later. The system acts thesurface as a holomorphic automorphism.
Again we investigate the linear system of the anti-canonical divisor class−KX = 2Hx + 2Hy −E1 − · · · −E8. For the example, dim−KX is zero anddim−2KX is one. Actually, we have
| − 2KX | =αx2y2 + β(2x2y3 + 2x3y2 + x2y4 + x4y2 − 2x3y3−2xy4 − 2x4y + x4 + y4 + 2t2(xy2 + x2y − y2 − x2) + t4) ≡ αf + βg,
and
k =g
f=(2x2y3 + 2x3y2 + x2y4 + x4y2 − 2x3y3 − 2xy4 − 2x4y
+ x4 + y4 + 2t2(xy2 + x2y − y2 − x2) + t4))/
(x2y2)
is the conserved quantity. So it belongs to Case ii-1.
Remark 6.3. We say a curve f(x, y) = 0 passes through a point (x0, y0) with
multiplicity m if ∂jf(x0,y0)∂xp∂yq
= 0 for any j ≤ m and p+ q = j. The calculation
of multiplicity at P8 is very sensitive. For example, for f(x, y) = x2y2, werelate F (X, Y ) = X2Y 2 so that the sum of degrees is (4, 4). Since this curvepasses through P7 with multiplicity 2, the proper transform of the curve inthe coordinate: (u, v) = (X, Y/X) is given by F (u, uv)/u2 = u2v2, whichpasses through P8 : (u, v) = (0, 1) with multiplicity 2.
Remark 6.4. The unique anti-canonical divisor −KX is decomposed as−KX =
∑4i=0Di by
D0 = Hy − E1 − E2, D1 = Hx − E6 − E7
D2 = E7 − E8, D3 = Hy − E5 − E6, D4 = Hx − E3 − E4,
which constitute A(1)4 -type singular fiber xy = 1. The orthogonal complement
of Di’s is generated by
α0 = Hx +Hy − E1 − E3 − E7 − E8, α1 = E1 − E2
α2 = Hx − E1 − E5, α3 = Hy − E3 − E6, α4 = E3 − E4,
45
which forms the Dynkin diagram of type D(1)4 . Let ω be the same as (6.5),
then the period map χ : Q→ C for Q =⊕4
i=0 Zαi is computed as
χ(α0) = log
(− a3a1a5
), χ(α1) = log
a1a2, χ(α2) = log
1
a1,
χ(α3) = log a3, χ(α4) = loga4a3,
and thereforeχ(−KX) = log
a3a4a1a2a5
+ πi
Hence we can take q as
q = − a3a4a1a2a5
and we have q = −1 for the original mapping.
Figure 2: Singular fiber and orthogonal compliment.
The mapping (6.6) can be deautonomized to one of qPV (qP (A(1)4 )) as
x =a5(x− a1)(x− a2)
y(x− 1)y = x
with
(a1, a2, a3, a4, a5, q) 7→ (a4q,a3q, a1, a2,−
1
a5, q),
46
which acts on the root lattice as
(α0, α1, α2, α3, α4) 7→ (α2 + α3 + α4,−α4,−α3, α0 + α3 + α4, α1).
While the original qPV mapping was
x =a5(x− a1)(x− a2)
y(x− 1)y = x
(same with the above) with
(a1, a2, a3, a4, a5, q) 7→ (a4q,a3q, a2, a1,−
1
a5, q),
which acts on the root lattice as
(α0, α1, α2, α3, α4)
7→ (α1 + α2 + α3 + α4,−α4,−α3, α0 + α1 + α3 + α4,−α1).
6.3 Case ii-2
We consider the following mapping ϕ:
ϕ :
x =x(−ix(x+ 1) + y(bx+ 1))
y(x(x− b) + iby(x− 1))
y =x(x(x+ 1) + iby(x− 1))
b(x(x+ 1)− iy(x− 1))
, (6.7)
which is obtained by specializing one of qP (A(1)5 ) equation. Notice that the
space of initial conditions for both qPIII and qPIV is the generalized Haphensurface of type of A
(1)3 [57], and thus we may not be able to say that a
translational element of the corresponding affine Weyl group is one of qPIII
equations or qPIV equations.The inverse of ϕ is
ϕ−1 :
x =y(bxy − bx− by + 1)
xy − x+ by − 1
y =−iy(bxy − bx− by + 1)(bxy + x− by + 1)
bx(xy − x− y − 1)(xy − x+ by − 1)
(6.8)
47
and the space of initial conditions is obtained by blow-ups from P1×P
1 at 8points:
P1 : (x, y) = (−1, 0), P2 : (x, y) = (0, 1/b)
P3 : (x, y) = (1,∞), P4 : (x, y) = (∞, 1)P5 : (x, y) = (0, 0), P6 : (x, y/x) = (0, i)
P7 : (x, y) = (∞,∞), P8 : (x, x/y) = (∞,−ib).
Then ϕ acts the surface as a holomorphic automorphism.For the above example, dim | − KX | is zero and dim | − 2KX | is one.
Actually, we have
| − 2KX | :0 = kf0(x, y)− f1(x, y)= kx2y2 −
(ix(x+ 1)2 − i(x+ i)(x2 − 1)y
+b(x− 1)2y2)(− ix(y − 1) + y(by − 1)
) .
By ϕ, the parameter
k =f1(x, y)
f0(x, y)
is mapped to −k. So,
k2 =
(f1(x, y)
f0(x, y)
)2
is the conserved quantity and ϕ belongs to Case ii-2.We found this example by observing the following facts:
• Let X be a generalized Halphen surface of multiplicative type, thenexp(χ(−KX)) is the parameter q of the corresponding q-discrete Painleveequation. From Theorem 5.2, if q is a primitivem-th root of unity, thendim | − kKX | is 0 for k = 1, 2, . . . ,m− 1 and 1 for k = m.
• Let ψ be an automorphism of the surface. If there exists another auto-morphism σ of the surface such that σ acts the base space of |−mKX |nontrivially, then ϕ = σ ψ belongs to the case ii-2 unless it is finiteorder.
48
First, we consider the family of generalized Halphen surfaces of type A(1)5 .
Those surfaces are obtained by blow-ups from P1×P
1 at 8 points:
P1 : (x, y) = (b1, 0), P2 : (x, y) = (0, 1/b2)
P3 : (x, y) = (1,∞), P4 : (x, y) = (∞, 1)P5 : (x, y) = (0, 0), P6 : (x, y/x) = (0, c)
P7 : (x, y) = (∞,∞), P8 : (x, x/y) = (∞, 1/(cb0)).
The anti-canonical divisor xy = 0 is decomposed by
Hx − E2 − E5, E5 − E6, Hy − E1 − E5,
Hx − E4 − E7, E7 − E8, Hy − E3 − E7,
and their orthogonal complement is generated by
α0 = Hx +Hy − E5 − E6 − E7 − E8
α1 = Hx − E1 − E3
α2 = Hy − E2 − E4
β0 = Hx +Hy − E1 − E2 − E7 − E8
(β1 = Hx +Hy − E3 − E4 − E5 − E6).
The period map χ : Q→ C for the same ω with (6.5) is computed as
χ(α0) = − log b0, χ(α1) = − log b1, χ(α2) = − log b2,
χ(β0) = − log(−cb0b1b2), χ(β1) = log(−c),
and therefore χ(−KX) = − log(b0b1b2). We set q = (b0b1b2)−1.
The following actions generate the group of automorphisms of the family
49
Figure 3: Singular fiber and orthogonal complement.
of surfaces, whose type is A(1)2 + A
(1)1 :
(x, y; b0, b1, b2, c) is mapped to
wα1:
(x
b1,y(x− 1)
x− b1; b0b1,
1
b1, b1b2, c
)
wα2:
(b2x(y − 1)
b2y − 1, b2y; b0b2, b1b2,
1
b2, c
)
π :
(y, x;
1
b0,1
b1,1
b2,1
c
)
ρ :
(1
y,cx
y; b1, b2, b0, c
)
wβ1:
(−cx(xy − x− y)
xy + cx− y ,− y(xy − x− y)(cxy − cx+ y)
; b0, b1, b2,1
c
)
σ :
(b1x,1
b2y; b0, b1, b2,
1
b0b1b2c
),
wα0= ρ−1 wα2
ρ and wβ0= σ wβ1
σ.Here, wαi
acts as the elementary reflection of the affine Weyl group of
type A(1)2 on
⊕i Zαi and trivially on
⊕j Z βj. Similarly, wβi
acts triviallyon⊕
i Zαi and as the elementary reflection of the affine Weyl group of type
50
A(1)1 on
⊕j Z βj. The generators (α0, α1, α2, β0, β1) are mapped by π, ρ, σ to
π :(α0, α2, α1, β0, β1)
ρ :(α2, α0, α1, β0, β1)
σ :(α0, α1, α2, β1, β0).
If q = (b0b1b2)−1 = −1, then χ(−Kx) = − log(−1) = −πi mod 2πiZ,
and | − 2KX |, i.e. the set of curves of degree (4, 4) passing through theblow-up points with multiplicity 2, is given by
k0x2y2 + k1
(c2x4(y − 1)2 + 2b1cx
2(cxy − cx+ y + b2y2(xy − x− y))+
b21(c2x2 + 2cxy(b2y − 1) + (y + b2y
2(x− 1))2))= 0.
Moreover, if c = i, then σ acts identically on the parameter space and mapsk1/k0 to −k1/k0.
Let ψ = (wα1wα2
ρ)2, where wα1wα2
ρ is the original qPIII equation
(x, y; b0, b1, b2, c)
7→(
cx− yb2y(b0cx− y)
,cx(b0(cx− y)− y(b0cx− y))y((cx− y)− b2y(b0cx− y))
;b0q, b1q, b2, c
)
and acts on the root lattice as
(α0, α1, α2, β0, β1)
7→ (α0 − δ, α1 + δ, α2, β0, β1)
(δ = α0 + α1 + α2 = β0 + β1).
Then the mapping ψ acts trivially on the parameter space. Since it is veryintricate mapping, we restrict the parameters to b0 = 1/b, b1 = −1 andb2 = b, then we have ϕ = σ ψ as (6.7), which acts on the root lattice as
(α0, α1, α2, β0, β1)
7→ (α0 − 2δ, α1 + 2δ, α2, β1, β0).
As a conclusion, the mapping σ wα1 wα2
ρ wα1 wα2
ρ with the
full parameter b1, b2, b3, c is one of qP (A(1)5 ) equation and gives the mapping
by specializing of the parameters.
51
7 Q4 mapping
We said in the beginning of the previous chapter that Sakai showed everydiscrete Painleve equation can be formulated as a translation in an affineWeyl group which acts on a family of generalised Halphen surfaces obtainedby nine blow-ups from P
2 (or eight blow ups from P1×P
1). This formulationshows how to classify all discrete Painleve equations using this algebraic-geometric framework. Naturally for the richest affine Weyl group (E
(1)8 ) the
translational component acting on a surface of type A(1)0 represents a kind
of ”master” equation for all the other Painleve equations. This has beenobtained by Sakai under the name of elliptic Painleve equation and initiallyhad a very complicated form. Later on, Ohta, Ramani and Grammaticos [43]found a regular form of an elliptic Painleve equation. We have to point outthat if one wishes to construct some examples of an equation associated to agiven affine Weyl group one has to specify a nonclosed periodically repeatedpattern in the appropriate space, and moreover since any such pattern wouldlead to a discrete Painleve equation the potential number of discrete Painleveequations in infinite.
On the other hand, since the continuous Painleve equations appeared assimilarity reductions of soliton equations, it is natural to think about thesame thing for similarity reduction of lattice equations. In [44] we discussedvarious travelling wave reductions of the deautonomised classical discretesoliton equations (KdV, mKdV, SG and Burgers). It was shown that in-deed various discrete Painleve equations are obtained. Also in [45] the sameapproach has been done on the Lax pair of nonautonomous mKdV and alot of Painleve equations appeared. In this direction the travelling wave re-duction applied to the famous Adler-Bobenko-Suris cube-consistent latticeequations [46] is quite tempting and a lot of results appeared. The case ofQ4 lattice equation is rather special. It is in fact the master equation for theABS-classification and moreover is an integrable discretisation of the famousKrichever-Novikov equation [47]. All the other equations in ABS class ap-pear as a result of a degeneration cascade. Sakai showed that correspondingto Kodaira’s elliptic singular fibers the discrete equations can be classifiedin elliptic, q-discrete and difference type equations. In particular the ellipticequations are related to automorphisms of surfaces of A
(1)0 - type (I0 in Ko-
daira classification). We are going to show that the travelling wave reductionof Q4 ABS- lattice equation can be lifted to an automorphism of a rationalelliptic surface having A
(1)1 type fibers. Accordingly the corresponding nonau-
52
tonomous equations can be only multiplicative or additive but not elliptic asit was suggested by deautonomisation using singularity confinement.
In order to get a clear description of quadrilateral lattice equations Adler,Bobenko and Suris proposed a classification based on a special symmetrynamely consistency around the cube[46]. This allows to construct immedi-ately the discrete zero curvature representation thus proving the integrability.Up to homographic and linear transformations a part((Q-list) of the quadri-lateral lattice equations were classified as follows: (for simplicity we use thenotations x = xn,m, x = xn+1,m, x = xn,m+1, etc.)
