continued fractions and hecke triangle groups€¦ · front hecke triangle groups continued...
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Front Hecke triangle groups Continued fractions Coding of geodesics Transfer operator Model system Conclusions
Front page
Continued fractions and Hecke triangle groups
Tobias MuhlenbruchJoint work with Dieter Mayer and Fredrik Stromberg
Lehrgebiet StochastikFernUniversitat Hagen
19 May 2010EU-Young and Mobile Workshop: Dynamical Systems and Number
Theory
Front Hecke triangle groups Continued fractions Coding of geodesics Transfer operator Model system Conclusions
Outline of the presentation
1 Hecke triangle groups
2 Continued fractions
3 Coding of geodesics
4 Transfer operator
5 Model system
6 Conclusions
Front Hecke triangle groups Continued fractions Coding of geodesics Transfer operator Model system Conclusions
Hecke triangle groups
Hecke triangle groups Gq
Gq =⟨S ,Tq|(STq)q = 1
⟩⊂ PSL2(R) with S =
(0 −11 0
),
Tq =
(1 λq
0 1
), λq = 2 cos
(π/q
)and q = 3, 4, 5, . . ..
Mobius transformations:
(a bc d
)z =
az+bcz+d if z ∈ C,ac if z =∞.
Translation Tq : z 7→ z + λq and rotation S : z 7→ −1z .
Gq is a Fuchsian group of the first kind.
Gq\H is finite and non-compact.
limit set Gq∞ ⊂ Q(λq) ⊂ P1R is dense.
Gq\H is non-arithmetic for q 6= 3, 4, 6.
(Closed) fundamental domain Fq =z ∈ H; |z | ≥ 1, |Re(z)| ≤ λq
2
.
Front Hecke triangle groups Continued fractions Coding of geodesics Transfer operator Model system Conclusions
Nearest λ-multiple continued fractions
Recall Gq =⟨
S, Tq |(STq )q = 1⟩
with S =
(0 −11 0
), Tq =
(1 λq0 1
), λq = 2 cos
(π/q
)and q = 3, 4, 5, . . ..
Nearest λ-multiple continued fractions (λCF)
We identify a sequence of integers, a0 ∈ Z, and a1, a2, . . . ∈ Z? with a point
x = T a0q ST a1
q ST a2q · · · 0 = a0λ+
−1
a1λ+ −1
a2λ+−1...
=: [a0; a1, a2, . . .]q
and say that it is a
regular λCF, [a0; a1, a2, . . .]q, if it does not contain “forbidden blocks”,and
dual regular λCF, [a0; a1, a2, . . .]q, if it does not contain reversed“forbidden blocks”.
Forbidden blocks are blocks of the form h =
q2− 1 if 2 | q,
q−32
if 2 - q.
q = 3 [1], [−1], [2,m] and [−2,−m],
even q [1h+1], [(−1)h+1], [1h,m] and [(−1)h,−m],
odd q ≥ 5 [1h+1], [(−1)h+1], [1h, 2, 1h,m], [(−1)h,−2, (−1)h,−m].
for all m ∈ N. Hecke continued fractions
Front Hecke triangle groups Continued fractions Coding of geodesics Transfer operator Model system Conclusions
Nearest λ-multiple continued fractions
Exampleq π e −λq/23 [3;−7, 16, 294, 3, 4, 5, 15, . . .]3 [3; 4, 2,−5,−2, 7, 2,−9,−2, . . .]3 [0; 2]3
4 [2;−2, 2, 8,−4,−5,−1, 2, 3, . . .]4 [2; 6,−1, 3, 1,−2,−1, 1,−7, . . .]4 [0; 1]4
5 [2; 7, 1, 2,−1, 1,−1, 1,−3, . . .]5 [2; 1,−2,−14, 5, 1, 9,−2,−1, . . .]5 [0; 1, 2, 1]5
6 [2; 2, 2, 1, 2, 1,−1,−3,−1, . . .]6 [2; 1, 1,−1,−1, 3, 5, 1, 1,−7, . . .]6 [0; 1, 1]6
Equivalent points
x and y are Gq-equivalent :⇐⇒ ∃ g ∈ Gq such that g x = y⇐⇒
the CF of x and y have the same tail or
the CF of x and y have tail
q = 3 [ 3 ]3 or [−3 ]3,
even q [ 1h−1, 2 ]q or [ (−1)h−1,−2 ]q, or
odd q ≥ 5 [ 1h, 2, 1h−1, 2 ]q or [ (−1)h,−2, (−1)h−1,−2 ]q.
