morse theory on spaces of braids and lagrangian dynamicsjanbouwe/pub/braids.pdf · morse theory on...

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MORSE THEORY ON SPACES OF BRAIDS

AND LAGRANGIAN DYNAMICS

R.W. GHRIST, J.B. VAN DEN BERG, AND R.C. VANDERVORST

ABSTRACT. In the first half of the paper we construct a Morse-type theory on certain

spaces of braid diagrams. We define a topological invariant of closed positive braids

which is correlated with the existence of invariant sets of parabolic flows defined on

discretized braid spaces. Parabolic flows, a type of one-dimensional lattice dynam-

ics, evolve singular braid diagrams in such a way as to decrease their topological

complexity; algebraic lengths decrease monotonically. This topological invariant is

derived from a Morse-Conley homotopy index.

In the second half of the paper we apply this technology to second order La-

grangians via a discrete formulation of the variational problem. This culminates in

a very general forcing theorem for the existence of infinitely many braid classes of

closed orbits.

1. PRELUDE

It is well-known that under the evolution of any scalar (uniform) parabolic equa-tion of the form

(1) ��

the graphs of two solutions � �� and �� evolve in such a way that the numberof intersections of the graphs does not increase in time. This principle, known invarious circles as “lap number” techniques, entwines the geometry of the graphs(�� is a curvature term), the topology of the solutions (the intersection number is alocal linking number), and the local dynamics of the PDE. This is a valuable approachfor understanding local dynamics for a wide variety of flows exhibiting parabolicbehavior with both classical [52] and contemporary [6, 19, 39] implications.

This paper is an extension of this local technique to a global technique. Onesuch well-established globaliziation first appears in the work of Angenent on curve-shortening [5]: evolving closed curves on a surface by curve shortening isolates theimmersion classes dynamically and implies a monotonicity with respect to numberof self-intersections.

In contrast, one could consider the following topological globalization. Superim-posing the graphs of a collection of functions �� gives something which resemblesthe projection of a topological braid onto the plane. Assume that the “height” of thestrands above the page is given by the slope �� , or, equivalently, that all of thecrossings in the projection are of the same sign (bottom-over-top): see Fig. 1[left].

Date: June 20, 2001.The first author was supported by NSF DMS-9971629. The second author was supported by an EPSRC

Fellowship. The third author was supported by grants ARO DAAH-0493G0199 and NIST G-06-605.

1

2 R.W. GHRIST, J.B. VAN DEN BERG, AND R.C. VANDERVORST

Evolving these functions under a parabolic equation (with, say, boundary endpointsfixed) yields a flow on a certain space of braid diagrams which has a topologicalmonotonicity: linking can be destroyed but not created. This establishes a partial order-ing on the semigroup of positive braids which is respected by parabolic dynamics.The idea of topological braid classes with this partial ordering is a globalization ofthe lap number (which, in braid-theoretic terms becomes the length of the braid inthe braid group under standard generators).

1.1. Parabolic flows on spaces of braid diagrams. In this paper, we initiate thestudy of parabolic flows on spaces of braid diagrams. The particular braids in ques-

tion will be (a) positive – all crossings are considered to be of the same sign; (b) closed1

– the left and right sides are identified; and (c) discretized – or piecewise linear withfixed distance between “anchor points,” so as to avoid the analytic difficulties ofworking on infinite dimensional spaces of curves. See Fig. 1 for examples of braiddiagrams.

FIGURE 1. Examples of smooth [left] and discretized [right] positivebraids on four strands. In a discretized isotopy, one slides the anchorpoints vertically.

The flows we consider evolve the anchor points of the braid diagram in such away that the braid class can change, but only so as to decrease complexity: local link-ing of strands may not increase with time. Due to the close similarity with parabolicpartial differential equations such systems will be referred to as parabolic recurrence

relations,2 and the induced flows as parabolic flows. These flows are given by�� (2)

where the variables � � represent the vertical positions of the ordered anchor pointsof discrete braid diagrams. The only conditions imposed on the dynamics is themonotonicity condition that every � � be increasing functions of � �� and � �� .

While a discretization of a PDE of the form (1) with nearest-neighbor interactionyields a parabolic recurrence relation, the class of dynamics we consider is signifi-cantly larger in scope (see, e.g., [38]). Parabolic recurrence relations are a sub-class ofmonotone recurrence relations as studied in [4] and [24].

1The theory works equally well for braids with fixed endpoints.2To be precise we refer to the stationary version of (2) as a parabolic recurrence relation.

MORSE THEORY ON SPACES OF BRAIDS 3

The evolution of braid diagrams yields a situation not unlike that considered byVassiliev in knot theory [57]: in our scenario, the space of all braid diagrams is parti-tioned by the discriminant of singular diagrams into the braid classes. The parabolicflows we consider are transverse to these singular varieties and are co-oriented in adirection along which the algebraic size of the braid (length of the word in the braidgroup) decreases.

Two types of noncompactness on spaces of braid diagrams must be repaired. Mostsevere is the problem of braid strands completely collapsing onto one another (a tan-gency of “maximal order” which is not removed by parabolic dynamics). To resolvethis type of noncompactness, we assume that the dynamics fixes some collection ofbraid strands, a skeleton, and then work on spaces of braid pairs: one free, one fixed.The relative theory then leads to forcing results of the type “Given a stationary braidclass, which other braids are forced as invariant sets of parabolic flows?” The sec-ond type of noncompactness in the dynamics occurs when the braid strands are freeto evolve to arbitrarily large values. In the PDE setting, one requires knowledgeof boundary conditions at infinity to prove theorems about the dynamics. In ourbraid-theoretic context, we convert boundary conditions to “artificial” braid strandsaugmented to the fixed skeleton.