Q4:
sn(α; k)(x˜x+ xx)− sn(β; k)(xx+ x˜x)−− sn(α−β; k)(xx+x˜x)+sn(α; k) sn(β; k) sn(α−β; k)(1+k2xxx˜x) = 0 (7.1)
In the case k → 0 then the elliptic sin goes to ordinary sin and Q4 →Q3(below)
Q3:
sinα(x˜x+ xx)− sin β(xx+ x˜x)−− sin(α− β)(xx+ x˜x) + sinα sin β sin(α− β) = 0 (7.2)
Q2:a(x− x)(x− ˜x) + b(x− x)(x− ˜x)+
+c(x+ x+ x+ ˜x) + d = 0 (7.3)
where c, d are expressed in terms of a and bQ1:
α(x− x)(x− ˜x) + β(x− x)(x− ˜x) + δ = 0 (7.4)
These equations form a degeneration cascade; If, sinα = a, sin β = b, x →1 + ǫx, sin(α− β) = −(a+ b) + ǫc, sin(α− β) sinα sin β = −2ǫc+ ǫ2d, thenQ3→ Q2
In order to obtain a mapping we make the so called (p,q)-reduction≡travelling wave reduction, namely:
xn,m = xpn+qm = xν
The simplest reduction appears for the travelling wave with speed 1: xn,m+1 =xn+1,m. In this case the Q4 mapping becomes:
53
(sn(α; k)− sn(β; k))(xx+ xx)−− sn(α− β; k)(xx+ x2) + sn(α; k) sn(β; k) sn(α− β; k)(1 + k2x2xx) = 0
It can be written as φ : P1×P1 → P
1×P1 in the form:
x = y
y =By2 −Gxy − A
Ak2xy2 − Bx+Gy
and also the inverse:y = x
x =Bx2 −Gxy − A
Ak2yx2 −By +Gx
where A,B,G are expressed in terms of elliptic Jacobi sines. The blow uppoints can be computed from the expressions but unfortunately are quitecomplicated. In order to get through this we change the parametrisation.Namely we introduce variables γ, z by α = γ + z, β = γ − z. Using additionformulas for elliptic functions we obtain:
A = (cn2(z; k)− cn2(γ; k)) cn(z; k) dn(z; k)
B = cn(z; k) dn(z; k)(1− k2 sn2(γ; k) sn2(z; k))
G = cn(γ; k) dn(γ; k)(1− k4 sn4(z; k))
7.1 Resolution of singularities and symmetry group
In this parametrisation we define the space of initial conditions as a rationalsurface X obtained after blow ups of the following 8 points (Ei, i = 1...4 areindeterminate points for φ and Ej, j = 5, ..., 8 for φ−1:
E1 : (x, y) =
(cn(γ)
dn(γ),cn(z)
dn(z)
), E2 : (x, y) =
(− cn(γ)
dn(γ),− cn(z)
dn(z)
)
E3 : (x, y) =
(dn(γ)
k cn(γ),dn(z)
k cn(z)
), E4 : (x, y) =
(− dn(γ)
k cn(γ),− dn(z)
k cn(z)
)
54
E5 : (x, y) =
(cn(z)
dn(z),cn(γ)
dn(γ)
), E6 : (x, y) =
(− cn(z)
dn(z),− cn(γ)
dn(γ)
)
E7 : (x, y) =
(dn(z)
k cn(z),dn(γ)
k cn(γ)
), E8 : (x, y) =
(− dn(z)
k cn(z),− dn(γ)
k cn(γ)
)
After the blowing up points Ei the mapping is lifted to φ : X → P1×P
1
which is free of any singularities. Also one can check by direct (and long)calculation that φ : X → X and its inverse are free of any singularities.Accordingly the mapping is an automorphism of a ratuional surface.
Now we are going to show the action on the Picard group. First of all itis easily seen that the image of the Ej, j = 5, ..., 8 are Ei, i = 1...4 namely(for convenience we note φ(Ei) as Ei):
E5 = E1, E6 = E2, E7 = E3, E8 = E4
For the the image of the total transform of the line x = 0.
(x, y)|x=0,y = (y,By2 − AGy
)
which is the curve Bx2−Gxy−A passing through E5, E6, E7, E8. Accordingly
Hx = 2Hx +Hy − E5 − E6 − E7 − E8
Hy = Hx
In the same way we get the following:
Hy − E1 → E5 → E1 → Hx − E5
Hy − E2 → E6 → E2 → Hx − E6
Hy − E3 → E7 → E3 → Hx − E7
Hy − E4 → E8 → E4 → Hx − E8
This is exactly the singularity confinement pattern. It shows a strictly con-fining shape (guaranteed by the integrability of the mapping)
Let us denote an element of the Picard lattice < Hx, Hy, E1...E8 >Z byA = h0Hx+h1Hy+
∑8i=1 eiEi (hi, ej ∈ Z), then the induced bundle mapping
55
is acting on it as
φ∗(h0, h1, e1, ..., e8)
=(h0, h1, e1, ..., e8)
2 1 0 0 0 0 −1 −1 −1 −11 0 0 0 0 0 0 0 0 01 0 0 0 0 0 −1 0 0 01 0 0 0 0 0 0 −1 0 01 0 0 0 0 0 0 0 −1 01 0 0 0 0 0 0 0 0 −10 0 1 0 0 0 0 0 0 00 0 0 1 0 0 0 0 0 00 0 0 0 1 0 0 0 0 00 0 0 0 0 1 0 0 0 0
From the eigenspace of value 1 eigenvector we get the invariant of the map-ping. It turns out that the anticanonical divisor is preserved
¯−KX = −KX = 2Hx + 2Hy −8∑
i=1
Ei
. The proper transform of the anticanonical divizor gives the following pencilof elliptic curves (λ parametrizes the elliptic fibration )
f0(x, y) + λg0(x, y) ≡ (1 + k2x2y2)(AB − λAG)− (x2 + y2)(B2 + λGB)+
+2xy(BG− λk2A2 + λB2) = 0
From the fact that:λ = −λ
the corresponding invariant is given by λ2 = (f0(x, y)/g0(x, y))2. Also be-
cause degf0=degg0 = (2, 2) we can conclude that the mapping belongs to thecase (ii-1) in our classification and exchange fibers in a Halphen surface ofindex one.
In order to see what is the type of the surface X we take the followingcurves:
D1 = −xyk(cn(γ)
dn(γ)− cn(z)
dn(z)
)+(x−y)
(cn(γ)
dn(γ)
cn(z)
dn(z)k − 1
)+cn(γ)
dn(γ)− cn(z)
dn(z)= 0
56
D2 = −xyk(cn(γ)
dn(γ)− cn(z)
dn(z)
)−(x−y)
(cn(γ)
dn(γ)
cn(z)
dn(z)k − 1
)+cn(γ)
dn(γ)− cn(z)
dn(z)= 0
Their proper transforms are given by:
D1 = Hx +Hy − E1 − E3 − E6 − E8
D2 = Hx +Hy − E2 − E4 − E5 − E7
and−KX = D1 +D2
Also the action of the mapping permutes the curves D1 = D2, D2 = D1
and the intersection form is D1 · D2 = 2. Accordingly the surface X is anA
(1)1 -type.The affine Weyl group symmetries are related to the orthogonal comple-
ment of Dred = D1, D2. In order to see this, we note that rank Pic(X) =rank 〈Hx, Hy, E1, ...E8〉Z = 10. The orthogonal complement of Dred:
〈D〉⊥ = α ∈ Pic(X)|α ·Di = 0, i = 0, 3
has 8-generators :
〈D〉⊥ = 〈α1, α1, ..., α8〉Z,withα1 = E1 − E3, α2 = E3 − E6, α3 = E6 − E8
α4 = E2 − E4, α5 = E4 − E5, α6 = E5 − E7
α7 = Hx −Hy, α8 = Hy − E2 − E8
Now the family 〈α1, α1, ..., α8〉 together with the intersection form of divisorscan be seen as a root lattice associated to a Weyl group. Computing theCartan matrix cij = 2(αi · αj)/(αi · αi) one gets the structure of affine Weyl
group E(1)7 . As usual the mapping can be written in terms of elementary re-
flections associated to the extended Weyl group W (E(1)7 . This fact suggests
that the mapping cannot be the autonomous of an elliptic Painleve equation.However because the decomposition of the anticanonical divisor in effectivedivisors is not necessarily unique it may be possible to find another decom-position which provides a fully elliptic surface A
(1)0 . To our knowledge this
has not been done so far and we strongly believe that our decomposition isthe only one.
57
8 Minimization of elliptic surfaces from bira-
tional dynamics
As we said [63] Diller and Favre showed that for any birational automorphismϕ on a projective smooth rational surface S, we can construct a rationalsurface S by successive blow-ups from S such that (i) σ ϕ = ϕ σ, where σdenotes the successive blow-downs σ : S → S, (ii) ϕ = ϕ on S in generic, and(iii) ϕ : S → S is analytically stable. In general, ϕ is said to be lifted from ϕif the condition (i) and (ii) are satisfied and a birational automorphism ϕ onS is said to be analytically stable if the condition ((ϕ)∗)n = ((ϕ)n)∗ holds onthe Picard group on S. The notion of analytical stability is closely related tothe singularity confinement. Indeed, this notion is equivalent to the conditionthat there is no curve C on S and a positive integer k such that ϕ(C) is apoint on S and ϕk(C) is an indeterminate point of ϕ, i.e. analytical stabilitydemands that singularities are not recovered by the dynamical system. Inother words, if a mapping ϕ satisfies singularity confinement criterion, i.e.for any curve such that ϕ(C) is a point, ϕk+1(C) recovers to a curve againfor some positive integer k, then if we blow up the phase space at ϕi(C) for1 ≤ i ≤ k, then the singularity would be relaxed and resolved by successiveapplications of this procedure. Finally we would obtain a surface where thelifted birational automorphism ϕ is analytically stable.
From Diller and Favre’s work, such birational automorphisms f are clas-sified as follows: Let f be a bimeromorphic automophism of a Kahler surfacewith the maximum eigenvalue of f ∗ is one. Up to bimeromorphic conjugacy,exactly one of the following holds.
• The sequence ||(fn)∗|| is bounded, and fn is an automorphism isotopicto the identity for some n, where ||·|| denotes the Euclidean norm w.r.t.some basis of the Picard group.
• The sequence ||(fn)∗|| grows linearly, and f preserves a rational fibra-tion. In this case, f can not be conjugated to an automorphism. (Wesay f is linearizable or linearizable in cascade in this case [56, 72]).
• The sequence ||(fn)∗|| grows quadratically, and f is an automorphismpreserving an elliptic fibration.
These three conditions are essential in analysing any birational dynami-cal system on P
2. They represent the rigurous formulation of the complexity
58
growth (algebraic entropy) criterion) exposed at the beginnig of the thesis.In addition these conditions show that a secific growth provides also themethod of integration. Namely if the growth is linear the system is linearis-able preserving a rational fibration. But because it cannot be conjugated toan automorphism the number of blowing ups can be infinite (we shall returnto the problem of linearisable systems). If the growth is quadratic then themapping can be integrated in terms of elliptic functions since preserves anelliptic fibration (and the number of blow ups is finite). For exponentialgrowth we have the chaotic case. So the whole analysis of complexity growthcan be done also using the bundle mapping on the Picard group.
Though in this paper we consider mainly autonomous case, the proce-dure constructing analytically stable mapping can be applied also for non-autonomous case such as linearizable mappings or discrete Painleve equa-tions. In this case, we start from a sequence of birational mappings ϕi :S → S and blow up successively at confined singular points whose posi-tions depend on i. Then, ϕi would be lifted to a sequence of birationalmappings ϕi : Si → Si+1 such that (i) σi+1 ϕi = ϕi σi, (ii) ϕi = ϕi
on S in generic, and (iii) ϕ : Si → Si+1i∈Z is analytically stable, i.e.ϕ∗i · · · ϕ∗
i+n = (ϕi+n · · · ϕi)∗ holds on the Picard group on Si+n+1 for
any i and non-negative integer n (see [58, 59, 72] about computation and therelation to the degree growth).
In the study of integrable systems, we often want to find conserved quan-tities or linearize a given integrable mappings, but the above construction ofanalytically stable mapping does not guarantees that(i) ϕ is an automorphism.(ii) S is relatively minimal, i.e. there does not exists a blow-down of S,π : S → S ′ such that ϕ′ is still analytically stable on S ′.In other words, the following possibilities remain.(ia) A singularity sequence consists of infinite sequences of points to bothsides and a finite sequence of curves:
· · · → point→ point→ curves→ · · · → curves→ point→ point→ · · · ,
where the image of a curve C, parametrized as (f(t), g(t)) on some coordi-nates, under ϕn is defined as the Zariski closure of limε→0 ϕ
n(f(t)+c1ǫ, g(t)+c1ǫ) with generic t, c1 and c2.(ib) A singularity sequence consists of an infinite sequence of points and that
59
of curves to each side:
· · · → point→ point→ · · · · · · → curves→ curves→ · · · .
(ii)’ A finite set of exceptional curves are permuted.
Proposition 1.7 and Lemma 4.2 of [63] (cf. [51]) says that curves in(ia) can be blown down, and that (ib) occurs only if f is not conjugate to anautomorphism, i.e. if f is linearizable or has a positive entropy. And if a curvein Case (ii)’ or Case (ib) is exceptional of the first kind, we can blow downthem. Hence theoretically, we can obtain relatively minimal analyticallystable surfaces and compute the action on the Picard group. However, forinvestigating properties of the mapping f such as conserved quantities, weneed to know coordinate change explicitly.
Our aim in this part is to develop a method to control blowing down struc-tures on the level of coordinates. We apply our method to various examples,including the newly studied discretization of reduced Nahm equations [55].In general, finding elliptic fibration for an elliptic surface is not easy if thesurface is not minimal, and we use information of singularity patterns of thedynamical systems for finding unnecessary (−1) curves. Accordingly, thismethod of minimization shows how an order-two mapping with complicatedsingularity structure can be brought to a simpler form which enables com-putations of conserved quantities.
In the next section, we recall some basic notions and blowing down struc-tures. Then, we investigate discrete versions of reduced Nahm equations,which preserve a rational elliptic fibration. We will show that the associ-ated surfaces are not minimal and by minimization one can transform themappings to simpler ones. Further on we investigate linearizable dynamicalsystems, including non-autonomous case.