λCF’s admit natural lexicographic order ≺.
Front Hecke triangle groups Continued fractions Coding of geodesics Transfer operator Model system Conclusions
Associated dynamical system
The generating map fq
(x)q ∈ Z denotes the nearest λ-multiple of x :
|x − (x)qλ| ≤ λ2
.
For Iq =[−λ
2, λ
2
]the generating map is
f : Iq → Iq; x 7→ −1x−(−1
x
)qλ.
If we set x0 = − 1x
then the CF x = [a0; a1, . . .]q arecomputed by
an = (xn)q and xn+1 = f (xn) = −1xn− anλ.
Natural extension of fq
The natural extension of fq is (x = [0; a1, . . .]q)
Ωq → Ωq; (x , y) 7→(f (x), −1
y+a1λ
).
Front Hecke triangle groups Continued fractions Coding of geodesics Transfer operator Model system Conclusions
λCF and coding of geodesics
Recall Gq = 〈S ,Tq〉.
We denote geodesics γ on H by their basepoints: γ = (γ−, γ+).
In the diagram we illustrate the closedgeodesic γ = ([0;−3,−4], [0;−4,−3]−1).
γ? is closed ⇐⇒ γ− = [0; . . . , a1, . . . , an]q is regular andγ−1
+ = [0; . . . , an, . . . , a1]q is dual regular.
Equivalent geodesics
γ and γ′ are Gq-equivalent :⇐⇒∃ g ∈ Gq such that (g γ−, g γ+) = (γ′−, γ
′+)
Front Hecke triangle groups Continued fractions Coding of geodesics Transfer operator Model system Conclusions
λCF and coding of geodesics
Theorem
Each geodesic γ′ in H is Gq-equivalent with a geodesic γ = (γ−, γ+)satisfying (γ−, γ
−1+ ) ∈ Ω.
If γ and υ satisfy (γ−, γ−1+ ), (υ−, υ
−1+ ) ∈ Ω and no base point is
Gq-equivalent with
q = 3 [0; 3 ]3 or [0;−3 ]3,
even q [0; 1h−1, 2 ]q or [0; (−1)h−1,−2 ]q, or
odd q ≥ 5 [0; 1h, 2, 1h−1, 2 ]q or [0; (−1)h,−2, (−1)h−1,−2 ]q.
then γ = υ.
Closed geodesics γ have at most 2 Gq-equivalent geodesics “in Ω”.
Front Hecke triangle groups Continued fractions Coding of geodesics Transfer operator Model system Conclusions
λCF and coding of geodesics
TheoremThe following statements are equivalent:
γ? is a closed geodesic on Gq\H.
γ = (γ−, γ+) with γ− = [0; a1, . . . , an]q is regular andγ−1
+ = [0; an, . . . , a1]q is dual regular.
The map γ? 7→ γ is bijective except if [a1, . . . , an] is Gq-equivalent to
q = 3 [ 3 ]3 or [−3 ]3,
even q [ 1h−1, 2 ]q or [ (−1)h−1,−2 ]q, or
odd q ≥ 5 [ 1h, 2, 1h−1, 2 ]q or [ (−1)h,−2, (−1)h−1,−2 ]q.
Front Hecke triangle groups Continued fractions Coding of geodesics Transfer operator Model system Conclusions
General form of a transfer operator
General form of a transfer operator
Given a set Λ and maps f : Λ→ Λ and g : Λ→ C,a transfer operator L acting on functions h : Λ→ C is defined by(
Lh)(x) =
∑y∈f−1(x)
g(y) h(y)
Remarks:
Usually, take g = |J|−1 if the Jacobian J of f exists.
L of the form Lh(x) =∑
y∈f−1(x)
(f ′(y)
)−1h(y) is also known as a
Perron-Frobenius Operator.
Relation of L to the dynamical zeta-function: more
ζ(z) =1
detc (1− zL).