Thus, working on spaces of braid pairs, the dynamics at the discriminant per-mits the construction of a Morse theory in the spirit of Conley to detect invariantsets of parabolic flows. Recall that, in analogy to the Morse index of a fixed pointin a gradient field (the dimension of the unstable manifold of the fixed point underthe flow), Conley’s extension of the Morse index associates to any sufficiently iso-lated invariant set a space whose homotopy type measures not only the dimensionsbut the coarse topological features of the unstable dynamics associated to this set.We obtain a well-defined Conley index for braid diagrams from the monotonicityproperties of parabolic flows. To be more precise, relative braid classes (equivalenceclasses of isotopic braid diagrams fixing some skeleton) serve as candidates for iso-lating neighborhoods to which the Conley index can be assigned. This approach isreminiscent of the ideas of linking of periodic orbits used by Angenent [3, 5] andLeCalvez [34, 35].

Our finite-dimensional approximations to the (infinite-dimensional) space ofsmooth topological braids conceivably alter the Morse-theoretic properties of thediscretized braid classes. One would like to know that so long as the discretizationis not degenerately coarse, the homotopy index is independent of both the discretiza-tion and the specific parabolic flow employed. This is true. The principal topologicalresult of this work is that the homotopy index is indeed an invariant of the topolog-ical (relative) braid class: see Theorems 19 and 20 for details. These theorems seemto evade a simple algebraic-topological proof. The proof we employ in �5 constructsthe appropriate homotopy by recasting the problem into singular dynamics and ap-plying techniques from singular perturbation theory.

In analogy with classical Morse theory, where certain homology classes of a mani-fold are in strong correlation with invariant sets of nondegenerate gradient flows onthe manifold, one has the relationship:


Homotopy index invariant for braids�Existence of invariant sets of parabolic flows

The remainder of the paper explores applications of the machinery to a broad classof Lagrangian dynamics.

1.2. Second order Lagrangian dynamics. Our principal application of the Morsetheory on discretized braids is to the problem of finding periodic orbits of secondorder Lagrangian systems: that is, Lagrangians of the form � �� where� � � � �� . An important motivation for studying such systems comes from thestationary Swift-Hohenberg model in physics, which is described by the fourth orderequation ��

�� (3)

This equation is the Euler-Lagrange equation of the second order Lagrangian� ��

Our applications of the machinery to this equation generalizes to extremely broadclasses of second-order Lagrangians. One begins with the conventional convexityassumption, � �� . The objective is to find bounded functions� � � � � which are stationary for the action integral � �� .Such functions � are bounded solutions of the Euler-Lagrange equations

(4)

��

��

� �� Due to the translation invariance � ��

, the solutions of (4) satisfy the energyconstraint

(5) ! ��

�� "�� # � constant �

where # is the energy of a solution. To find bounded solutions for given values of # ,

we employ the variational principle �$ %& � &' (� �� # )�� , which forcessolutions of (4) to have energy # . The Lagrangian problem can be reformulatedas a two degree-of-freedom Hamiltonian system; in that context, bounded periodicsolutions are closed characteristics of the (corresponding) energy manifold * � + �� .Unlike the case of first-order Lagrangian systems, the energy hypersurface is not ofcontact type in general [7], and the recent stunning results in contact homology areinapplicable.

The variational principle can be discretized for a certain considerable class of sec-ond order Lagrangians: those for which monotone laps between consecutive ex-trema ,� � - are sufficiently unique. We give a precise definition in �8, denoting theseas (second order Lagrangian) twist systems. Due to the energy identity (5) the extrema,� � - are restricted to the set ./ � ,� � � �� # � �-, connected componentsof which are called interval components and denoted by 0/ . An energy level is called


regular if ��$ �� for all � satisfying � �� # � �. In order to deal with

non-compact interval components 0/ certain asymptotic behavior has to be speci-fied, for example that “infinity” is attracting. Such Lagrangians are called dissipative,and are most common in models coming from physics, like the Swift-HohenbergLagrangian. For a precise definition of dissipativity see �9. Other asymptotic behav-iors may be considered as well, such as “infinity” is repelling, or more generally thatinfinity is isolating, implying that closed characteristics are a priori bounded in �� .

Closed characteristics are either simple or non-simple depending on whether � ��,represented as a closed curve in the �� -plane, is a simple closed curve or not.This distinction is a sufficient language for the following general forcing theorem:

Theorem 1. Any dissipative twist system possessing a non-simple closed characteristic � �� at a regular energy value # such that � �� 0/ , must possess an infinite number of (non-isotopic) closed characteristics at the same energy level as � �� .

This is the optimal type of forcing result: there are neither hidden assumptionsabout nondegeneracy of the orbits, nor restrictions to generic behavior. Sharpnesscomes from the fact that there exist systems with finitely many simple closed char-acteristics at each energy level.

The above result raises the following question: when does an energy manifoldcontain a non-simple closed characteristic? In general the existence of such charac-teristics depends on the geometry of the energy manifold. One geometric propertythat sparks the existence of non-simple closed characteristics is a singularity or near-singularity of the energy manifold. This, coupled with Theorem 1, triggers the exis-tence of infinitely many closed characteristics. The results that can be proved in thiscontext (dissipative twist systems) give a complete classification with respect to theexistence of finitely many versus infinitely many closed characteristics on singularenergy levels. The first result in this direction deals with singular energy values forwhich 0/ � � .

Theorem 2. Suppose that a dissipative twist system has a singular energy level # with0/ � �, which contains two or more rest points. Then the system has infinitely many closed

characteristics at energy level # .3

Complementary to the above situation is the case when 0/ contains exactly onerest point. To have infinitely many closed characteristics, the nature of the rest pointwill come into play. If the rest point is a center (four imaginary eigenvalues), then thesystem has infinitely many closed characteristics at each energy level sufficiently close to # ,including # . If the rest point is not a center there need not exist infinitely many closedcharacteristics as results in [55] indicate.

Similar results can be proved for compact interval components (for which dissi-pativity is irrelevent) and semi-infinite interval components 0/ � �� .

3From the proof of this theorem in �9 it follows that the statement remains true for energy values � � ,with � small.


Theorem 3. Suppose that a dissipative twist system has a singular energy level # with aninterval component 0/ � �� , or 0/ � �� , which contains at least one rest point of saddle-focus/center type. Then the system has infinitely many closed characteristics at energy level# .

If an interval component contains no rest points, or only degenerate rest points (0eigenvalues), then there need not exist infinitely many closed characteristics, com-pleting our classification.