60
8.1 Blowing down structure
Notations: cf. [52, 53]
S : a smooth rational surface
D : the linear equivalent class of a divisor D
D ·D′ : the intersection number of divisors D and D′
O(D) : the invertible sheaf corresponding to D
Pic(S) = the group of isomorphism classes of invertible sheaves on S
≃ the group of linear equivalent classes of divisors on S
E : the total transform of divisor class of a line on P2
Hx,Hy : the total transform of divisor class of a line x = constant
(or y = constant) on P1×P
1
Ei : the total transform of the exceptional divisor class of the i-th blow-up
|D| ≃(H0(S,O(D))− 0)/C× : the linear system of D
KS : the canonical divisor of a surface S
g(C) : the genus of an irreducible curve C, given by the genus formula
g(C) = 1 + 12(C2 + C ·KS) if C is smooth.
Let S = Sm be a surface obtained by successive m times blowing up fromP2 (or any rational surface) at indeterminate or extremal point of ϕ, i.e. the
Jacobian ∂(x, y)/∂(x, y) in some local coordinates is zero, such that ϕ on §is analytically stable. Let Fm be a curve on S with self-intersection −1 andFm be the corresponding divisor class. Our strategy to write the blow-downSm along Fm by coordinates is as follows.
Take a divisor class F such that there exists a blowing down structure (thisterminology is due to [50]): S = Sm → Sm−1 → Sm−2 → · · · → S1 → P
2,where Sm → Sm−1 is a blow-down along Fm and each Si → Si−1 is a blow-down along an irreducible curve, such that the divisor class of lines in P
2 isF. Let |F| = α0f0+α1f1+α2f2 = 0. Then (f0 : f1 : f2) gives P
2 coordinates.In order to find such F we note the following facts.It is necessary for the existence of such a blow-down structure that there
exists a set of divisor classes F1, . . . ,Fm such that
F2 = 1 F
2i = −1, Fi · Fj = 0, F · Fi = 0
61
for (1 ≤ i, j ≤ m), and further that (i) the genus of divisor F is zero; (ii)the linear system of F does not have a fixed part in the sense of Zariskidecomposition and its dimension is two.
If the linear system of F does not have fixed part, then by Bertini theorem,its generic divisor is smooth and irreducible (this follows from the fact thattwo divisors defines a pencil by blowing up at the unique intersection and P.137 of [53]), and its genus is given by the formula
g = 1 +1
2(F 2 + F ·KS).
From this fact and Condition (ii), 1 + 12(F 2 + F ·KS) should be zero.
Example 8.1. If degree of F is less than 6, then F is given by one of thefollowing forms.
E
2E− Ei1 − Ei2 − Ei3
3E− 2Ei1 − Ei2 − Ei3 − Ei4 − Ei5
4E− 2Ei1 − 2Ei2 − 2Ei3 − Ei4 − Ei5 − Ei6
4E− 3Ei1 − Ei2 − Ei3 − Ei4 − Ei5 − Ei6 − Ei7
5E− 2Ei1 − 2Ei2 − 2Ei3 − 2Ei4 − 2Ei5 − 2Ei6
5E− 3Ei1 − 2Ei2 − 2Ei3 − 2Ei4 − Ei5 − Ei6 − Ei7
5E− 4Ei1 − Ei2 − Ei3 − Ei4 − Ei5 − Ei6 − Ei7 − Ei8 − Ei9 ,
(8.1)
where ij’s are all distinct with each other. All the above F admit blow-downstructure if the positions of blow-up points are generic. For example, forF = 2E− Ei1 − Ei2 − Ei3 , Fi’s are given by
E− Ei − Ej(i, j|i 6= j ⊂ i1, i2, i3), Ej(j 6= i1, i2, i3)
and for F = 3E− 2Ei1 − Ei2 − Ei3 − Ei4 − Ei5 , Fi’s are given by
E−Ei1−Ej(j ∈ i2, . . . , i5), 2E−Ei1−Ei2−Ei3−Ei4−Ei5 , Ej(j 6= i1, . . . , i5).
If we want to blow down to P1×P
1 instead of P2, our strategy becomesas follows.
Let Fm−1 be a curve on S with self-intersection −1 and Fm−1 be thecorresponding divisor class. Take a divisor class Hu and Hv such that thereexists a blow-down structure: S = Sm−1 → Sm−2 →→ · · · → S1 → P
1×P1,
62
where Sm−1 → Sm−2 is a blow-down along Fm−1 and each Si → Si−1 is ablow-down along an irreducible curve, such that the divisor class of linesu = const and v = const are Hu and Hv. Let |Hu| = α0f0 + α1f1 = 0and |Hv| = β0g0 + β1g1 = 0. Then (u, v) = (f0/f1, g0/g1) gives P
1×P1
coordinates.In this case, it is necessary that there exits a set of divisor classes F1, . . . ,Fm−1
such that
H2u = H
2v = 0,Hu ·Hv = 1, F
2i = −1,
Fi · Fj = 0, Hu · Fi = Hv · Fi = 0
for (1 ≤ i 6= j ≤ m− 1), and further that (i) each genus of divisor Hu or Hv
is zero; (ii) each linear system of Hu or Hv does not have a fixed part andits dimension is one. Consequently, 1+ 1
2(F 2 +F ·KS) should be zero again.
Example 8.2. If S is obtained by successive blow-ups from P2, and the sum
of degree of Hu or Hv is less than 6, then each Hu or Hv is given by F−Ek,where F is in the list (8.1).
If S is obtained by successive blow-ups from P1×P
1, each Hu or Hv isgiven by
Hx
Hx +Hy − Ei1 − Ei2
2Hx +Hy − Ei1 − Ei2 − Ei3 − Ei4
2Hx + 2Hy − 2Ei1 − Ei2 − Ei3 − Ei4 − Ei5
3Hx +Hy − Ei1 − Ei2 − Ei3 − Ei4 − Ei5 − Ei6
3Hx + 2Hy − 2Ei1 − 2Ei2 − Ei3 − Ei4 − Ei5 − Ei6
4Hx +Hy − Ei1 − Ei2 − Ei3 − Ei4 − Ei5 − Ei6 − Ei7 − Ei8
(8.2)
and those with exchange of Hu and Hv. Not all, but many pairs of thesedivisor classes admit a blow-down structure for generic blow-up points. Forexample, for Hu = Hx and Hu = Hx +Hy − Ei1 − Ei2 , Fi’s are given by
Hx − Ei1 , Hx − Ei2 Ej(j 6= i1, i2)
and for Hu = Hx +Hy − Ei1 − Ei2 and Hu = Hx +Hy − Ei1 − Ei3 , Fi’s aregiven by
Hx − Ei1 , Hy − Ei1 , Hx +Hx − Ei1 − Ei2 − Ei3 , Ej(j 6= i1, i2, i3).
63
Remark 8.3. There is another way to obtain relatively minimal surface forelliptic surface case, though it needs heavy computation. Let S be a rationalelliptic surface (not necessarily minimal) where the mapping ϕ is lifted to anautomorphism. Compute a R-divisor θ by
θ := limn→∞
ϕn∗ (E)
||ϕn∗ (E)||
,
where || · || denotes the Euclidian norm of a divisor w.r.t. a fixed basis, andlet k > 0 be a minimum number such that kθ ∈ Pic(S). Then, the linearsystem |mkθ| gives an elliptic fibration for some integer m ≥ 1 (m is notalways one, (cf. Step 1 of Appendix of [63] and [2]). Let C be a curve in thelinear system |kθ| (such C exists [62]). By applying van Hoeji’s algorithm[69] (cf. [60]), we obtain a birational transformation S → S ′, (x, y) 7→ (u, v)such that C is transformed into Weierstrass normal form v2 = u3− g2u− g3.Since the degree of this curve is three, S ′ is obtained by 9 blow-ups fromP2. This implies S ′ is a minimal elliptic surface (the fibration is given by the
linear system | −mKS′ |).Remark 8.4. If ϕ is an automorphism of a non-minimal rational ellipticsurface, the invariant does not corresponds to the anti-canonical divisor, be-cause the self-intersection of the anti-canonical divisor is negative in this case,while θ2 of the above remark should be zero.
8.2 A simple example which needs blowing down
Let us show first a simple example which needs change of blow-down structureto obtain relatively minimal surface. This example is due to Diller and Favre’spaper [63]: (for simplicity we note xn = x, x = xn+1, x = xn−1 and so forth)
x = y +1
2
y =x(2y − 1)
2y + 2
. (8.3)
This system can be lifted to an automorphism on a surface S by blowing upP1×P
1 at the singularity points of the dynamical systems:
E1 : (x, y) = (1, 0), E2(1/2,−1/2), E3(0,−1), E4(−1/2,∞),
E5(∞,−1/2), E6(0,∞), E7(∞, 0), E8(1/2,∞), E9(∞, 1/2).
64
Immediately one can see the action on the Picard group from the followingsingularity patterns:
Hy−E3 → E4 → E5 → E6 → E7 → E8 → E9 → Hx−E1
Hy−E9 → E1 → E2 → E3 → Hx−E4
and also the invariant divisor classes Hx+Hy−E1−E2−E3 andHx+Hy−E4−E5−E6−E7−E8−E9. The presence of invariant divisorcalsses imposes making blow-down along the curve which corresponds tothe divisor class Hx+Hy−E1−E2−E3 (it is the only one which has self-intersection -1, the other has self-intersection -3). Hence we take the basis ofblow-down structure as
Hu = Hx+Hy−E2−E3,Hv = Hx+Hy−E1−E2,
Hx +Hy−E1−E2−E3,F1 = Hx−E2, F2 = Hy−E2,
Fi = Ei+1 (i = 3, 4, 5, 6, 7, 8),
where the linear systems of Hu and Hv are given by
|Hu | : u0(x− y − 1) + u1(2xy + x) = 0,
|Hv | : v0(x− y − 1) + v1(2xy − y) = 0.
Using these, we take the following change of variables:
u =2xy + x
x− y − 1, v =
2xy − yx− y − 1
,
then our dynamical system (3) and (4) becomes
u =2uv − u− v − 1
u− 3v + 1
v =−2uv
u+ v + 1
. (8.4)
This system has the following blow-up points:
F1 : (u, v) = (−1, 0), F2(0,−1), F3(1, 2), F4 : (u, (v + 1)/u) = (0, 1).
F5(0, 1), F6(1, 0), F7 : ((u+ 1)/v, v) = (1, 0), F8(2, 1).
and the linear system of the anti-canonical divisor gives the invariant
K =uv(2uv − u− v − 1)
(u− v)2 − 1=x(2x− 1)y(2y − 1)(2xy − x+ y + 1)
(x− y − 1)2
65
and the invariant two form
ω =du ∧ dv
(u− v)2 − 1=
dx ∧ dy1− x+ y
.
8.3 Discrete Nahm equations with tetrahedral symme-try
In [55], Petrera, Pfadler and Suris proposed the following discretization ofthe reduced Nahm equations with tetrahedral symmetry
x− x = ǫ(xx− yy)y − y = −ǫ(xy + yx)
. (8.5)
Here ǫ is related to the step of discretization. The integrability can be provedby the existence of the following conserved quantity and invariant two-form
K =y(3x2 − y2)
−1 + ǫ2(x2 + y2), ω =
dx ∧ dyy(3x2 − y2) . (8.6)
In this case one can easily transform the system into a QRT one by thefollowing variable transformation
u =1− ǫxy
, v =1 + ǫx
y. (8.7)
Immediately we get u = v. From the equation (8.5) we get a QRT mapping
3uu− u(u+ u)− u2 + 4ǫ2 = 0
with the invariant:
K =−3(u− v)2 + 4ǫ2
2ǫ2(u+ v)(uv − ǫ2) , ω =du ∧ dv
3(u− v)2 − 4ǫ2,
which are precisely (8.6) in the variables x and y.Now we are going to study the singularity structure and its space of initial
conditions and recover the invariants. The fact that the conserved quantityis expressed by a ratio of a cubic polynomial implies that we have better tostart with P
2 than P1×P
1.On P
2 : (X : Y : Z) = (x : y : 1), we blow up the following points
66
E1(−1 : −√3 : 2ǫ), E2(1 :
√3 : 2ǫ), E3(−1 :
√3 : 2ǫ),
E4(1 : −√3 : 2ǫ), E5(1 : 0 : ǫ), E6(−1 : 0 : ǫ),
E7(1 : 0 : 0), E8(1 : 1 : 0), E9(1 : −1 : 0).
In order to blow down to P1×P1, we take the basis of blow-down structure
Hx, Hy, F1, . . . ,F8 as
Hx = E−E5, Hy = E−E6, Fi = Ei(i = 1, 2, 3, 4),
F5 = E7, F6 = E8, F7 = E8, F8 = E − E5−E7 .
The curves corresponding to the divisor classes Hx and Hy are:
α0(ǫX − Z) + α1Y = 0, β0(ǫX − Z) + β1Y = 0.
They give immediately the change of variable
u =ǫx− 1
y, v =
ǫx+ 1
y,
which is essentially (8.6) up to rescaling factors.