Related to the Ising 1d spin model. Ising Model
Front Hecke triangle groups Continued fractions Coding of geodesics Transfer operator Model system Conclusions
The transfer operator for continued fractions
Recall respectively define:
S =(0 −1
1 0
)and T =
(1 10 1
)∈ G3 = PSL2(Z),(
h∣∣
2s
(a bc d
))(z) :=
((cz + d)2
)−sh(
az+bcz+d
)and
f :[− 1
2 ,12
]→[− 1
2 ,12
]; x 7→ −1
x (mod 1) = T−nS x for some n ∈ Z.
The associated transfer operator is formally given by
Lsh(x) :=∑
y∈f−1(x)
(f ′(y)
)−sh(y) (x ∈ [−1/2, 1/2])
Banach space (with sup-norm) V := C(D) ∩ Cω(D), D = z ; |z | ≤ 1.
Transfer operator for continued fractions (q = 3)
The operator Ls : : V × V → V × V , Re(s) > 1, is defined as
Ls~h(z) =
(∑∞n=3 h1
∣∣2sST n +
∑∞n=2 h2
∣∣2sST−n∑∞
n=2 h1
∣∣2sST n +
∑∞n=3 h2
∣∣2sST−n
)for ~h =
(h1h2
)∈ V × V . Numerical example
Front Hecke triangle groups Continued fractions Coding of geodesics Transfer operator Model system Conclusions
The transfer operator for continued fractions
Recall
Ls : V 2 → V 2; Ls~h(z) =
(∑∞n=3 h1
∣∣2sST n +
∑∞n=2 h2
∣∣2sST−n∑∞
n=2 h1
∣∣2sST n +
∑∞n=3 h2
∣∣2sST−n
).
TheoremThe transfer operator Ls is nuclear of order 0.
The transfer operator Ls allows a meromorphic continuation into thecomplex s-plane.
Corollary
For almost all ~h ∈ V 2 the limit limn→∞ L1~h converges to the unique
invariant measure.
Front Hecke triangle groups Continued fractions Coding of geodesics Transfer operator Model system Conclusions
Setup
Model system
1 particle freely moving on Gq\H
Quantum-mechanical interpretation Maass cusp formsMaass cusp forms
Classical mechanical picture (closed) geodesics
Selberg trace formula:Quantum mechanical system ←→ Classical mechanical system∑
test function over spectral values ←→∑
test function over closed geodesics
∃ Maass cusp form ←→ Zeros of Selberg zeta-function
Front Hecke triangle groups Continued fractions Coding of geodesics Transfer operator Model system Conclusions
Geodesics on Gq\H
We denote geodesics γ on H by their basepoints: γ = (γ−, γ+).
The diagram illustrates a closed geodesic γ
Classical mechanics =⇒ geodesics on Gq\H.
The classical system is chaotic =⇒ closed geodesics are dense.
Each geodesic γ induces an geodesic γ? on Gq\H.
Each geodesic γ? on Gq\H induces class of geodesics Gq γ on H.
Coding of geodesics.(γ? on Gq\H is closed ⇐⇒ γ− = [0; a1, . . . , an]q , γ−1
+ = [0; an, . . . , a1]q
Front Hecke triangle groups Continued fractions Coding of geodesics Transfer operator Model system Conclusions
The Selberg zeta-function and the transfer operator
Selberg zeta-function
Z (s) :=∏ω
∞∏m=0
(1−
[e l(ω)
]−s−m)
(Re(s) > 1)
where ω runs over distinct primitive periodic geodesics ω andl(ω) is its length.
Properties of Z (s)
Z (s) can be analytically continued to an entire function.
The non-trivial zeros of Z (s) are located at s = 1, 2s is Riemannzero or s is a spectral parameter.
The trivial zeros of Z (s) are located at s = −l , l = 0,−1,−2, . . ..
Z(s)Z(1−s) = Φ(s)Ψ(s) with scattering matrix Φ(s) =
√π
Γ(s− 12 )ζ(2s−1)
Γ(s)ζ(2s)
(for q = 3) and a (computable) function Ψ.
Front Hecke triangle groups Continued fractions Coding of geodesics Transfer operator Model system Conclusions
The Selberg zeta-function and the transfer operator
Recall the Selberg zeta-function
Z (s) =∏ω
∞∏m=0
(1−
[e l(ω)
]−s−m).