This classification immediately applies to the Swift-Hohenberg model (3), whichis a twist system for all parameter values � � � . We leave it to the reader to applythe above theorems to the different regimes of � .

1.3. Additional applications. The framework of parabolic recurrence relations thatwe construct is robust enough to accommodate several other important classes ofdynamics.

1.3.1. First-order nonautonomous Lagrangians. Finding periodic solutions of first-order Lagrangian systems of the form � � � �� , with � being 1-periodicin � , can be rephrased in terms of parabolic recurrence relations of gradient type.The homotopy index can be used to find periodic solutions � �� in this setting, eventhough a globally defined Poincare map on � �

need not exist.

1.3.2. Monotone twist maps. A monotone twist map (compare [4, 46]) is a (not neces-sarily area-preserving) map on � �

of the form

�� $ � � �� $ � � �� $ � � Periodic orbits , �� $ � �- are found by solving a parabolic recurrence relation for the�-coordinates derived from the twist property.

1.3.3. Uniform parabolic PDE’s. The study of the invariant dynamics of Equation (1)can also be formulated in terms of parabolic recurrence relations. The results aboutthe braid invariants that link the invariants of discrete braids to invariants of topo-logical braids allows one to develop a Morse-type theory for Equation (1) via discretebraid invariants. For this application of the theory we refer to [2].

1.3.4. Lattice dynamics. The form of a parabolic recurrence relation is precisely thatarising from a set of coupled oscillators on a [periodic] one-dimensional lattice withnearest-neighbor attractive coupling. A similar setup arises in Aubry-LeDaeron-Mather theory of the Frenkel-Kontorova model [8]. In this setting, a nontrivial ho-motopy index yields existence of invariant states (or stationary, in the exact context)within a particular braid class. Related physical systems (e.g., charge density waveson a 1-d lattice [43]) are also often reducible to parabolic recurrence relations.


1.4. Brief outline. The history of our approach is the convergence of ideas fromknot theory, the dynamics of annulus twist maps, and curve shortening. We have al-ready mentioned the similarities with Vassiliev’s topological approach to the spaceof immersed knots. From the dynamical systems perspective, the study of parabolicflows and gradient flows in relation with embedding data and the Conley indexcan be found in work of Angenent [3, 4, 5] and Le Calvez [34, 35] on area perse-vering twist maps. More general studies of dynamical properties of parabolic-typeflows appear in numerous works: we have been inspired by the work of Smillie [51],Mallet-Paret and Smith [37], Hirsch [24], and, most strongly, the work of Angenenton curve shortening [5]. Many of our applications to finding closed characteristics ofsecond order Lagrangian systems share similar goals with the programme of Hoferand his collaborators (see, e.g., [16, 25, 26]), with the novelty that our energy surfacesare all non-compact and not necessarily of contact type [7].

Clearly there is a parallel between the homotopy index theory presented here andthe Nielsen-Thurston theory for braid types of surface diffeomorphisms [10]. An im-portant difference is that we require compactness only at the level of braid diagrams,which does not yield compactness on the level of the domains of the return maps [ifthese indeed exist]. Another important observation is that the recurrence relationsare sometimes not defined on all of � �

, which makes it very hard if not impossi-ble to rephrase the problem of finding periodic solutions in terms of fixed points of2-dimensional maps.

There are three components of this paper: (a) the precise definitions of the spacesinvolved and flows constructed, covered in �2-�3; (b) the establishment of existence,invariance, and properties of the index for braid diagrams in �4-�7; and (c) appli-cations of the machinery to second order Lagrangian systems �8-�10. Finally, �11contains open questions and remarks.

Acknowledgements. The authors would like express special gratitude to SigurdAngenent and Konstantin Mischaikow for numerous enlightening discussions. Alsospecial thanks to Madjid Allili for his computational work in the earliest stages ofthis work.

CONTENTS

1. Prelude 12. Spaces of discretized braid diagrams 73. Parabolic recurrence relations 114. The homotopy index for discretized braids 145. Stabilization and invariance 206. Duality 277. Morse theory 298. Second order Lagrangian systems 329. Multiplicity of closed characteristics 3610. Computation of the homotopy index 4611. Postlude 49


References 52Appendix A. Construction of parabolic flows 54

2. SPACES OF DISCRETIZED BRAID DIAGRAMS

2.1. Definitions. Recall the definition of a braid (see [9, 23] for a comprehensiveintroduction). A [geometric] braid � on � strands is a collection of embeddings ,� � �� -�� with disjoint images such that (a) � � �� ; (b) � � �� for some permutation � ; and (c) �

� � � � �� for all � � �� . The lastcondition implies that the braid is to be “read” from left to right. Two such braidsare said to be of the same topological braid class if they are homotopic in the space ofbraids: one can deform one braid to the other without any intersections among thestrands. There is a natural group structure on the space of topological braids with �strands, �� , given by concatenation. Using generators � � which interchange the �and � � �� strands (with a positive crossing) yields the presentation for �� :

(6) �� Braids find their greatest applications in knot theory via taking their closures.

Algebraically, the closed braids on � strands can be defined as the set of conjugacy

classes4 in �� . Geometrically, one quotients out the range of the braid embeddingsvia the equivalence relation �� and alters the restriction (1) and (2)of the position of the endpoints to be � � �� . Thus, a closed braid is a

collection of disjoint embedded loops in � �� which are everywhere transverse to

the � �-planes.

The specification of a topological braid class (closed or otherwise) may be accom-plished unambiguously by a labeled projection to the �� -plane: a braid diagram.Any braid may be perturbed slightly so that pairs of strand crossings in the projec-tion are transversal: in this case, a marking of �� or �� serves to indicate whetherthe crossing is “bottom over top” or “top over bottom” respectively. Fig. 1[left] illus-trates a topological braid with all crossings positive.

2.2. Discretized braids. In the sequel we will restrict to a class of closed braid dia-grams which have two special properties: (a) they are positive — that is, all crossingsare of �� type; and (b) they are discretized, or piecewise linear diagrams with con-straints on the positions of anchor points. We parameterize such diagrams by theconfiguration space of anchor points.