8.4 Discrete Nahm equations with octahedral symme-try:
The second Nahm equation is the one corresponding to octahedral symmetry.The system has the following form
x− x = ǫ(2xx− 12yy)y − y = −ǫ(3xy + 3yx+ 4yy)
, (8.8)
which is again integrable by the invariants:
K =y(2x+ 3y)(x− y)2
1− 10ǫ2(x2 + 4y2) + ǫ4(9x4 + 272x3y − 352xy3 + 696y4)
ω =dx ∧ dy
y(x− y)(2x+ 3y). (8.9)
67
Inspired by the transformation (8.7) we can simplify the system by the fol-lowing transformations:
x =1
3(χ− 2y), x =
1
3(χ− 2y)
and u = (1 − ǫχ)/y, v = (1 + ǫχ)/y. Finally we get a simpler equation butnon-QRT type:
8uu− 2u(u+ u) + 20ǫ(u− u)− 4u2 + 400ǫ2 = 0,
which can be written as a system on P1×P
1
u = v
v =(u+ 2v − 20ǫ)(v + 10ǫ)
4u− v + 10ǫ
. (8.10)
The space of initial conditions is given by the P1×P
1 blown up at thefollowing nine points:
E1 : (u, v) = (−10ǫ, 0), E2(0, 10ǫ), E3(10ǫ, 5ǫ),
E4(5ǫ, 0), E5(0,−5ǫ), E6(−5ǫ,−10ǫ)E7(∞,∞), E8 : (1/u, u/v) = (0,−1/2), E9 : (1/u, u/v) = (0,−2).
The action on the Picard group is the following:
Hu = 2Hu+Hv−E1−E3−E7−E8, Hv = Hu
E1 = E2, E2 = Hu−E3, E3 = E4, E4 = E5, E5 = E6,
E6 = Hu−E1, E7 = Hu−E8, E8 = E9, E9 = Hu−E7 .
From this action one can see immediately that we have three invariant divisorclasses:
α0 = Hu +Hv−E1−E2−E7, α1 = Hu +Hv−E1−E2−E8−E9,
α2 = E7−E8−E9, α3 = Hu+Hv−E3−E4−E5−E6−E7 .
The curve corresponding to α0 is a (-1) curve which must be blown down.Let Ha = Hu+Hv−E2−E7 and Hb = Hu +Hv−E1−E7, then their linearsystems are given by
a1u+ a2(v − 10ǫ) = 0, b1(u+ 10ǫ) + b2v = 0
68
and the basis of blow-down structure is given by
Ha, Hb, α0, F1 = Hu−E7, F2 = Hv−E7,
F3 = E3, F4 = E4, F5 = E5, F6 = E6, F7 = E8, F8 = E9 .
So if we set:
a =v − 10ǫ
ub =
u+ 10ǫ
v,
our dynamical system becomes
a =3ab− 2a+ 2
a− 4
b =4− a2a+ 1
. (8.11)
This system has the following space of initial conditions which define aminimal rational elliptic surface:
F1 : (a, b) = (0,∞), F2 : (a, b) = (∞, 0),F3 : (a, b) = (−1/2, 4), F4 : (a, b) = (−2,∞)
F5 : (a, b) = (∞,−2), F6 : (a, b) = (4,−1/2),F7 : (a, b) = (−2,−1/2), F8 : (a, b) = (−1/2,−2).
The invariants can be computed from the anti-canonical divisor as
K =(ab− 1)(ab+ 2a+ 2b− 5)
4ab+ 2a+ 2b+ 1, ω =
da ∧ db(ab− 1)(ab+ 2a+ 2b− 5)
which are equivalent to the invariants (8.9).
8.5 Discrete Nahm equations with icosahedral symme-try
The last example of discrete reduced Nahm equations refers to icosahedralsymmetry. It is given by
x− x = ǫ(2xx− yy)y − y = −ǫ(5xy + 5yx− yy) (8.12)
69
and is integrable as well. However the invariants here are more complicated.They are reported also by [55] as1
K =y(3x− y)2(4x+ y)3
1 + ǫ2c2 + ǫ4c4 + ǫ6c6, ω =
dx ∧ dyy(3x− y)(4x+ y)
(8.13)
where
c2 = −7(5x2 + y2)
c4 = 7(37x4 + 22x2y2 − 2xy3 + 2y4)
c6 = −225x6 + 3840x5y + 80xy5 − 514x3y3 − 19x4y2 − 206x2y4.
Again we can make first the following change of variable
x =1
5(X +
y
2), x =
1
5(X +
y
2),
then we divide by yy both equations and call again a = X/y, b = 1/y, u =b− ǫa, v = b+ ǫa and finally we get a simpler equation but non-QRT type:
6uu− u(u+ u)− 7ǫ
2(u− u)− 4u2 + 49ǫ2 = 0.
We can apply our procedure to this last non-QRT mapping. However,here we demonstrate that our procedure works well even for the originalmapping.
The space of initial condition is given by the P1×P
1 blown up at thefollowing 12 points:
E1 : (x, y) = (∞,∞), E2(−1/7ǫ,−3/7ǫ), E3(−1/7ǫ, 4/7ǫ),E4(1/7ǫ, 3/7ǫ), E5(1/7ǫ,−4/7ǫ) E6(1/5ǫ, 0),
E7(1/3ǫ, 0), E8(1/ǫ, 0), E9(−1/ǫ, 0),E10(−1/3ǫ, 0), E11(−1/5ǫ, 0),E12 : (1/x, x/y) = (0, 1/3).
On this surface the dynamical system is neither an automorphism noranalytically stable due to the following topological singularity patterns:
Hy−E1 (y =∞)→ point→ · · · (4 points) · · · → point→ Hy−E1
· · · → point→ point→ Hx−E1 (x =∞)→ point→ point→ · · · ,1a sign in c2 was corrected by information from the authors of that paper
70
where the image of a curve under ϕn is defined as (ia) in Section 1. Moreover,the curve 4x + y = 0 : Hx +Hy−E1−E3−E5 is invariant. We blow downalong these three curves with the blow-down structure
Hu = Hx +Hy−E1−E3, Hv = Hx+Hy−E1−E5,
Hx−E1, Hy−E1, Hx+Hy−E1−E3−E5,
F1 = E12, F2 = E2, F3 = E4, F4 = E6,
F5 = E7, F6 = E8, F7 = E9, F8 = E10, F9 = E11,
where the linear systems of Hv and Hv are given by
|Hu | :u0(1 + 7ǫx) + u1(4x+ y)
|Hv | :v0(1− 7ǫx) + v1(4x+ y).
If we take the new variables u and v as
v =2(1 + 7ǫx)
ǫ(4x+ y), v =
2(1− 7ǫx)
ǫ(4x+ y),
then we have
F1 : (u, v) = (2,−2),F2 : (0,−4),F3 : (4, 0),F4 : (6,−1),F5 : (5,−2),F6 : (4,−3),F7 : (3,−4),F8 : (2,−5),F9 : (1,−6).
The dynamical system becomes an automorphism having the following topo-logical singularity patterns
Hv−F9 → F2 → F1 → F3 → Hu−F4
Hv−F3 → F4 → F5 → F6 → F7 → F8 → F9 → Hu−F2
and Hu → Hu +Hv−F2−F4. Hence we find the invariant (−1) curveHu +Hv−F1−F2−F3, which should be blown down. Again we take theblow-down structure as
Hs = Hu+Hv−F1−F2, Ht = Hu+Hv−F1−F3,
Hu +Hv−F1−F2−F3, F′1 = Ha−F1, F
′2 = Hb−F1
F′3 = F4, F
′4 = F5, F
′5 = F6, F
′6 = F7,
F′7 = F8, F
′8 = F9,
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where the linear systems of Hs and Ht are given by
|Hs | :s0u(v + 2) + s1(u− v − 4)
|Ht | :t0v(u− 2) + t1(u− v − 4)
and hence we take the new variables s and t as
s = − 3u(v + 2)
2(u− v − 4), t = − 3v(u− 2)
2(u− v − 4).
Then we have
F′1 : (s, t) = (3, 0), F′
2(0, 3), F′3(−3, 2), F′
4 : (s
t− 3, d− 3) = (5, 0),
F′5(2, 3), F
′6(3, 2), F
′7 : (u− 3,
t
s− 3) = (0, 5), F′
8(2,−3)
and
s =2st− 3s− 3t+ 9
s+ t− 3
t =2(s− 3)(t+ 3)
3s− t− 9
.
The invariants can be computed by using the the anticanonical divisor as
K ′ =(s− t)2 + 4(s+ t)− 21
(s− 2)(t− 2)(2st− 5s− 5t+ 15)=−56ǫ6y(−3x+ y)2(4x+ y)3
d1d2d3(8.14)
and
ω =2ǫds ∧ dt
(s− t)2 + 4(s+ t)− 21=
dx ∧ dyy(3x− y)(4x+ y)
, (8.15)
where
d1 = −3− 12ǫx+ 15ǫ2x2 − 3ǫy − 17ǫ2xy + 4ǫ2y2
d2 = −3 + 12ǫx+ 15ǫ2x2 + 3ǫy − 17ǫ2xy + 4ǫ2y2
d3 = −3 + 27ǫ2x2 + 10ǫ2xy + 10ǫ2y2.
The denominator of K ′ is related to K of (8.13) as
d1d2d3 = 160ǫ6(numerator of K)− 27(denominator of K).
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9 Linearizable mappings
In this section we are going to discuss about linearisable mappings. Roughlyspecaking linearisable means that exists a nonlinear transformation of thedependent variable which bring down the mapping to a linear equation. Themain problem is that such a transformation is complicated and it may havemany steps. So one can wonder if the singularity analysis can be implementedhere. The bad news is that linearisable systems possesses nonconfined sin-gularities so in principle one has to perform an infinite number of blow upsHowever here we demonstrate that our method works well also for lineariz-able mappings. The first example is a simple non-autonomous linearizablemapping studied in [72]. We show our method is different from that paperand [63]. The second example is also a linearizable mapping proposed againby [55] as a discretization of the Suslov system.
9.1 A non-autonomous linearizable mapping
Here we consider the following very simple mappingx = yy =
(− y
x+ an
)y, (9.1)
where an is an arbitrary sequence of complex numbers. This dynamical sys-tem is a linearizable mapping studied in [72] and the degree of this dynamicalsystem grows linearly and it is lifted to an analytically stable mapping byblowing up at the following points:
E1 : (x, y) = (0, 0), E2 : (∞,∞).
The topological singularity patterns are
(x
y, y) = (0, 0)→ Hx−E1 → Hy−E2 → (
1
x,x
y) = (0, 0)
(point on E2)→ Hx−E2 → (curve)
(curve)→ Hy−E1 → (point on E2).
These are not confined at all. Moreover, we can compute the action on thePicard group as
Hx = 2Hx+Hy−E1−E2
Hy = Hx, E1 = Hx, E2 = Hx .
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However, since the dynamical system is not an isomorphism, we need tocompute very carefully for this result. One can see detail of such computationin [72]. Anyway here we are going to linearize the dynamical system usingsingularity patterns instead of the action on the Picard group.
From the singularity pattern, we can blow down the surface alongHx−E1,keeping analytical stability. Then we can easily find a basis of blow-downstructure as
Hu = Hx, Hv = Hx+Hy−E1−E2, F1 = Hx−E1, F2 = Hx−E2 .
where the linear systems of Hu and Hv are
|Hu | : u0x+ u1 = 0, |Hv | : v0x+ v1y = 0.
Taking new variables u and v as u = x and v = y/x, we haveu = uvv = v + an
. (9.2)
9.2 Discrete Suslov system
The discrete Suslov system proposed in [55] is a linearizable mapping:x− x = ǫa(xy + xy)y − y = −2ǫxx . (9.3)
Again, the degree of this dynamical system grows linearly and it is lifted toan analytically stable mapping by blowing up at the following points: (weput a = −b2 for simplicity)
E1 : (x, y) =
(− 1
bǫ,1
b2ǫ
), E2 :
(1
bǫ,1
b2ǫ
),
E3 :
(− 1
bǫ,− 1
b2ǫ
), E4 :
(1
bǫ,− 1
b2ǫ
), E5 : (∞,∞).
The topological singularity patterns are
x =∞→ (0,− 1
b2ǫ)
y =∞→ y =∞(2bex+ b2ǫy + 1 = 0)→ E3 → E2 → (2bǫx− b2ǫy + 1 = 0)
(−2bǫx+ b2ǫy + 1 = 0)→ E4 → E1 → (−2bǫx− b2ǫy + 1 = 0)
(2b2ǫ2x2 − b2ǫy − 1 = 0)→ E5 → (2b2ǫ2x2 + b2ǫy − 1 = 0),
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where divisor classes are
x =∞ : Hx−E5
x =∞ : Hy−E5
2bǫx− b2ǫy + 1 = 0 : Hx+Hy−E4−E5
− 2bǫx− b2ǫy + 1 = 0 : Hx +Hy−E3−E5
2b2ǫ2x2 + b2ǫy − 1 = 0 : 2Hx+Hy−E3−E4−E5 .
At first, we blow down along Hx−E5 and Hy−E5. For that purpose wetake the blow-down structure as
Hs := Hx+Hy−E1−E5, Ht := Hx +Hy−E2−E5,
Hx−E5, Hy−E5, Hx +Hy−E1−E2−E5, E3, E4 .
Then we have a surface whose Picard group is generated by Hs, Ht, E3,E4 where the dynamical system is still analytically stable. We abbreviatedetail, but again we find effective (-1) divisor classes Hs−E3 and Hs−E4 insingularity pattern which can be blown down preserving analytical stability.Hence we take a basis of blow-down structure as
Hu := Hs +Ht−E3−E4 = 2Hx+2Hy−E1−E2−E3−E4−2E5,
u0(x2 − b2y2) + u1(1− b2ǫ2x2) = 0,
Hv := Hs = Hx +Hy−E1−E5 : v0(1 + bǫx) + v1(x+ by) = 0
Hs−E3 = Hx+Hy−E1−E3−E5
Hs−E4 = Hx+Hy−E1−E4−E5 .