Theorem
det(1− Ls) = Z (s) det(1−Ks)
where Ks is a simple operator with s → det(1−Ks) has no poles andsimple zeros in sn,k = n + 2πik
const , n ∈ Z≤0, k ∈ Z.
Corollary
For 0 < Re(s) < 1, 2s 6= 1:
Ls has eigenvalue 1 if and only if Z (s) = 0 or s = sn,k .
(for q = 3) Ls has eigenvalue 1 only in spectral parameters s,ζ(2s) = 0.
Remark: Connection between 1-eigenfunctions of Ls and period functions (for q = 3)
Period functions
Front Hecke triangle groups Continued fractions Coding of geodesics Transfer operator Model system Conclusions
A simplified overview
Maass cusp forms period functions
zeros of Z (s) eigenfunctions of Ls
with eigenvalue 1
- [LZ01] [BLZ]
- [MMS]
6
?
[Mar03, §14]
6
?
[BM09]
Advantages of the transfer operator approach:
Eigenfunctions of Ls can be calculated numerically, Z (s) is hard tocalculate in general.
Extension to (non-arithmetic) Hecke triangle groups.
Direct connection form eigenfunctions to period functions.
Gives different approach to study problems in “quantum chaos”.
Changes interpretation of s: from spectral value to a “weight-type”parameter.
Front Hecke triangle groups Continued fractions Coding of geodesics Transfer operator Model system Conclusions
Thank you!
Thank you for your time!
References Abstract Additional things
References
[BLZ]: R.W. Bruggeman, J. Lewis and D. Zagier,
Period functions for Maass wave forms. II: Cohomology,in preparation.
[BM09]: R.W. Bruggeman and T. Muhlenbruch,
Eigenfunctions of transfer operators and cohomology ,J. Number Theory 129 (2009) 158–181.
[DFG]: DFG Research Project
“Transfer operators and non arithmetic quantum chaos” (Ma 633/16-1).
[He83]: D.H. Hejhal,
The Selberg Trace Formula for PSL(2,R), Vol.2,Lecture Notes in Mathematics 1001, Springer-Verlag, 1983.
[He92]: D.H. Hejhal,
Eigenvalues of the Laplacian for Hecke triangle groups,Mem. Amer. Math. Soc. 97 (1992).
[Hu89]: A. Hurwitz,
Uber eine besondere Art der Kettenbruch-Entwicklung reeller Grossen,Acta Mathematica 12 (1889), 367–405.
[KU07]: S. Katok,and I. Ugarcovic,
Symbolic dynamics for the modular surface and beyond ,Bulletin of the American Mathematical Society 44 (2007), 87–132.
References Abstract Additional things
References
[LZ01]: J. Lewis and D. Zagier,
Period functions for Maass wave forms. I ,Annals of Mathematics 153 (2001), 191–258.
[Mar03]: J. Marklof,
Selberg’s trace formula: an introduction,Proceedings of the International School “Quantum Chaos on Hyperbolic Manifolds” (SchlossReisensburg, Gunzburg, Germany, 4-11 October 2003).
[Mar06]: J. Marklof,
Arithmetic quantum chaos,Encyclopedia of Mathematical Physics, editors J.-P. Francoise, G.L. Naber and Tsou S.T.Oxford, Elsevier, 2006, Volume 1, pp. 212–220.
[Ma03]: D. Mayer,
Transfer operators, the Selberg-zeta function and Lewis-Zagier theory of period functions,Lecture notes of a course given in Gunzburg, Germany, 4-11 October 2003.
[MM]: D. Mayer, T. Muhlenbruch,
Nearest λq-multiple fractions.
[MMS]: D. Mayer, T. Muhlenbruch, F. Stromberg,
The transfer operator for the Hecke triangle groups.
[MS08]: D. Mayer and F. Stromberg,
Symbolic dynamics for the geodesic flow on Hecke surfaces,Journal of Modern Dynamics 2 (2008), 581–627.
References Abstract Additional things
References
[Na95]: H. Nakada,
Continued fractions, geodesic flows and Ford circles,in Algorithms, Fractals, and Dynamics, Edited by T. Takahashi, Plenum Press, New York,1995.