Definition 4. The space of discretized period�

braids on � strands, denoted � �� , is thespace of all pairs �� where � � �� is a permutation on � elements, and � is an unorderedcollection of � strands, � � ,�� -�� , satisfying the following conditions:

(a) Each strand consists of� � �

anchor points: �� ' � �� .

4Note that we fix the number of strands and do not allow the Markov move commonly used in knot

theory.


(b) For all � � � � � � , one has

�� ' (c) The following transversality condition is satisfied: for any pair of distinct strands� and � � such that �� for some ,

(7) (�� ) (�� ) � � The topology on � �� is the discrete topology with respect to the permutation � and the stan-dard topology of ��

on the strands. Specifically, two discretized braids �� and � �� are close iff for some permutation � � �� one has �� close to �� (as points in ��

) forall �, with � � �� .

Remark 5. In Equation (7), and indeed throughout the paper, all expressions involv-ing coordinates � � are considered mod the permutation � at

�; thus, for every � � �

, werecursively define

(8) �� As a point of notation, subscripts always refer to the spatial discretization and su-perscripts always denote strands. For simplicity, we will henceforth suppress the �portion of a discretized braid � . We also often suppress the indexing in our notationand write � or � � to denote the space of discretized braids.

One associates to each configuration � � � �� the braid diagram � �� , given as fol-lows. For each strand �� , consider the piecewise-linear (PL) interpolation

(9) � � �� for � � �� . The braid diagram � �� is then defined to be the superimposed graphsof all the functions � � , as illustrated in Fig. 1[right] for a period six braid on fourstrands (crossings are shown merely for suggestive purposes).

This explains the transversality condition of Equation (7): a failure of this equationto hold implies that there is a PL-tangency in the associated braid diagram. Since allcrossings in a discretized braid diagram are PL-transverse, the map � �� sends � toa topological closed braid diagram once a convention for crossings is chosen. Forreasons explained in the Prelude, we label all crossings of � �� as positive type.This can be thought of as using the slope of the PL-extension of � as the “height”of the braid strand (though this analogy breaks down at the sharp corners). Withthis convention, then, the space � � embeds into the space of all closed positive braiddiagrams on � strands.

Definition 6. Two discretized braids � � � � � � �� are of the same discretized braid class,denoted �� , if and only if they are in the same path-component of � �� . The topolog-ical braid class, ,� -, denotes the path component of � �� in the space of topological braiddiagrams.

The proof of the following lemma is essentially obvious.

Lemma 7. If �� in � �� , then the induced positive braid diagrams � and � � correspondto isotopic closed topological braids.


The converse to this Lemma is not true: two discretizations of a topological braidare not necessarily connected in � �� .

Since one can write the generators � � of the braid group �� as elements of � � , itis clear that all positive topological braids are captured by elements of � . Likewise,the relations for the groups of positive closed braids can be accomplished by movingwithin the space � ; hence, this setting suffices to capture all the relevant braid theorywe will use.

2.3. Singular braids. The appropriate discriminant for completing the space � ��consists of those “singular” braid diagrams admitting tangencies between strands.

Definition 8. Denote by �� the space of all discretized braid diagrams � which satisfy prop-erties (a) and (b) of Definition 4. Denote by

�� the set of singular discretizedbraids.

We will often suppress the period and strand data and write�

for the space ofsingular discretized braids. It follows from Definition 4 and Equation (7) that the set�� is a semi-algebraic variety in �� . Specifically, for any singular braid � � �

there

exists an integer � ,� � � � - and indices � �� such that �� , and

(10) (�� ) (�� ) � � where the subscript is always computed mod the permutation � at

�. The number

of such distinct occurrences is the codimension of the singular braid diagram � � �.

We decompose�

into the union of strata� �� graded by � , the codimension of the

singularity.Any closed braid (discretized or topological) is partitioned into components by

the permutation � . Geometrically, the components are precisely the connected com-ponents of the closed braid diagram. In our context, a component of a discretizedbraid can be specified as ,�� -��, since, by our indexing convention, “wrapsaround” to the other side of the braid when �� ,� � � � -.

For singular braid diagrams of sufficiently high codimension, entire componentsof the braid diagram can coalesce. This can happen in essentially two ways: (1)a single component involving multiple strands can collapse into a braid with fewernumbers of strands, or (2) distinct components can coalesce into a single component.We define the collapsed singularities,

��, as follows:

�� ,� � � � �� - + � Clearly the codimension of singularities in

��is at least

�. Since for braid diagrams

in��

the number of strands reduces, the subspace��

may be decomposed into a

union of the spaces �� for � � � � ; i.e.,�� . If � � �

, then�� .

2.4. Relative braid classes. Evolving certain components of a braid diagram whilefixing the remaining components motivates working with a class of “relative” braiddiagrams.

Given � � �� and � � �� , the union � � � � �� is naturally defined as theunordered union of the strands. Given � � �� , define� �� REL � �� ,� � � � � � � �� - � � ��


fixing � and imposing transversality. The path components of � �� REL � , comprisethe relative discrete braid classes, denoted �� REL � �, which give a partitioning of � �� .The braid � is called the skeleton in this setting. The set of singular braids

�REL �

are those singular braids in� + �� of the form � � � , fixing � . The associated

collapsed singular braids are denoted by��

REL � . As before, the set �� REL � � ��

REL � � is the closure of � �� REL � , and is denoted �� REL � . By ,� REL � - weindicate the topological relative braid class (for some � � ,� - fixed).

Two relative braid classes �� REL � � and �� REL � � � in � �� REL � and � �� REL � �

respectively, are called equivalent, notation: �� REL �� REL �� , if �� and �� , and thus �� REL �� denotes the set of all equivalent braid

classes �� REL � �. By �� REL ,� -� we denote the set of all equivalent topologicalrelative braid classes ,� REL � -.

3. PARABOLIC RECURRENCE RELATIONS

We consider the dynamics of vector fields given by recurrence relations on thespaces of discretized braid diagrams. These recurrence relations are nearest neighborinteractions — each anchor point on a braid strand influences anchor points to theimmediate left and right on that strand — and resemble spatial discretizations ofparabolic equations.