If we take the new variables u and v as
u =x2 − b2y21− b2ǫ2x2 , v =
1 + bǫx
x+ by,
then the dynamical system becomes
u = u
v =bǫ+ v
1− bǫuv. (9.4)
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Remark 9.1. The action of the mapping on the Picard group on the firstsurface is given by
Hx = 2Hx +Hy−E3−E4−E5
Hy = 2Hx +2Hy−E3−E4−2E5
E1 = 2Hx+Hy−E3−E5
E2 = 2Hx+Hy−E4−E5
E3 = E2, E4 = E1
E5 = 2Hx+Hy−E3−E4−E5
and Hu is the invariant divisor class whose self-intersection is zero.
A finer classification may be done by the types of singular fibers and theautomorphism of surfaces. Indeed, the symmetries of generalized Halphensurfaces have a close relationship with the Mordell-Weil lattice of rationalsurfaces. However, there are too many types of surfaces and we gave a coarsebut useful classification in this paper.
9.3 Other new linearisable systems
In this section we shall analyse new types of linearisable mappings. Theirforms are inspired by the canonical forms of the QRT mapping twisted inthe logic of replacing products in QRT with ratios. Because the algebraicgeometry here is rather difficult we shall implement the euristic arguments ofdegree growth and impose also the same growth for deautonomisation. Alsomany of the mappings have transcendental invariant which is not clear howto extract from the structure of singularities. Practically all o the examplesbelow will be linearised using the so called Gambier mappings which arecoupled discrete Riccatti equations.
We start withxn+1 + xnxn−1 + xn
= fx2n + axn + b
x2n + cxn + d(9.5)
The investigation of the integrability of (9.5) is carried out using the algebraicentropy criterion, since we expect some integrable subcases to be linearisable.We shall not present here the details of this analysis but just the end result.We find that the only integrable case corresponds to f = 1, c = −a andd = b. Its degree growth is 1, 2, 3, 4, 5,. . . and thus we expect the mapping
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to be linearisable. Indeed by considering the Gambier mapping
yn+1 = yn + a (9.6)
xn =b+ ynxn−1
a− yn + xn−1
(9.7)
and eliminating y we recover the linearisable form of (9.5)
xn+1 + xnxn−1 + xn
=x2n + axn + b
x2n − axn + b(9.8)
The mapping (9.8) possesses a transcendental conserved quantity. Indeed,from the solution of (9.6) we have that yn = na+ y0 and thus tan(πyn/a) =cnst. Solving (9.7) for y we find thus
tan
(π
a
xnxn−1 + axn − bxn + xn−1
)= K (9.9)
As a consequence of the linearisability some of the parameters of (9.5)may be functions of the independent variable. We are thus led to examine(9.5) afresh, keeping f = 1 but allowing for some less stringent constraint ona, b, c, d. We require that the degree growth be the same as in the autonomouscase. We find now that the constraints on the parameters are dn = bn−1 andcn = −an−1. In order to linearise the mapping we consider now the Gambiermapping
yn+1 = yn (9.10)
xn =bn−1 + (yn − gn)xn−1
gn−1 − yn + xn−1
(9.11)
and eliminating y we find
xn+1 + xnxn−1 + xn
=x2n + (gn − gn+1)xn + bnx2n + (gn − gn−1)xn + bn−1
(9.12)
where we have introduced the auxiliary variable g through an = gn − gn+1.The case where the polynomials in the numerator and denominator of the
rhs of (9.5) are linear is also interesting. We start from
xn+1 + xnxn−1 + xn
= cxn + a
xn + b(4.7)
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The application of the algebraic entropy integrability criterion leads to c freewhile b = −a, and the degree growth is the same a for (9.5). The extensionto a non autonomous case is straightforward: a and c are free functions ofthe independent variable n. Thus the linearisable form of the mapping is
xn+1 + xnxn−1 + xn
= cxn + anxn − an−1
(9.13)
The linearisation of (9.13) is given by the Gambier mapping
yn+1 = yn + gn+1 (9.14)
xn =gnan−1 + ynxn−1
gn − yn(9.15)
Elimination of y leads to (9.13) with cn = −gn+1/gn. It is interesting to pointout here that even in the autonomous case of constant c the correspondingGambier mapping is explicitly nonautonomous since in that case we havegn = g0(−c)n. We should also remark that the linearisable case (9.13) canbe obtained from (9.12) by taking x → 0 and an appropriate redefinition ofthe auxiliary variables.
Next we analyse the mapping
xn+1xn − 1
xnxn−1 − 1= f
x2n + axn + b
x2n + cxn + d(9.16)
Again we start by the purely autonomous case. We find that one linearisablecase exists of the form
xn+1xn − 1
xnxn−1 − 1= λ2
x2n + axn + 1/λ
x2n + aλxn + λ(9.17)
Its linearisation is given by the Gambier mapping
yn+1 = yn/λ (9.18)
xn = λxn−1 + yn + a
λynxn−1 − 1(9.19)
At this point it is interesting to exhibit a case where (9.17) possesses aconserved quantity. If we take λ as a root of unity, say λp = 1, then from(9.18) we have ypn+1 = ypn. Solving (9.19) for y we have
(xn−1 + a+ axn/λ
xnxn−1 − 1
)p
= K (9.20)
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Since p may be any integer we have here an invariant of arbitrary degree.In order to proceed to the deautonomisation it is preferable to start with
the full freedom of (9.16). We find again that the mapping is integrable inone linearisable case which has the form
xn+1xn − 1
xnxn−1 − 1=
bn+1x2n + anxn + bn
bn−1x2n + an−1xn + bn(9.21)
Its linearisation is given by the Gambier mapping
yn+1 = yn (9.22)
xn =bnxn−1 + yn + an−1
ynxn−1 − bn−1
(9.23)
Next we turn to the case where the right hand side of (9.16) is not a ratio ofquadratic but rather of linear polynomials. Two cases can be distinguishedhere. The first correspond to a degenerate case of (9.21) where the numeratorand denominator have one common factor. This happens whenever a and bsatisfy the constraint
(an − an−1)(an−1bn+1 − anbn−1)− bn(bn+1 − bn−1)2 = 0 (9.24)
in which case (9.21) degenerates to
xn+1xn − 1
xnxn−1 − 1=bn+1(an − an−1)xn + bn(bn+1 − bn−1)
bn−1(an − an−1)xn + bn(bn+1 − bn−1)(9.25)
The autonomous limit of (9.25) can be easily obtained. We find that in thiscase the constraint is just a = ±(1 + λ) and the mapping becomes
xn+1xn − 1
xnxn−1 − 1=
1± xnλ1± xn/λ
(9.26)
However a second integrable case does exist which cannot be obtained fromthe quadratic one through some limiting procedure. It has the autonomousform
xn+1xn − 1
xnxn−1 − 1=
1− axn1 + axn
(9.27)
The degree growth of the iterates of (4.19) is again linear, 1, 2, 2, 3, 3, 4, 4,5, 5, . . . , an indication that this mapping should be linearisable. This turnsto be the case since (4.19) is equivalent to the Gambier mapping
yn+1 + yn = 0 (9.28)
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xn =a+ yn + xn−1
1 + axn−1
(9.29)
The deautonomisation of (9.27) is straightforward. We find
xn+1xn − 1
xnxn−1 − 1=
1− anxn1 + an+1xn
(9.30)
where an is a free function of the independent variable. The associatedGambier mapping is exactly (??) where a is now the function an and notsimply a constant.
9.4 Linearisable mappings of Q4 family
In this section we are going to study the mappings given by tavelling wavereduction of (7.2) and (7.3). It is indeed our experience that when a mappingis linearisable its coefficients after deautonomisation can be expressed in termsof some completely arbitrary function (we do not have a poof of this fact,rather we have observed this in practically all examples we did). This isindeed the case for projective mappings as well as for the Gambier one. Weare going to work with a general mapping which generalises the reduction of(7.2)
axn+1xn−1 + b(xn+1 + xn−1)xn + cx2n = 1 (9.31)
i.e. a form similar to travelling wave of (7.2) but where the relative coefficientof the xn+1xn−1 and x2n terms is not 1 any more. The parameters a, b, c arenow functions of the independent variable.
We are not going to go into all the details of the derivation. It suffices tosay that the linearisation can be obtained in terms of a Gambier mapping.We subtract (9.31) from its upshift (i.e. taking its discrete derivative) andreduce the order of the remaining homogeneous mapping by introducing theauxiliary variable yn = xn+1/xn. We find the mapping
bn+1y2nyn+1yn−1+cn+1y
2nyn−1+an+1ynyn+1yn−1+(bn+1−bn)ynyn−1−anyn−cnyn−1−bn = 0
(9.32)This mapping is again a Gambier one. Indeed it can be written as a systemof two discrete Riccati in cascade
yn =α + zn(β + yn−1)
yn−1
(9.33)
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zn+1 = −δ −zn
γ + κzn(9.34)
where α, β, γ, δ and κ are functions of the independent variable. In orderto simplify the presentation of the results we introduce the (free) functiongn = bn/an. A detailed calculation shows that it is possible to express theparameters of the Gambier mapping as follows
αn =gn−1
gn+1
βn = gn−1
γn =gn−1bn+1
bngn
κn =bn+1gngn−1
bngn+1
δn =gn+1
1 + gngn+1
(gn + gn+2
gn+2
− bnbn+1
gn+1 + gn−1
gn−1
)
Moreover the three functions a, b and c can be expressed in terms of the freefunction g. From the definition of g we have
an =bngn
(9.35)
and moreover we find
cn = gnbngn−1gn−2(gn+1 + gn−1) + bn−1gn+1(1− gn−1gn−2)
gn+1gn−1gn−2(1 + gngn−1)(9.36)
while b is given by the linear equation
bn+1gn−1gn−2(gn−1gn + 1)(gn+1gn+2 − 1) + bngn−2gn+2(g2n−1 − g2n+1)
+bn−1gn+1gn+2(gn+1gn + 1)(1− gn−1gn−2) = 0 (9.37)
Thus equation (9.31) is linearisable and as expected its general nonautonomousform does involve a free function.
Before concluding this section it would be interesting, as an aside, toconsider the degeneration of the mapping (9.31). As already shown by Adler,Bobenko and Suris the integrable lattice Q3 does, under the appropriate
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limiting procedure, degenerate to the lattice these authors of have dubbedQ2. In [48] the following reduced form is presented:
(xn+1 − xn)(xn − xn−1) + α(xn+1 + 2xn + xn−1) + β = 0 (9.38)
and have shown that it is linearisable in the same way as the mapping ob-tained from the reduction of Q3. It would be interesting to present here itsdeautonomisation. For the linearisation of the autonomous form of (9.38)we had started by subtracting it from its upshift and reducing the order ofthe remaining mapping by introducing the auxiliary variable yn = xn+1−xn.Here we start by consider the Gambier mapping:
yn = yn−1zn + gn(zn + 1) (9.39)
zn+1zn =fnfn+1
(9.40)
Eliminating z and introducing the variable x we obtain a mapping which canbe written as fn+1Mn+1 − fnMn, where Mn = 0 defines a mapping which isthe nonautonomous form of (9.38). We find that f can be explicitly given interms of the free function g:
fn =κgn + 2k(−1)n
(gn + gn−1)(gn + gn+1)(9.41)
where κ and k are two arbitrary constants. The mapping M has now theform
(xn+1−xn)(xn−xn−1)+xn+1gn−1+xn(gn−gn+1+γn(gn+gn−1))+xn−1gn+1+βn = 0(9.42)
where γn = (κgn+1 − 2k(−1)n)/(κgn + 2k(−1)n), βn = −gn−1gn+1 + (c +k(−1)n)/fn and c is another free constant. It is clear from the expressionof (9.42) that this nonautonomous form could not have been obtained bysimply allowing the parameters α and β in (9.38) to depend on n.
So one can say that the case of Q2 mapping is more challenging: itsnon-autonomous form was obtained from the appropriate limit of the (non-autonomous form of the) Q3 mapping. In this case the straightforward deau-tonomisation, i.e. allowing the parameters of the mapping to depend on theindependent variable, would not have given the desired result. This shouldbe an indication for future deautonomisation investigations: in some casesone must extend the autonomous form, introducing a priori superfluous pa-rameters, in order to ensure a parametrisation rich enough, to be amenableto deautonomisation.
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10 Ultradiscrete (tropical) mappings
What is an ultradiscrete system? The name ultradiscrete is used to designatesystems where the dependent variables as well as the independent ones as-sume only discrete values. In this respect ultradiscrete systems are nothingbut generalised cellular automata. The idea of ‘ultradiscretisation’ comesfrom the following question which is crucial for any analysis of a complexsystem; how simple can a nonlinear system be and still be genuinely non-linear? The nonlinearities which we are accustomed with involving simpleinteger powers are not the simplest. It turns out that the simplest nonlinearfunction of x one can think is |x|. It is linear for both x > 0 and x < 0but the nonlinearity comes from different determinations. Accordingly anyequation involving nonlinearities only in terms of absolute values will be thesimplest. In fact it will be an equation which is piecewise linear. The ul-tradiscrete limit converts a nonlinear discrete equation into one where onlyabsolute value nonlinearities appear. Of course now the dynamics will besimpler but retains the ‘nonlinear skeleton’ of the initial discrete one. Theonly drawback is the positivity requirement for any dependent variable andparameters.
In order to obtain the ultradiscrete limit we start with an equation forx, introduce X through x = eX/ǫ and then take appropriate limit ǫ → 0+.Clearly the substitution x = eX/ǫ requires x to be positive. The key relationis:
limǫ→0+
ǫ ln(1 + eX/ǫ) = max(0, X) =X + |X|
2
which can be easily generalised to the following basic formulas for sums andproducts:
limǫ→0+
ǫ ln(N∑
j=1
eXj/ǫ) = max(X1, X2, ..., XN ) (10.1)
limǫ→0+
ǫ ln(N∏
j=1
eXj/ǫ) = X1 +X2 + ...+XN (10.2)
In physics this procedure has been applied for the first time in 1996 [75]in the case of soliton equations and it was shown that indeed the ultradis-crete soliton equations posses multisoliton solution and they behave as inthe discrete case. Also other properties appeared, which are specific to ul-tradiscrete framework and these are related to the problem of integrability.