[Ro54]: D. Rosen,
A class of continued fractions associated with certain properly discontinuous groups,Duke Mathematical Journal 21 (1954), 549–563.
[Ru78]: D. Ruelle,
Thermodynamic formalism,2nd edition, Cambridge University Press, 2004.
[Ru02]: D. Ruelle,
Dynamical zeta functions and transfer operators,Notices Amer. Math. Soc. 49 (2002), no. 8, 887–895.
[SS95]: T.A. Schmidt and M. Sheingorn,
Length spectra of the Hecke triangle groups,Mathematische Zeitschrift 220 (1995), 369–397.
[Str]: F. Stromberg,
Computation of Selberg Zeta Functions on Hecke Triangle Groups,Preprint.
References Abstract Additional things
Abstract of the talk
Classical mechanics and Maass cusp forms
I present the Nakada continuous fraction expansion which is of the form
λa0 +−1
λa1 + −1λa2+ −1
...
, λ = 2 cos
(π
q
), q = 3, 4, 5, . . . .
The coefficients ai are non-zero integers, satisfying certain conditions.We present some properties of these continued fractions and show thatthey can be used to code closed geodesics on the surface H/Gq, H beingthe upper complex plane and Gq the qth Hecke triangle group.As an application we demonstrate the construction of a transfer operator.It turns out this transfer operator can be used to express the Selberg zetafunction associated to H/Gq.
References Abstract Additional things
Maass cusp forms
Maass cusp form u
u : H→ C real-analytic function,
∆u = s(1− s) u with ∆ = −y 2(∂2
x + ∂2y
),
u(M z) = u(z) for all M ∈ Gq,
u(x + iy) = O(yC)
as y →∞ for all C ∈ R.
∆ admits self-adjoined extension in L2(Gq\H
).
s is called spectral parameter.
s(1− s) ≤ 14 , i.e., s ∈ 1
2 + iR?.
Discrete spectrum.(For G3 multiplicity 1 conjectured.)
Precise locations of eigenvalues are “not known”.
u(x + iy) =√y∑
n∈Z 6=0
an Ks− 12(2π |n| y) e
2πinxλq
Maass cusp form at
s = 12 + i13.78 . . . for G3.
Maass cusp form at
s = 12 + i9.533 . . . for G3.
References Abstract Additional things
Maass cusp forms
L2(Gq\H
)are realized by functions h satisfying
h : H→ C measurable,
h(M z) = f (z) for all M ∈ Gq and∫F |h(x + iy)|2 y−2dxdy <∞.
(s = 12 + i12.173 . . ., G3)
Some properties of ∆
The Laplace-Beltrami operator on L2(M) is the self-adjoined extension of ∆.
∆ can be viewed as a time-independent Schrodinger operator.
Spectrum of ∆ is discrete.
Eigenvalues λ = s(1− s) obey Weyl’s law: ]λ ≤ Λ ∼ Area(H)4π
Λ.
Conjectures about Eigenvalue statistics.
Quantum unique ergodicity.
Arithmetic properties for G3, G4 and G6:Hecke operators, associated L-series satisfy a GRH.
Model system
References Abstract Additional things
Hurwitz continued fractions
Hurwitz continued fractions (HCF)
We identify a sequence of integers, a0 ∈ Z, and a1, a2, . . . ∈ Z? with a point
x = T a0 ST a1 ST a2 · · · 0 = a0 +−1
a1 + −1
a2+−1...
=: [a0; a1, a2, . . .]
and say that it is a
formal CF, [a0; a1, a2, . . .] in general.
regular CF, [a0; a1, a2, . . .], if it does not contain “forbidden blocks”:no ±1 appear and if ai = ±2 then ai+1 ≶ 0.
π = [3;−7, 16, 294, 3, 4, 5, 15, . . .] and e = [3; 4, 2,−5,−2, 7, 2,−9, . . .]
Equivalent points
x and y are equivalent :⇔ there exists a g ∈ PSL2(Z) such that gx = y ⇔the CF of x and y have the same tail or
the CF of x and y have tail [ 3 ] and [−3 ].
References Abstract Additional things
Associated dynamical system
The generating map f
(x) ∈ Z denotes the nearest integer of x , i.e., |x − (x)| ≤ 12.