3.1. Axioms and exactness. Denote by � the sequence space � �� .

Definition 9. A parabolic recurrence relation � on � is a sequence of real-valued �

functions � � �� satisfying

(A1): [monotonicity]5 � � � � � and �� for all � �(A2): [periodicity] For some

� � � , � �� for all � �.

For applications to Lagrangian dynamics a variational structure is necessary. Atthe level of recurrence relations this implies that � is a gradient:

Definition 10. A parabolic recurrence relation on � is called exact if

(A3): [exactness] There exists a sequence of � �generating functions, �� , satisfy-

ing

(11) � � ��

for all � �.

In discretized Lagrangian problems the action functional naturally defines thegenerating functions � �. This agrees with the “formal” action in this case: � �� . In this general setting, � � �� .

3.2. The induced flow. In order to define parabolic flows we regard � as a vectorfield on � : consider the differential equations

�� (12)

5Equivalently, one could impose � � � and �� for all �.


Equation (12) defines a (local) � flow �

�on � (see [3] for details in the aperiodic

case� � � ). To define flows on the finite dimensional spaces �� , one considers the

same equations:

(13)

��

where the ends of the braid are identified as per Remark 5. Axiom (A2) guaranteesthat the flow is well-defined. Indeed, one may consider a cover of �� by takingthe bi-infinite periodic extension of the braids: this yields a subspace of periodicsequences in � � �� invariant under the product flow of (12) thanks toAxiom (A2).

Any flow ��

generated by (13) for some parabolic recurrence relation � is calleda parabolic flow on discretized braids.

3.3. Monotonicity and braid diagrams. The monotonicity Axiom (A1) in the previ-ous subsection has a very clean interpretation in the space of braid diagrams. Recallfrom �2 that any discretized braid � has an associated diagram � �� which can beinterpreted as a positive closed braid. Any such diagram in general position can be

expressed in terms of the (positive) generators ,�� -�� of the braid group �� . Whilethis word is not necessarily unique, the length of the word is, as one can easily seefrom the presentation of �� and the definition of � �� . The length of a closed braid inthe generators �� is thus precisely the word metric �� word from geometric group theory.The geometric interpretation of �� word for a braid � is clearly the number of pairwisestrand crossings in the diagram � �� .

The primary result of this section is that the word metric acts as a discrete Lya-punov function for any parabolic flow on �� . This is really the braid-theoretic ver-sion of the lap number arguments that have been used in several related settings[3, 5, 6, 19, 22, 37, 39, 51]. The result we prove below can be excavated from thesecited works; however, we choose to give a brief self-contained proof for complete-ness.

Proposition 11. Let ��

be a parabolic flow on �� .

(a) For each point � � � ��, the local orbit ,� � �� - intersects

�

uniquely at � for all � sufficiently small.(b) For any such �, the length of the braid diagram �

� �� for � � in the word metricis strictly less than that of the diagram �

� �� , � �.

Proof. Choose a point � in�

representing a singular braid diagram. We induct onthe codimension � of the singularity. In the case where � � � �� (i.e., � � �

), there

exists a unique and a unique pair of strands � �� such that �� and

�� Note that the inequality is strict since � � �

. We deduce from (13) that��

��' � � � ��


From Axiom (A2) one has that

SIGN !� � �� " �SIGN

�� Therefore, as � �, the two strands have two local crossings, and as � �� , thesetwo strands are locally unlinked (see Fig. 2): the length of the braid word in the wordmetric is thus decreased by two, and the flow is transverse to

� ��.

�

��

� � � � � � �

FIGURE 2. A parabolic flow on a discretized braid class is transverseto the boundary faces. The local linking of strands decreases strictlyalong the flowlines at a singular braid �.

Assume inductively that (a) is true for every point in� �� for � � * . Choose� � � �* �. There are thus exactly * distinct pairs of anchor points of the braid

which coalesce at the braid diagram �. Since the vector field � is defined by nearestneighbors, singularities which are not strandwise consecutive in the braid behaveindependently to first order under the parabolic flow. Thus, it suffices to assume

that for some , �, and � � one has ,�� -� � � �' and ,�� -� � � �' chains of consecutive

anchor points for the braid diagram � such that �� if and only if� � * .

(Recall that the addition � � is always done modulo the permutation � at�). Then

since�� ' � � ��

it follows that for all � � � � � �* ��, the anchor points �� and �� are notseparated to first order. At the left “end” of the singular braid, where � � �,

� � �� so that the vector field � is tangent to

�at � but is not tangent to

� �* �: the flow-line through � decreases codimension immediately. By the induction hypothesis, theflowline through � cannot possess intersections with

� �� for � � * which accu-mulate onto �; thus the flowline intersects

�locally at � uniquely. This concludes

the proof of (a).


It remains to show that the length of the braid word decreases strictly at � in�. The previous analysis shows that (b) is true for � � � ��. Choose � � �

��not necessarily in

� ��. By (a), the flow ��

is nonsingular in a neighborhoodof � ; thus, by the Flowbox Theorem, there is a tubular neighborhood of local �

�-

flowlines about �� . The beginning and ending points of these local flowlines all

represent nonsingular diagrams with the same word lengths as the beginning andendpoints of the path through � , since the complement of

�is an open set. Since

�

is a codimension-�

algebraic semi-variety in �� , it follows from transversality thatmost of the nearby orbits which through

�, pass only through

� ��, along whichbraid word length strictly decreases.

To put this result in context with the literature, we note that the monotonicity in[22, 37] is one-sided: translated into our terminology, �� for all . One canadapt this proof to generalizations of parabolic recurrence relations appearing in thework of Le Calvez [34, 35]: namely, compositions of twist symplectomorphisms ofthe annulus reversing the twist-orientation (see �11).

A parabolic flow on �� REL � is a special case of a parabolic flows on �� with afixed skeleton � � �� , and the analogue of the above proposition for relative classesthus follows as a special case.

FIGURE 3. Two relative braids with the same linking data but dif-ferent homotopy indices. The free strands are in grey.