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Mathematically ultradiscretisation procedure is older and it appears for thefirst time in computer science. Since then it was developed up to now into fullfleshed topic called tropical mathematics. In order to have a more accurateunderstanding we shall define the things more rigurously. We follow the book[99]. Calling Rmax = R∪−∞ we introduce the semiring Rmax,⊕,⊗, ε, ethrough the following definitions:
• a⊕ b := max(a, b), a⊗ b := a+ b
• ε := −∞, e := 0
The following properties are easily verified:
• x⊕ (y⊕ z) = (x⊕ y)⊕ z and x⊗ (y⊗ z) = (x⊗ y)⊗ z, ∀, x, y, z ∈ Rmax
• commutativity: (trivial)
• distributivity x⊗ (y ⊕ z) = x⊗ y ⊕ x⊗ z
• zero elements: x⊕ ε = ε⊕ x = x, x⊗ e = e⊗ x = x
• multiplicative inverse: if x 6= ε, ∃!y, x⊗ y = e
• absorbing element: x⊗ ε = εx = ε
• idempotency: x⊕ x⊕ x...⊕ x = x and in general (x⊕ y)n = xn ⊕ yn
In addition we have the following proposition:
Proposition 10.1. For any a ∈ Rmax,⊗,⊕ with a 6= ε there is no additiveinverse for it
Proof. Suppose a ∈ Rmax has an additive inverse b and a⊕ b = ε. Then a⊕(a⊕b) = a⊕ε = a = (a⊕a)⊕b =︸︷︷︸
idempotency
a⊕b = ε so a = ε contradiction!
Instead of the cumbersome notations ⊗,⊕ one can use the usual additionand multiplication signs although it may induce confusions. For instance thetropical polinomial −2x3 − x2 + x + 5 is max(3x− 2, 2x− 1, x + 1, 5). Alsowe have the following definition:Definition: the tropical (ultradiscrete) hypersurface V (F ) defined by thetropical polinomial F in n-variables is the nondiferentiable locus in R
n. We
84
will see next section that this locus is essential for defining singularity con-finement.
We have seen at the beginning that one can tropicalize any polynomial(with no minus sign) using the limiting procedures given by the logarithms.However one can implement a more general procedure using valuation of afield K∗, val : K∗ → R. More precisely for any Laurent polynomial f =∑
ν∈I cνxν ∈ K[x±1
1 , ..., x±1n ] its ultradiscrete version is given by
trop(f)(z) = max(val(cν) + zν)
. In the next part we are going to use only real valuation for positive numbersand we implement the usual notation with max.
Remark 10.1. In [99] the tropicalization is taken using the min function in-stead of max. However things are equivalent sincemin(a, b) = −max(−a,−b)
It is thus natural at this point to ask how the integrability-related proper-ties of discrete systems carry over to cellular automata obtained from discretesystems following the ultradiscretisation procedure. The ultradiscretisationprocedure preserves any integrable character of the initial system. One wouldthus naturally expect the ultradiscrete analogue of integrability-related prop-erties, like the singularity confinement of the discrete case, to exist. Thiswould allow one to formulate ultradiscrete integrability conjectures and pro-pose integrability detectors. This question has been already addressed byJoshi and Lafortune [76] who proposed a singularity analysis approach whichis perceived as the ultradiscrete equivalent of singularity confinement. In thischapter we shall critically examine this approach and show that the situationis more complicated than what one would initially expect. In particular weshall show that, just as in the discrete case, there exist integrable ultradis-crete systems with unconfined singularities but also nonintegrable systemswith confined singularities.
10.1 Ultradiscrete singularities and their confinement
Before proceeding to the analysis of ultradiscrete systems let us recall thenotion of singularity. Given a mapping of the form xn+1 = f(xn, xn−1) weare in the presence of a singularity whenever ∂xn+1
∂xn−1= 0 i.e., xn+1 “loses”
its dependence on xn−1. When this is due to a particular choice of initialconditions we are referring to this singularity as a movable one. Movable
85
singularities may be bad, for integrability, because they may lead, after a fewmapping iterations, to an indeterminate form (0/0,∞−∞, . . . ) or propagateindefinitely. In the former case, provided we can lift the indeterminacy whilerecovering the lost degree of freedom (using an argument of continuity withrespect to the initial conditions), we are talking about a confined singularity.As explained in the introduction, mappings which are integrable throughspectral methods have confined singularities. The typical singularity patternin this case is the following: the solution is regular for all values of the indexn up to some value ns, then a singularity appears and propagates up to nc
whereupon it disappears and the solution is again regular for all values ofthe index larger than nc. In some cases we are in presence of the reciprocalsituation. The solution is singular for all values of n < ns, becomes regularbetween ns and nc and is again singular for n > nc. This singularity iscalled weakly confined by Takenawa [77] and is considered to be compatiblewith integrability. At the limit where there exists no interval where thesolution may be regular, and the solution is singular throughout, we are inthe presence of what we call a “fixed” singularity (which again does nothinder integrability).
How can these notions be transposed to the ultradiscrete setting? Thisis a question that has been addressed by Joshi and Lafortune [76] who pro-posed an analogue to the singularity confinement property for ultradiscretemappings. In the ultradiscrete systems the nonlinearity is mediated byterms involving the max operator. Typically one is in presence of termslike max(Xn, 0). When, depending on the initial conditions, the value of Xn
crosses zero, the result of the max(Xn, 0) operation becomes discontinuous:when X is slightly smaller than 0 the result is zero, while for X > 0 the resultis X. It is this discontinuity that plays the role of the singularity. Typicallyif we put X = ǫ, a term µ = max(ǫ, 0) propagates with the iterations ofthe mapping and perpetuates the discontinuity unless by some coincidenceit disappears. This disappearance is the equivalent of the singularity con-finement for ultradiscrete systems. Joshi and Lafortune [76] have introducedan algorithmic method for testing the confinement property for ultradiscretesystems, linked it to integrability and reproduced results on ultradiscretePainleve equations by initially deautonomising ultradiscrete mappings.
Before proceeding to a critical analysis of the method let us give an illus-trative example. In [78] there were introduced three different forms for the
86
ultradiscrete Painleve I equations starting from the QRT mapping
xn+1xn−1 = a1 + xnxσn
σ = 0, 1, 2 (10.3)
and its nonautonomous form. In order to illustrate the singularity analysisapproach we shall limit ourselves to the autonomous case and moreover takeσ = 2. Ultradiscretising (10.3) (putting x = eX/δ, a = eA/δ and taking δ → 0)we find
Xn+1 +Xn−1 = A+max(0, Xn)− 2Xn (2.2)
The singularity corresponds to the discontinuity induced by the term max(0, Xn)when the value of Xn crosses zero. We shall thus examine the behaviour ofa singularity appearing at, say, n = 1 where X1 = ǫ, while X0 is regular andlook at the propagation of this singularity both forwards and backwards. Inwhat follows we introduce the notation µ ≡ max(ǫ, 0) and the presence of µindicates that the value of X is singular. Below we present only the resultscorresponding to A > 0, those corresponding to A < 0 leading to similarconclusions. First we examine the case X0 < 0 and |X0| < A where one cansee a regular zone between X−3 and X1 and a singular pattern from X2 onas well as until X−4.
...X−13 = X−7 − 2X−5
X−12 = X−6 − 2X−5
X−11 = X−5
X−10 = X−7 −X−5
X−9 = X−6 −X−5
X−8 = X−5
X−7 = A+ ǫX−6 = −X0 − 2ǫ+ µX−5 = X0 + ǫ− µX−4 = A−X0 − ǫ+ µX−3 = −ǫX−2 = X0 + ǫX−1 = A− 2X0 − ǫX0
X1 = ǫX2 = A−X0 − 2ǫ+ µX3 = X0 + ǫ− µ
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X4 = −X0 + µX5 = A− ǫX6 = X3
X7 = X4 −X3
X8 = X5 +X3
X9 = X3
X10 = X4 − 2X3
X11 = X5 + 2X3...This is a weakly confined case, in the sense that a (small) regular region
exist surrounded by singular values extending all the way to infinity in bothdirections. As we explained already such a behaviour is deemed compatiblewith integrability. The cases 0 < X0 < A and X0 < −A lead to similar,weakly confined, patterns. The last case is X0 > A where the solution isregular until X1 then singular, confined, between X2 and X4 and regularfrom X5 on.
...X−3 = A− ǫX−2 = X0 − A+ 2ǫX−1 = −X0 + A− ǫX0 = X0
X1 = ǫX2 = A−X0 − 2ǫ+ µX3 = 2X0 − A+ 3ǫ− 2µX4 = A−X0 − ǫ+ µX5 = −ǫX6 = X0 + 2ǫ...Thus in all cases we have either a confined singularity (a central singular
zone with regular behaviour outside) or a weakly confined singularity (acentral regular zone with singular behaviour outside). Both behaviours aredeemed compatible with integrability. The two points which we considerimportant in this analysis are that a) one must study all possible sectors ofinitial conditions and/or parameters and b) one must consider the possibilityof weakly confined solutions.
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10.2 Nonintegrable systems with confined singularitiesand integrable systems with unconfined singular-ities
As we explained in the introduction there exist discrete systems which whilebeing nonintegrable still posses confined singularities (for instance the Hietarinta-Viallet mapping [30]). This discovery has as a consequence that singularityconfinement alone cannot be used a discrete integrability detector. As weshall show now the same problem appears in an ultradiscrete setting. In [79]there was found a mapping which did pass the confinement test while havinga positive algebraic entropy
xn+1 = xn−1
(xn +
1
xn
)(10.4)
The main advantage of this mapping over the examples of [30] is that it ismultiplicative and by choosing the appropriate initial data one can restrictthe solution to positive values. In that case the ultradiscretisation of (10.4)is straightforward. We find
Xn+1 = Xn−1 + |Xn| (10.5)
where we have preferred to introduce the absolute value of X instead of itsequivalent max(X, 0) + max(−X, 0). We shall examine the behaviour of asingularity appearing at, say, n = 1 where X1 = ǫ, while X0 is regular.We again use the identity µ ≡ max(ǫ, 0) = (|ǫ| + ǫ)/2 and distinguish twodifferent sectors X0 < 0 and X0 > 0. In the first case (X0 < 0) we find thesequence
...X−3 = 3X0
X−2 = 2X0 − ǫX−1 = X0 + ǫX0
X1 = ǫX2 = X0 − ǫ+ 2µX3 = −X0 + 2ǫ− 2µX4 = ǫX5 = −X0 + ǫ
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...We can see readily that the singularity, indicated by the presence of µ
is confined (to X2 and X3 only). Turning to the case X0 > 0 we find thesequence
...X−4 = −X0 + 2µ+ ǫX−3 = −X0 + 2µX−2 = ǫX−1 = −X0 + ǫX0
X1 = ǫX2 = X0 + 2µ− ǫX3 = −X0 + 2µX4 = 2X0 + 4µ− ǫ...In this case we are in presence of a weakly confined solution: a regular
part around n = 0 is surrounded by unconfined singularities both for largepositive and large negative n’s. Thus the ultradiscrete mapping (10.4) hasconfined singularities and is not integrable. (A stronger indication concerningthis nonintegrability, based on growth properties, rather than the analogywith the discrete case, will be presented in section 5). In this sense system(10.4) is an ultradiscrete analogue of the equation discovered by Hietarintaand Viallet [30].
The converse situation, of a mapping which while integrable does notpossess confined singularities does also exist. As expected an example is tobe sought among linearisable systems. In [79] it was discovered the “multi-plicative” linearisable mapping
xn+1
xn−1
= axn + a
xn + 1(10.6)
It is straightfoward to check that the parameter a can be always taken largerthan unity. (Indeed it suffices to reverse the direction of the evolution inwhich case a goes to 1/a). We can now ultradiscretise (10.6) to
Xn+1 = Xn−1 + A+max(Xn, A)−max(Xn, 0) (10.7)
where A > 0. The complete description of the solution would require exam-ining several sectors exist but in order to show that there exist unconfined
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singularities it suffices to exhibit such a situation in one sector. It turns outthat the case where X0 has a large negative value is one leading to unconfinedsingularities.
...X−4 = −X0 − 4AX−3 = −4A+ ǫX−2 = X0 − 2AX−1 = −2A+ ǫX0
X1 = ǫX2 = X0 + 2A− µX3 = 2A+ ǫX4 = X0 + 3A− µX5 = 4A+ ǫX6 = X0 + 4A− µX7 = 6A+ ǫ...We remark readily that while for negative indices the solution is regular,
a singularity, mediated by µ, appears for positive n’s and is never confined.We will analyse mapping (10.7) from the point of view of the growth of thesolutions as well.
Thus in perfect parallel to the discrete situation there exist ultradiscretesystems where despite the nointegrable character we have confined singular-ities while for ultradiscrete systems obtained from linearisable mappings thesingularities are not confined.