For I =[− 1
2, 1
2
]the generating map for the CF of x is
f : I → I ; x 7→ −1
x−(−1
x
).
The coefficients a0, a1, . . . computed by
a0 = (x) and xn+1 = f (xn) = −1xn− an+1
satisfy x = [a0; a1, . . .] and the CF is regular.
Natural extension of fThe natural extension of f is
Ω→ Ω; (x , y) 7→(f (x),
−1
y + a1
)with x = [0; a1, . . .].
Reference: [Hu89] Nearest λ-multiple continued fractions
References Abstract Additional things
The Ising model and transfer matrices
Ernst Ising (1900 – 1998) discussed an 1d lattice spin model.
Ising-Model
Config. space S = ±1N; left-shift τ : S → S , (τξ)i := ξi+1.Total energy: E = −J
∑∞i=1 ξiξi+1 + B
∑∞i=1 ξi
with spin interaction J and magnetic field interaction B.)Ernst Ising ≈ 1925.
Partition function Zm(A, s)
Zm(A, s) =∑ξ∈S ;
m−periodic
e−s∑m−1
k=0A(τkξ) with s = 1
Temp.and A(ξ) = J ξ0ξ1 + Bξ0.
free energy = limm→∞m−1 logZm(A, s).
Ising rewrote Zm as
Zm(A, s) =∑
ξ1,...,ξm∈±1
e(ξ1, ξ2) · · · e(ξm, ξ1) with e(ξi , ξj ) = e−s(Jξiξj−Bξi ).
Transfer matrix Ls
Ls :=
(e(+1,+1) e(+1,−1)e(−1,+1) e(−1,−1)
)satisfies Zm(A, s) = trace
(Lm
s
).
General form of a transfer operator
References Abstract Additional things
The dynamical zeta function and the the counting determinant
Recall the transfer operator LΦ(x) =∑
y∈f−1(x) g(y) h(y).
Define the counting trace tracec (L) =∑
x∈Fix(f ) g(x) and
the counting determinant detc(1− zL
)= exp
(−∑∞
m=1zm
m
(tracec (L)
)m).
Dynamical zeta function
The dynamical zeta function for a dynamical system (S , f ) is
ζ(z) = exp
∞∑m=1
zm
m
∑x∈Fix(f m)
m−1∑k=0
g(f k (x)
) .
ζ(z) =1
detc(1− yL
)Reference: [Ru02] Transfer operators
References Abstract Additional things
Spectrum of the transfer operator
Spectrum of the transfer operator Ls for Hecke triangle group G5 ands = 1
2 + iR, R ∈ [6, 14]:
Movie
Transfer operators
References Abstract Additional things
Period functions and the transfer operator
Period function P
P : C′ := Cr (−∞, 0]→ C holomorphic,
P(z) = P(z + 1) + (z + 1)−2sP(−1z+1
)(three-term equation) and
P(z)
|Im(z)|−C
(1 + |z |2C−2Re(s)
)if Re(z ≤ 0),
1 if Re(z) > 0, |z | ≤ 1 and
|z |−2Re(s) if Re(z) > 0, |z | ≥ 1.
The period function P depends implicitly on s.
Theorem ([LZ01])
For Re(s) > 0:
P is period function ⇐⇒ s is spectral parameter of a Maass cusp form.
References Abstract Additional things
Period functions and the transfer operator
Let ~h =(
h1
h2
)be an eigenfunction of Ls with eigenvalue 1.
∃g ∈ Cω(−r − 1, r), r = 1+√
52 , s.th. g = h1, g
∣∣2sT−1 = h2 on [−1, 1].
g satisfies the relation
g = g∣∣2s
∑∞n=3 ST
n + g∣∣2s
∑∞n=2 T
−1ST−n
and on (−r , r) the 4-term equation
g∣∣2s
(1 + ST 2
)= g
∣∣2s
(T−1 + T−1ST−2
).
Theorem ([BM09])
Assume q = 3, 0 < Re(s) < 1 and 2s 6= 1.Eigenfunctions of Ls with eigenvalue 1 correspond to (Lewis-Zagier)period functions.
Selberg zeta-function and transfer operator