Remark 12. The information that we derive from relative braid diagrams is morethan what one can obtain from lap numbers alone (cf. [34]). Fig. 3 gives examplesof two closed discretized relative braids which have the same set of pairwise linkingnumbers of strands (or lap numbers) but which force very different dynamical be-haviors. The homotopy invariant we define in the next section distinguishes thesebraids. The index of the first picture can be computed to be trivial, and the index forthe second picture is computed in �10 to be nontrivial.

4. THE HOMOTOPY INDEX FOR DISCRETIZED BRAIDS

Technical lemmas concerning existence of certain types of parabolic flows are re-quired for showing the existence and well-definedness of the Conley index on braidclasses. We relegate these results to Appendix A.


4.1. Review of the Conley index. We include a brief primer of the relevant ideasfrom Conley’s index theory for flows. For a more comprehensive treatment, we referthe interested reader to [45].

In brief, the Conley index is an extension of the Morse index. Consider the caseof a nondegenerate gradient flow: the Morse index of a fixed point is then the di-mension of the unstable manifold to the fixed point. In contrast, the Conley indexis the homotopy type of a certain pointed space (in this case, the sphere of dimen-sion equal to the Morse index). The Conley index can be defined for sufficiently“isolated” invariant sets in any flow, not merely for fixed points of gradients.

Recall the notion of an isolating neighborhood as introduced by Conley [11]. Acompact set � is an isolating neighborhood for a flow �

�if the maximal invariant set

INV�� ,� � � � ��,� � �� -�� + � - is contained in the interior of � . The

invariant set INV�� is then called a compact isolated invariant set for �

�. In [11] it is

shown that every compact isolated invariant set INV�� admits a pair �� such

that (following the definitions given in [45]) (i) INV�� INV

�� with� � �

a neighborhood of INV�� ; (ii) � �

is positively invariant in � ; and (iii) � �

is an exit set for � : given � � � and � � such that �� , then there exists

a ' � �� for which ,� � �� ' �- + � and ��

. Such a pairis called an index pair for INV

�� . The Conley index, � �� , is then defined as thehomotopy type of the pointed space (� �� ), which is abbreviated �� .This homotopy class is independent of the defining index pair, making the Conleyindex well-defined.

A large body of results and applications of the Conley index theory exists. Werecall two foundational results which will be of repeated use in the remainder of thispaper.

(a) Stability of isolating neighborhoods: Any isolating neighborhood � for aflow �

�is an isolating neighborhood for all flows sufficiently � '

-close to ��.

(b) Continuation of the Conley index: If �� , � �� is a continuous family

of flows with �� a continuous family of isolating neighborhoods, then theindex �� is invariant under .

Since the homotopy type of a space is notoriously difficult to compute, one often

passes to homology or cohomology. One defines the Conley homology6 of INV��

to be � � �� , where � is singular homology. To the homologicalConley index of an index pair �� one can also assign the characteristic poly-nomial � � � �� ' � , where � is the free rank of � �� . Note that, in

analogy with Morse homology, if � � �� , then there exists a nontrivial invari-ant set within the interior of � . For more detailed description see �7.

4.2. Proper and bounded braid classes. From Proposition 11, one readily sees thatcomplements of

�yield isolating neighborhoods, except for the presence of the col-

lapsed singular braids��

, which is an invariant set in�

. For the remainder of thispaper we restrict our attention to those relative braid diagrams whose braid classesprohibit collapse.

6In [14] Cech cohomology is used. For our purposes ordinary singular (co)homology always suffices.


Fix � � � �� , and consider the relative braid classes ,� REL � - (topological) and�� REL � � (discretized).

Definition 13. A topological relative braid class ,� REL � - is proper if it is impossibleto find a continuous path of braid diagrams � ��

REL � for � �� such that � �� ,� ��REL � defines a braid for all � �� , and � ��

REL � is a diagram where an entirecomponent of the braid has collapsed onto itself or onto another component. A discretizedrelative braid class �� REL � � is called proper if the associated topological braid class isproper.

A braid class is called improper if it is not proper. Examples appear in Fig. 4.

Definition 14. A topological relative braid class ,� REL � - is called bounded if there existsa uniform bound on all the representatives � of the equivalence class, i.e. on the strands � �� (in � ' �� ). A discrete relative braid class �� REL � � is called bounded if the set �� REL � �is bounded.

Note that if a topological class ,� REL � - is bounded then the discrete class�� REL � � is bounded as well for any period. The converse does not always hold.The bounded braid classes have the compactness necessary to implement the Con-ley index theory without further assumptions.

��

��

��

��

��

��

��

��

�

�

�

�

��

�

�

�

�

FIGURE 4. Improper [left] and proper [right] relative braid classes.Both are bounded.

4.3. Existence and invariance of the Conley index for braids. Bounded properbraid classes yield a well-defined Conley index.

Theorem 15. Suppose �� REL � � is a bounded proper relative braid class and ��

is a para-bolic flow fixing � . Then the following are true:

(a) � �� REL � � is an isolating neighborhood for the flow ��, which thus yields a

well-defined Conley index � ��REL � � �� ;

(b) The index � ��REL � � is independent of the choice of parabolic flow �

�so long as

�� ;

(c) The index � ��REL � � is independent of the choice of � within its discretized braid

class �� , so long as it is kept constant.

Definition 16. The homotopy index of a bounded proper discretized braid class�� REL �� in � �� REL �� is defined to be � ��

REL � �, the Conley index of the braid class

MORSE THEORY ON SPACES OF BRAIDS 17�� REL � � with respect to some (hence any) parabolic flow fixing any representative � of theskeletal braid class �� .

Proof. Isolation is proved by examining ��

on the boundary �� . By Definition13 the set � is compact, and �� + ��

. Choose a point � on �� . Proposition11 implies that the parabolic flow �

�locally intersects �� at � alone and that fur-

thermore its length in the braid group strictly decreases. This implies that under ��,

the point � exits the set � in either forwards or backwards time (if not both). Thus,� �� INV�� and (a) is proved.