10.3 A family of integrable mappings and their ultra-discrete counterparts
In this section we shall pursue the study of the singularities of ultradiscretesystems which come as limits of mappings of the QRT family and discusstheir special properties. In particular we shall examine a mapping of theform:
(xn+1xn − 1)(xnxn−1 − 1) =x4n + ax2n + 1
(1 + xn/b)σσ = 0, 1, 2 (10.8)
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Mapping (10.8) is a special subcase of the autonomous limit of q-discretePainleve V. When σ = 0 the mapping was shown in [80] to be linearisable.All three cases belong to the QRT family and do possess a conserved quantity.We introduce yn = xn+1xn − 1 and (with obvious notations) we obtain theultradiscrete form of (10.8)
Xn+1 = −Xn +max(Yn, 0) (4.2)
Yn = −Yn−1 +max(4Xn, 2Xn + A, 0)− σmax(Xn − B, 0)Let us concentrate first on the σ = 0 case. The singularity corresponds hereto the value of Y crossing 0. We thus put Y0 = ǫ and iterate (??) starting fromX0 both backwards and forwards. We examine the branch 0 < X0 < A/2.This is the sequence we find for n < 0
Xn = X0 + n(A− ǫ)
Yn = Xn +Xn+1 (10.9)
At n = 0 we have by definition X0 and Y0 = ǫ. At n = 1 we find a singularvalue
X1 = −X0 + µ (10.10)
and iterating for positive n we obtain
Xn+1 = X1 + n(A− ǫ)
Yn = Xn +Xn+1 (10.11)
Since Xn+1 contains X1, the singularity which appeared at n = 1 propagatesad infinitum. On the other hand since (10.8) with σ = 0 is a member of theQRT family it does have an invariant:
K =x2n + x2n−1 + a
xnxn−1 − 1(10.12)
Ultradiscretising (10.12) is straightforward
K = max(4X, 2X + A, 2max(Y, 0))− 2X − Y (10.13)
We can check that (10.13) is indeed conserved by (10.8) and at no point doesthe singularity hinder this conservation.
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Thus we are here in the presence of an integrable mapping with unconfinedsingularities. This counterexample to the integrability criterion of [76] is evenmore serious than the examples of the pevious subsection since the mappinghere possesses an explicit invariant. It is thus natural to wonder what doeshappen in the remaining cases of (??), σ = 1 and 2. Presenting exhaustiveresults, as in the case of section 2, would be prohibitively long. Below wepresent a few typical numerical examples. We start with the case σ = 2, takeparameters A = 100 and B = 11, and initial condition X0 = 7. We obtainthe sequence:
...X−3 = −15 + ǫ− µY−3 = ǫX−2 = 15Y−2 = 122X−1 = 107Y−1 = 114X0 = X0
Y0 = ǫX1 = −7 + µY1 = 86− ǫ+ 2µX2 = 93− ǫ+ µY2 = 122− ǫX3 = 29− ǫ+ µY3 = ǫX4 = −29 + 2µY4 = 42 + 3ǫ− 4µ...We remark that this is a weakly confined singularity. A regular pattern
exists between Y−3 and Y0 and the singularity extends all the way to ±∞ onthe outside. What is more interesting is that the value of Y comes backs tozero, up to a quantity of O(ǫ), repeatedly albeit not in a periodic way. As amatter of fact the values of n for which Y is of order ǫ do show some regularity:. . . ,-26, -22, -19, -16, -13, -10, -6, -3, 0, 3, 7, 10, 13, 16, 19, 23, 26. . . . Weremark that the interval between two successive zeros is either 3 or 4 but asfar as we can tell there is no particular regularity in the succession of these twonumbers. Similar results can be obtained in the σ = 1 case. Again we finda weakly confined singularity and the zeros of Y appear at values: . . . , -34,
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-29, -25, -21, -17, -13, -8, -4, 0, 4, 9, 13, 17, 21, 25, 30, 34. . . . By studyingthe variation of the (mean) length of the intervals between two successivezeros of Y , which, we point out again here, give also the length of the regularzone, we arrive at the following conclusion. For fixed (appropriate) valuesof X0 and A and increasing values of B, with 2B/A integer, the length isexactly 2B/A + 3 for σ = 2 and 2B/A + 4 for σ = 1. If Y0 takes exactlythe value 0 then the solution is strictly periodic. If 2B/A is not integer thenthese quantities give the mean length of the interval. We can now see whatis happenning in the σ = 0 case. We can obtain this case by starting fromσ = 1 or 2 and take B →∞. Thus at the limit the length of the regular zonebecomes infinite and we go from a situation of weakly confined singularitiesto one of an unconfined singularity.
At this point one can wonder what is happening in the case where themapping is not integrable. We take (??) with σ = 3 and choose the sameparameters as for the case analysed just above, namely A = 100, B = 11 withinitial conditions X0 = 7 and Y0 going through zero. Iterating the mappingwe find that the solution does not recur to O(ǫ) although it does repeatedlycross zero to change sign. So for negative n the solution is regular while forpositive values of n the singularity continues indefinitely. Thus in this casewe have unsurprisingly an unconfined singularity.
In our analysis above we have presented the “interesting” singularity pat-terns. There also exist ranges of parameters in combination with the initialvalueX0 for which the solution has strictly confined singularities. Their studydoes not bring any new element: it suffices that one unconfined singularitypattern exist for confinement to be violated.
10.4 Complexity growth of ultradiscrete systems
As we have seen in the previous sections the situation concerning the in-tegrability criterion of [76] is far from clear. Counterexamples exist bothas to its sufficient and as to its necessary character. This does not meanthat the criterion is not useful. As was shown by Joshi and Lafortune thereexist many instances where the criterion can be put to use and succesfullypredict integrable deautonomisations. Still, because of the counterexamples,one is tempted to look for auxiliary or complementary criteria. Since in thediscrete case growth arguments turned out to be crucial for integrability itmakes sense to try to adapt these arguments to the case of ultradiscretesystems.
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Clearly the complexity argument used in the case of discrete systems(and its implementation through the algebraic entropy techniques) cannotbe transposed as such to the ultradiscrete case. Still the growth of the valuesof the variable can be of interest as we shall see in what follows.
We start with the integrable ultradiscrete system (??) and iterate it back-wards and forwards for parameter A = 7 and initial values X0 = −100 andX1 = 0. We find the following sequence of values: . . . , -100, 107, 0, -100,207, -100, 0, 107, -100, 100, 7, -100, 200, -93, -7, 114, -100, 193, -86, -14, 121,-100, 86, 21, . . . . We remark that the solution does not grow but oscillatesaround zero. As a matter of fact the absolute value of the solution neverexceeds the value 2|X0|+ |A|. Similar results can be obtained for other val-ues of the parameter and initial conditions. Another integrable ultradiscretesystem with an explicit conserved quantity is (??). We have given abovenumerical values of the iterates of the case σ = 2,with parameters A = 100,B = 11, and initial condition X0 = 7. Again the solution is not growing butbouncing between values which in this case never exceed 2B + A.
It may turn out that this property of bounded, bouncing solution is char-acteristic of a certain class of integrable ultradiscrete systems. Clearly moredetailed studies are needed before one can make a more affirmative statement.What is clear at this stage is that not all integrable ultradiscrete systems dohave such solutions. Analysing the growth of (??) with σ = 0 (which in thediscrete case is not just QRT-integrable but in fact linearisable) we find thesequence of values, for A = 100 and X0 = 7, Y0 = 0. We have for X: . . . ,207, 107, 7, -7, 93, 193, 293. . . and values that grow linearly by steps of 100away from zero in both positive and negative directions. Similarly for Y wefind: . . . , 314, 114, 0, 86, 286, . . . and linear growth in steps of 200 away fromzero in both directions. In order to investigate whether this linear growthis a property of ultradiscrete systems coming from linearisable mappings weanalyse the solutions of (10.7), taking A = 10, X0 = 0 and X1 = 7. Wefind the sequence: . . . -60, -53, -40, -33, -20, -16, 0, 7, 13, 17, 23, 27, 33, 37,. . . . Again we have a linear growth of the solution. Towards negative n thesolution grows with alternating steps of 7 and 13 while for positive n we havealternating steps of 4 and 6. Another example can be given by the mapping
Xn+1 = −Xn−1 +Xn +max(Xn, 0) (10.14)
which comes from the linearisable discrete system xn+1xn−1 = xn(xn + 1).Again starting from initial conditions X0 = 0 and X1 = 1 we find Xn = n,obvisouly a linear growth.
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While integrable mappings have moderate growth nonintegrable ones like(3.2) may grow much faster. By inspection we conclude that the solutions of(10.5) form a Fibonacci sequence and thus grow exponentially fast. On theother hand exponential growth is not the only possible one. For instance ifwe consider the ultradiscrete analogue of (10.3) with σ = −1, which is notintegrable, we find that the growth of the solutions is quadratic. What ismaking the situation even more complicated is for (??) with σ = 3, which isclearly a nonintegrable case, we find a bounded, bouncing solution.
In view of the above here are the (few) conclusions one can draw withrespect to growth properties of ultradiscrete systems. If one finds an expo-nential growth of the values of the iterates this is an indication of noninte-grability, while a linear growth indicates linearisability. However one mustbear in mind the fact that even in these cases a slower growth may be pos-sible. Thus the growth properties for ultradiscrete systems can be of someassistance in the detection of integrability but they do not constitute a pow-erful tool as in the discrete case. Probably a setting up in terms of tropicalalgebraic geometry would be more helpful.
10.5 Linearisable ultradiscrete dynamics: example froma biological model
In this section we discuss the tropicalization of a system of partial discreteequations which is linearisable. This system models a modular genetic net-work and it was published in [6]. We will show that it supports travellingwave solutions which exists also at the tropical level. Moreover a new periodicsolution it is shown to exist at the tropical limit.
The model itself has the following form:
p3n(t) =α + βp3n−1(t− τ)1 + p3n−1(t− τ)
− λp3n(t) (10.15)
p3n+1(t) =a+ bp3n(t− τ)1 + p3n(t− τ)
− λp3n+1(t) (10.16)
p3n+2(t) =A+ Bp3n+1(t− τ)1 + p3n+1(t− τ)
− λp3n+2(t) (10.17)
where (α, β), (a, b), (A,B) are the parameters characterising promoters ofthe genes in the group. We are going to consider here λ to be high - which
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happens only in artificially circuits by means of specific peptide sequencesappended to the proteins to make them targets for proteases in the cell.
Because we have nonlinear partial differential discrete equations with de-lay is more convenient make them fully discrete by writting time derivativeas a finite difference.
p3n(t+ δ)− p3n(t)δ
=α + βp3n−1(t− τ)1 + p3n−1(t− τ)
− λp3n(t) (10.18)
p3n+1(t+ δ)− p3n+1(t)
δ=a+ bp3n(t− τ)1 + p3n(t− τ)
− λp3n+1(t) (10.19)
p3n+2(t+ δ)− p3n+2(t)
δ=A+ Bp3n+1(t− τ)1 + p3n+1(t− τ)
− λp3n+2(t) (10.20)
Since λ can be made artificially big we can choose the time step δ to balanceit, namely λ = 1/δ. Taking for conveninence the notations with specificstaggering p3n−1 := zn−1, p3n := xn, p3n+1 := yn, p3n+2 := zn... we have thefollowing tractable form of rate equations:
xn(t+ σ) =α + βzn−1(t)
1 + zn−1(t)(10.21)
yn(t+ σ) =a+ bxn(t)
1 + xn(t)(10.22)
zn(t+ σ) =A+Byn(t)
1 + yn(t)(10.23)
where σ = τ + δ.In order to solve the system of equations (10.21), (10.22) and (10.23)
we eliminate yn from (10.22) and (10.23) and then plug into (10.21). Theresulting equation will be:
xn(t+ σ) =µ+ νxn−1(t− 2σ)
ρ+ γxn−1(t− 2σ)(10.24)
where:µ = α(1 + a) + β(A+ aB)
ν = α(1 + b) + β(A+ bB)
ρ = 1 + a+ A+ aB
γ = 1 + b+ A+ bB
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In order to show the modularity of the whole network we consider:
xn(t)→ρ
γxn(t), αe =
µ
γ, βe =
νρ
γ2
With these substitution the above equation is transformed into
xn(t+ 3σ) =αe + βexn−1(t)
1 + xn−1(t)(10.25)
The equation (10.25) can be immediately linearised by the Cole-Hopf typetransform,
xn(t) = −1 +Fn(t+ 3σ)
Fn−1(t)
to the following linear equation
Fn(t+ 6σ)− (αe − βe)Fn−2(t)− (1 + βe)γ)Fn−1(t+ 3σ) = 0 (10.26)
We check for travelling wave signals, i.e., Fn(t) = F (ξ) where ξ = n + vt,and v is the signal velocity. Equation (10.26) becomes
F (ξ + h)− (1 + βe)F (ξ)− (αe − βe)F (ξ − h) = 0 (10.27)
where h = 1 + 3vσ is the step. The speed v is free but the product vσ mustbe an integer. The solution can be easily computed and has the followingform:
F (ξ) = C1
(1 + βe +
√∆
2
)ξ/h
+ C2
(1 + βe −
√∆
2
)ξ/h
(10.28)
where ∆ = (1 + βe)2 + 4(αe − βe) and C1,2 are integration constants. Since
the solution must be positive, both terms in the right-hand-side of equation(10.28) should be as well. So we must have (1 + βe) >
√∆ which leads to
αe < βe ⇔ (a − b)(α − β)(A − B) < 0. Another condition comes from the
fact that pn(t) = −1 + F (ξ+h)F (ξ)
must be positive. Now if αe = 0, βe > 1 thenthe solution has the following kink-type shape
pn(t) = −1 +1 + Cβη+1
e
1 + Cβηe
= C(βe − 1)βηe
1 + Cβη+1e
(10.29)
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where η = (n+ vt)/h, C and v are arbitrary. This solution shows that alongthe transcriptional modular cascade, we have a successive gene expression, allgenes being sequentially expressed as the signal kink goes on (for biologicalrelevance see [6], [97]
Now we are going to analyse the ultradiscrete limit of our discrete equa-tions. Since the modular cascade is equivalent with a network having samegene we shall treat only the equation,
xn(t+ 3σ) =αe + βexn−1(t)
1 + xn−1(t).
Of course now the dynamics will be simpler but retains the ‘nonlinearskeleton’ of the initial discrete one. As we have seen in the previous chapterthe method of ultradiscretisation is algorithmic and extremely simple. Wehave applied the ultradiscrete approach to many biological models [85, 86,87, 88]. The only drawback is the positivity requirement for any dependentvariable and parameters, but here this is not a problem since all the biologicalquantities are positive. We are going to apply this method to show thelinearisability and and how kink nonlinear wave survives at ultradiscretelimit. In addition we shall show that a peiodic solution exists.
In order to obtain the ultradiscrete limit we start with an equation forx, introduce X through x = eX/ǫ and then take appropriate limit ǫ → 0+.Clearly the substitution x = eX/ǫ requires x to be positive.
For our equation we put xn(t) = eXn(t)/ǫ, αe = eAe/ǫ, βe = eBe/ǫ and obtainfinally:
Xn(t+ 3σ) = max(Ae, Be +Xn−1(t))−max(0, Xn−1(t))
which can be written in a more convenient form (using the distributivity ofmax-operation with respect to addition) as:
Xn(t+ 3σ) = max(Ae −Be, Xn−1(t))−max(0, Xn−1(t)) + Be (10.30)
One can see that if the parameters Ae, Be and initial conditions Xn(0) areintegers then the evolution will produce only integer results, so our equationis indeed a generalised cellular automaton. In addition, the variable Xn(t) isno longer positive, since it is related to the logarithm of the initial one xn(t)
In order to discuss the solution we impose the travelling wave ansatzν = Kn + Ωt with n, t,K,Ω ∈ Z. In this way one obtains a discrete
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piecewise linear equation in one integer variable ν shifted by the integervalue µ = K + 3σΩ. Calling Se = Ae −Be we have:
X(ν + µ) = max(Se, X(ν))−max(0, X(ν)) + Be (10.31)
This equation can be solved by reducing to linear discrete equations of orderµ on various sectors defined by the signs of Se, Be or X. Since µ is free wehave a lot of possible solutions. For simplicity we take µ = 1 and show thesolutions:
• Case Se < 0 and Be < 0; for Se < X < 0 we have X(ν+1) = X(ν)+Be
with the solution X(ν) = Beν+c1 (the initial condition is related to theconstant c1). But Se < X < 0 gives [Ae/Be−1−c1/Be] < ν < −[c1/Be].Because Ae/Be > 0 we have 0 ≤ Ae/Be ≤ 1 and accordingly n willhave only one or maximally two values. So the solution is trivial. ForX < Se < 0 we have X(ν + 1) = Se + Be = Ae which gives a constantsolution X(ν) = Ae.
Now for X > 0 the only sector is given by Se < 0 < X, X(ν + 1) = Be
with the constant solution X(ν) = Be > 0 - contradiction. So, forSe < 0 and Be < 0 we have only constant solution.
• Case Se < 0 and Be > 0. Again for X < Se < 0 we have X(ν + 1) =Se + Be = Ae giving a constant solution X(ν) = Ae which can bepositive or negative. Also for Se < X < 0 we haveX(ν+1) = X(ν)+Be
with the solution X(ν) = Bν + c1 and again we have an interval forν as above. But now Be is positive and in this case we can make thesolution to be nontrivial choosing −Ae to be huge namely Ae = −∞(this is not a problem; the biological parameter αe = 0 in this case).So indeed X(ν) = Beν+c1 for all ν < [−c1/Be]. Now for X > 0 the wehave X(ν + 1) = Be with the constant solution X(ν) = Be > 0. Thesesectors can be unified to give a sigle form of the solution which is nottrivial and has a travelling wave form:
X(ν) = Be(ν + 1) + c1 −max(0, Beν + c1)
This solution is nothing but the ultradiscrete limit of the discrete kinksolution (10.29) obtained in the case αe = 0 ⇔ Ae = −∞ and βe >1⇔ Be > 0
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• all other sectors give trivial solution except the following; 0 ≤ X ≤Se, Be < 0. We have X(ν + 1) = Ae −X(ν) with the solution
X(ν) =Ae
2+ (−1)νc2 (10.32)
This solution is a periodic one which has meaning only in the case ofinteger ν. For an appropriate choice of c2 (for instance c2 = Ae/4) thesolution is smaller than Se and bigger than zero. This solution showsthat we have a signal propagating also in periodic networks.
However, the fact that we have a periodic solution in the ultradiscretelimit does not guarantee the existence of such solution in the initial discreteequation. It may correspond not only to a periodic but also to a dampedoscillating solution. Moreover we first linearised our discrete equation by aCole-Hopf type transform and then compute solutions. It may happen thatsome solutions do not belong to the linearisable sector (not captured by theCole-Hopf). As we said at the beginning of the section, the ultradiscrete limitretains only the skeleton of the initial discrete equation and accordingly noteverything in the discrete case have an unique correspondent in the ultra-discrete one. Also the reverse is possible, for instance multiple ultradiscretelimit cycles which correspond to only one in the discrete case or negativeultradiscrete solitons with no counterpart in the discrete case as well [98].
We are going to end this section showing how the linearisability and Cole-Hopf transform works in the ultradiscrete case. Here we have a problem. TheCole-Hopf transform involves a negative minus one term
xn(t) = −1 + Fn(t+ 3σ)/Fn−1(t).
In order to eliminate this problem we will rewrite the equation in the variablewn(t) = 1 + xn(t) as :
wn(t+ 3σ) =αe − βewn−1(t)
+ 1 + βe (10.33)
The main drawback now is that only αe−βe > 0 is compatible with ultradis-cretisation. With the substitution wn(t) = eWn(t)/ǫ, ae − be = eSe/ǫ, 1 + be =eQ/ǫ, Fn(t) = eΦn(t)/ǫ we have:
Wn(t+ 3σ) = max(Se −Wn−1(t), Q) (10.34)
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Now, put the ultradiscrete Cole-Hopf Wn(t) = Φn(t + 3σ) − Φn−1(t) in(10.33). The equation goes down to:
Φn(t+ 6σ) = max(Se + Φn−2(t), Q+ Φn−1(t+ 3σ)) (10.35)
which is nothing but the ultradiscrete limit of the linear discrete equation(10.27) Of course the term linearisability is somehow invisible for ultradiscreteequation inasmuch as they are already piecewise linear. But the equation(3.25) has an additional symmetry with respect to Φn(t)→ Φn(t)+hn(t) forany function hn(t). This is the way of manifestiation of the linear characterat the ultradiscrete level.
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11 General conclusions
The main topic we covered in this thesis is the role of singularities in es-tablishinjg the integrable character of a mapping (discrete or ultradiscrete)and in integrating it effectively. Even though at the begging the criteriawere introduced from the physicist point of view namely pure euristic, grad-ually it was realised that the rigurous approach based on algebraic geometrytools can improve tremedously their effectiveness. We tried our best in thisthesis to underline the following aspects whenever one deals with a discretemapping: the integrable character can be established either by computing(numerically) complexity growth, or by analysing rigurously the singularities.Now for complexity growth,
• if the complexity growth is linear then the mapping is a linearisableone
• if the complexity growth is quadratic then the mapping is still inte-grable by means of spectral methods (Lax pairs) and it involves ellipticfunctions.
• if the complexity growth is exponential then the system has positivealgebraic entropy which by the theorem of Gromow-Yomdin [92] im-plies a positive topological entropy i.e. no-integrable one. Howeverthere are systems which escapes from this namely the ones that canbe linearised by non-rational or transcendental transformations. Forinstance, xn+1xn−1 − xpn = 0, p ≥ 3 has nonconfined singularities andpositive algebraic entropy. But by means of zn = log xn the system canbe linearised to zn+1+ zn−1− pzn = 0. Still the integrability of this lin-ear equation is problematic due to multivaluedness of the logarithnmicsubstitution. In addition the chaotic character of the original mappingis not rigurousluy established beacuse from the numerical experimentsseems rather an ergodic behaviour. Accordingly we do not take intoaccount these type of systems.
Even though the complexity growth is very effective as an integrabilitycriterion the main problem is to integrate effectively the mapping. Here thesingularity confinement enters on the stage by giving the pattern of singular-ities. Blowing up this singularities one obtains (if the number of singularitiesis finite) a rational elliptic surface (which must be minimised in case not min-imal). If the orthogonal complement of the associated singular fibers Dynkin
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diagram is a affine Weyl group then the systems is an integrable one. Theinvariant can be computed from the proper transform of the divisor classof eigenvalue one of the action on the associated Picard group (or echiva-lently Neron-Severi lattice). If the corresponding Weyl group is not affinethen the system is non-integrable (even though realises an automorphism ofan algebraic surface). In addition singularity confinement produces deau-tonomisation by imposing the same singularity pattern in case of a mappingwith unknown coefficients
The case of linearisable systems is more complicated. However still fromsingularity pattern it is possible to find the linearisation and even to deau-tonomise the system. The most enigmatic domain is the integrability oftropical (ultradiscrete) mappings. Being piecewise linear one is tempted tosay that they cannot be chaotic at all. However there is no algorithm tofind their invariants. The presence of nondetermiantion points given by themax function suggested that a kind of confinement can be idone namely dis-apearance of such nondetermination with initial data recovering. Howeverwe have shown that this criterion although instrumental in establishing inte-grability for simple QRT-like mappings and lattice soliton equations, fails inmany other cases and what is worst there is no analog of complexity growth.We expect that a deep understanding of notion of singularity based on theconcepts of toric and tropical algebraic geometry will shed light on the clar-ification of such problems.
We are ending the section by saying that discrete mappings or discretesoliton equations can have important applications in molecular biologicalmodels. We have written many papers on this topic and we intend to anal-yse mathematically various equations coming from quantitative molecularbiology. T
12 Future research directions
As we have seen the instruments of algebraic geometry are very effective inanalysis of the dynamics of two dimensional mappings. Even though manyresults are already known we intend to continue this approach to unveil otherbeautiful features of integrability and possible applications.1. Higher order mappings. This is the natural step which we intend to tacklein the near future. Of course here exists a major drawback. Since now wehave to work on P
3 or P1×P1×P
1 we can have singularities which are not
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only points but also curves - and this fact will overburden enormously thecomputations. By developing certain software techniques (in MATHEMAT-ICA or McCAULEY) we will consider the singularity patterns here and seehow to construct the invariants. Also nothing has been done on the systemshaving invariants parametrised by hyperelliptic curves.2. Ultradiscrete mappings. As we have seen in the last chapter integrabilityand invariants represents open problems. Also construction of a tropical QRTmapping is still problematic despite the results obtained by [91]. We intendto rely on the properties of tropical elliptic curves and try ”brute force” tofind some extension and improvement of ultradiscrete singularity confinementpresented in the last chapter. Also we intend to study the connection withtoric varieties although because the of the C
∗ action, they are not abelianso it will be difficult to imagine an integrable mapping with toric level sets.On the other hand we intend to study more carefully the symmetries of theultradiscrete Painleve equations using the recently introduced ultradiscreteHirota bilinear formalism [94] (this approach was used for discrete Painleveequations before the algebraic geonmetric one). In the same direction thesoliton dynamics for partial ultradiscrete equations is still at the beginningand apart from Korteweg de Vries and Toda systems there are very fewstudies on others [95].3. Connection between geometry and Lax pairs (represented here by isomon-odromic deformation). For instance the q-Painleve I equation
xn+1xn−1 =znzn+1(1 + xn)
x2n, zn = αqn/2
has the following Lax pairs
Ln =
0 0 zn/xn 00 0 xn−1 qxn−1
λxn 0 1 q0 λzn−1/xn−1 0 0
and
Mn =
0 xn/zn(1 + xn) 0 00 0 1 00 0 1/xn q/xnλ 0 0 0
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The deformation of the q-linear difference system is given by:
φn(qζ) = Ln(ζ)φn(ζ)
φn+1(ζ) =Mn(ζ)φn(ζ)
Complatibility of these two equations is:
Mn(qζ)Ln(ζ)− Ln+1(ζ)Mn(ζ) = xn+1xn−1 −znzn+1(1 + xn)
x2n
The existence of such compatibility for a deformation of a linear systemsis a fundamental aspect of integrability. For the moment there are very fewstudies [93],[?] about connection between the emergent space of initial con-ditions (geometry of a rational surface) and construction of such operators.We intend to extend the results of [93] to any type of Halphen surface.4. Isomonodromic deformations and space for initial conditions for delay-equations. Delay mappings are a hybrid between discrete and continuoussystems. For instance the well known delay-Painleve II equation:
d
dtw(t)w(t+ 1) = w2(t)− w2(t+ 1)
which can be deduced from the travelling wave reduction of the famous soli-ton equation sine-Hilbert Hut(x, t) = sin u(x, t) (H is the Hilbert transformwith respect to x) . Very few things are known for such systems. There areonly two papers concerning them one by Ramani et al. in 1992 [96] wherethe singularity confinement+Painleve test were mixed and Carstea 2010 [90]where the Hirota bilinear forms were obtained. We intend to study more care-fully these systems relying on the fact that their Lax pairs can be obtainedby reduction of Lax pairs of integro-differential soliton equations involvingsingular integral operators.5. Fermionic extensions of lattice soliton equations Although we did not treatin this thesis the supersymmetric integrability, we do have many significantresults concernig dynamics of supersymmetric solitons. For almost 10 yearsthe topic has been focusing on continuous systems like supersymmetric KdVhierarchy and modified KdV and we practically initiate the domain related toHirota super-bilinear formalism [81, 82, 83, 84]. But very recently appearedresults concerning fermionic extensions of partial discrete equations. Weintend to study the Hirota bilinear formalism in this context and interaction
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of fermionic lattice solitons. Even though Painleve analysis can be appliedto such equations (and we also constructed supersymmetric extensions ofPainleve I and II equations) nothing has been done in the discrete context.We intend at the beginning to understand what means a singularity in latticeequations with values in Grassmann algebra.
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