Denote by � ��REL � � the index of INV

�� . To demonstrate (b), consider two par-abolic flows �

�' and �� that satisfy all our requirements, and consider the isolating

neighborhood � valid for both flows. Construct a homotopy �� , � �� , by con-

sidering the parabolic recurrence functions � � � �� ' � � , where � '

and �

give rise to the flows ��' and �

� respectively. It follows immediately that �� ,

for all � �� ; therefore � is an isolating neighborhood for �� with � �� .

Define INV� �� , � �� , to be the maximal invariant set in � with respect to theflow �

�� . The continuation property of the Conley index completes the proof of (b).Let � � � � �� be in the same path component as � , i.e. �� , and define � � �

�� REL � � �. By the previous arguments � ��REL � � � is well-defined. To complete the

proof of the theorem we show that � ��REL � � � � ��

REL � � �.We know from Lemma 57 in Appendix A that there exists a homotopy � � � � �� ,

� �� , with � �� , and � �� , and a continuous family of flows �� , such

that �� , for all � �� . Item (a) ensures that �� REL � �� is

an isolating neighborhood for all � �� . The continuity of � �� implies that ��varies continuously with . Thus via the continuation property of the Conley index� ��

REL � � �� is independent of � �� , which completes the proof of Item (c).

4.4. An intrinsic definition. For any bounded proper relative braid class �� REL � �we can define its index intrinsically, independent of any notions of parabolic flows.Denote as before by � the set �� REL � � within �� . The singular braid diagrams

�

partition �� into disjoint cells (the discretized relative braid classes), the closures ofwhich contain portions of

�. For a bounded proper braid class, � is compact, and�� avoids

��.

To define the exit set � �, consider any point � on �� + �

. There exists a smallneighborhood � of � in �� for which the subset � �

consists of a finite numberof connected components ,� � -. Assume that � ' � � � � . We define � �

to be theset of � for which the word metric is locally maximal on � ' , namely,

(14) � � �� ' �word� ��

word

�� From the previous we deduce that �� is an index pair for any parabolic

flow for which �� , and thus by the independence of �

�, the homotopy type

�� gives the Conley index. The index can be computed in practice by choosinga representative � � �� and determining � and � �

. A computer assisted approachexists for computing the homological index using cube complexes and digital ho-mology [2].


For a definition of the braid index for improper and unbounded braid classes werefer to �11.

4.5. Three simple examples. It is not obvious what the homotopy index is mea-suring topologically. Since the space � has one dimension per free anchor point,examples quickly become complex.

Example 1: Consider the proper period-2 braid illustrated in Fig. 5[left]. (Note thatdeleting any strand in the skeleton yields an improper braid.) There is exactly onefree strand with two critical points (recall that these are closed braids and the leftand right sides are identified). The critical point in the middle, � , is free to movevertically between the fixed points on the skeleton. At the endpoints, one has asingular braid in

�which is on the exit set since a slight perturbation sends this

singular braid to a different braid class with fewer crossings. The end critical point,�� (� � ') can freely move vertically in between the two fixed points on the skeleton.The singular boundaries are in this case not on the exit set since pushing �� acrossthe skeleton increases the number of crossings.

��

��

��

��

��

�

�

FIGURE 5. The braid of Example 1 [left] and the associated config-uration space with parabolic flow [middle]. On the right is an ex-panded view of � � REL � where the fixed points of the flow corre-spond to the four fixed strands in the skeleton � . The reader shouldnote that the braid classes adjacent to these fixed points are notproper.

Since the points � and �� can be moved independently, the configuration space� in this case is the product of two compact intervals. The exit set � �

consists ofthose points on �� for which � is a boundary point. Thus, the homotopy index ofthis relative braid is ��

.

Example 2: Consider the proper relative braid presented in Fig. 6[left]. Since there isone free strand of period three, the configuration space � is determined by the vectorof positions �� ' � � � �� of the anchor points. This example differs greatly from theprevious example. For instance, the point �' (as represented in the figure) may passthrough the nearest strand of the skeleton above and below without changing the


braid class. The points � and �� may not pass through any strands of the skeletonwithout changing the braid class unless � ' has already passed through. In this case,either � or �� (depending on whether the upper or lower strand is crossed) becomesfree.

To simplify the analysis, consider �� ' � � � � � � as all of �� (allowing for the momentsingular braids and other braid classes as well). The position of the skeleton inducesa cubical partition of �� by planes, the equations being � � � � �� for the variousstrands � � of the skeleton � . The braid class � is thus some collection of cubes in�� . In Fig. 6[right], we illustrate this cube complex associated to � , claiming that itis homeomorphic to �

� � � . In this case, the exit set � �

happens to be the entireboundary �� . It is an extremely helpful exercise for the reader to show that thefigure is correct, and that the associated quotient space is homotopic to the wedge-sum � � � �

.

��

� �

��

��

FIGURE 6. The braid of Example 2 and the associated configurationspace � .

Example 3: To introduce the spirit behind the forcing theorems of the latter half ofthe paper, we reconsider the period two braid of Example 1. Take an �-fold coverof the skeleton as illustrated in Fig. 7. By weaving a single free strand in and out ofthe strands as shown, it is possible to generate numerous examples with nontrivialindex. A moment’s meditation suffices to show that the configuration space � forthis lifted braid is a product of

�� intervals, the exit set being completely determinedby the number of times the free strand is “threaded” through the inner loops of theskeletal braid as shown.

For an �-fold cover with one free strand we can select a family of �� possible braidclasses describes as follows: the even anchor points of the free strand are always inthe middle, while for the odd anchor points there are three possible choices. Twoof these braid classes are not proper. All of the remaining ��

braid classes arebounded and have homotopy indices equal to a sphere �

for some � � � . Sev-eral of these strands may be superimposed while maintaining a nontrivial homotopyindex for the net braid: we leave it to the reader to consider this interesting situation.


FIGURE 7. The lifted skeleton of Example 1 with one free strand.

Stronger results follow from projecting these braids back down to the period twosetting of Example 1. If the free strand in the cover is chosen not to be isotopic to aperiodic braid, then it can be shown via a simple argument that some projection ofthe free strand down to the period two case has nontrivial homotopy index. Thus,the simple period two skeleton of Example 1 is the seed for an infinite number ofbraid classes with nontrivial homotopy indices. This certainly suggests that any par-abolic recurrence relation (� � �) admitting this skeleton is forced to have positivetopological entropy, as will follow from the Morse type theory described in �7: cf.the related results from the Nielsen-Thurston theory of disc homeomorphisms [10].

5. STABILIZATION AND INVARIANCE

5.1. Free braid classes and the extension operator. Via the results of the previoussection, the homotopy index is an invariant of the discretized braid class: keepingthe period fixed and moving within a connected component of the space of relativediscretized braids leaves the index invariant. The topological braid class, as defined in�2, does not have an implicit notion of period. The effect of refining the discretizationof a topological closed braid is not obvious: not only does the dimension of the indexpair change, the homotopy types of the isolating neighborhood and the exit set maychange as well upon changing the discretization. It is thus perhaps remarkable thatany changes are correlated under the quotient operation: the homotopy index is aninvariant of the topological closed braid class.

On the other hand, given a complicated braid, it is intuitively obvious that a cer-tain number of discretization points are necessary to capture the topology correctly.If the period

�is too small � �� REL � may contain more than one path component

with the same topological braid class:

Definition 17. A relative braid class �� REL � � in � �� REL � is called free if

(15) �� REL � � � ,� REL � - � �� REL � � that is, if any other discretized braid in � �� REL � which has the same topological braid classas �

REL � is in the same discretized braid class �� REL � �.A braid class �� is free if the above definition is satisfied with � � �. Not all

discretized braid classes are free: see Fig. 8.


FIGURE 8. An example of two non-free discretized braids which areof the same topological braid class but define disjoint discretizedbraid classes in � � REL � .

Define the extension map � � �� via concatenation with the trivial braidof period one:

(16) ��

The reader may note (with a little effort) that the non-equivalent braids of Fig. 8become equivalent under the image of � . Note that under the action of � bound-edness of a braid class is not necessarily preserved, i.e. �� REL � � may be bounded,and �� REL � � � unbounded. For this reason we will prove a stabilization result fortopological bounded proper braid classes.

5.2. A topological invariant. Consider a period�

discretized relative braid pair�REL � which is not necessarily free. Collect all (finite number) discretized braids� �� such that the pairs � ��

REL � are all topologically isotopic to �REL �

but not pairwise discretely isotopic. For the case of a free braid class, � � �.

Definition 18. Given �REL � and � �� as above, denote by H ��

REL � � thewedge of the homotopy indices of these representatives,

(17) H��REL � � �� ' � ��

REL � � �

where�

is the topological wedge which, in this context, identifies all the constituent exit setsto a single point.

This wedge product is well-defined by Theorem 15 by considering the isolatingneighborhood � � ��

REL � �. In general a union of isolating neighborhoodsis not necessarily an isolating neighborhood again. However, since the word metricstrictly decreases at

�the invariant set decomposes into the union of invariant sets

of the individual components of � . Indeed, if an orbit would exists between twocomponents it would have to pass through

�.

The principal topological result of this paper is that H is an invariant of the topo-logical bounded proper braid class �� REL ,� -�, i.e.


Theorem 19. Given �REL � � � �� REL � and ��

REL�� REL

�� which are topologically

isotopic as bounded proper braid pairs, then

(18) H��REL � � � H��

REL��

The proof of Theorem 19 follows immediately upon showing that the homotopyindex is invariant under the operator � (Theorem 20), and that �

� is free for

�sufficiently large. The latter is proved in Proposition 27.

Theorem 20. For �REL � any bounded proper discretized braid pair, the wedged homotopy

index of Definition 18 is invariant under the extension operator:

(19) H�� REL � � � � H��REL � �

Proof. By the invariance of the index with respect to the skeleton � , we may as-

sume that � has all intersections generic (� �� for all strands � �� ). Thus, fromthe proof of Lemma 55 in Appendix A, there is a recurrence relation � having � asfixed point(s) for which � � ' � �.

For � � � consider the one-parameter family of augmented recurrence functions7

� � � �� ' :

� ��

Because of our choice of � ' �� ' �� as being independent of the first vari-

able, � �' is decoupled from the extension of the braid as �� wraps around to � � �� ' .By construction the above system satisfies Axioms (A1)-(A2) for all � � � with, inparticular, the strict monotonicity of (A1) holding only on one side. One thereforehas a parabolic flow �

�� on �� for all � � �. In the singular limit � � �, this forces�� , and one obtains the flow �

�' � � � ��. Notice that by construction

�� , for all � � �.Denote by

� �� + � �� the subset of relative braids which are topologically iso-topic to � � REL � � . Likewise, denote by

� � + � �� the image under � of the subsetof relative braid pairs in � �� which are topologically isotopic to �

REL � . In otherwords,(20)� �� REL ,� � -� � � �� REL � � � � �� !�� REL ,� -� � � �� REL �" As per the paragraph preceding Definition 18, there are a finite number of connectedcomponents of each of these sets. Clearly,

� � is a codimension-� subset of� �� .

By performing an appropriate change of coordinates (cf. [12]), we can recast theparabolic flow as a singular perturbation problem. Let � � �� , with �� , and let � � �� , with � � �� . Upon rescaling time as� �� , the vector field induced by our choice of � � is of the form

(21)

��

7Recall the indexing conventions: for a period � � braid, � � �� , and � � �� .


for some (unspecified) vector fields � and � with the functional dependence in-dicated. The product flow of this vector field (21) is denoted � �� — the flow �

��

in the new coordinates — and is well-defined on �� . In the case � � �, the set� �� ,� � �- + �� is a submanifold of fixed points containing� � for which

the flow � �' is transversally nondegenerate (since here � � � �). By construction,� � � � �� — to be interpreted in the new coordinates —, as illustrated in Fig. 9

[in the simple case where all braid classes are free and� �� is thus connected].

�

�

��

��

�

��

FIGURE 9. The rescaled flow acts on� �� , the period

� � �braid

classes. The submanifold�

is a critical manifold of fixed points at� � �. Any appropriate isolating neighborhood � ' in� � thickens to

an isolating neighborhood in� �� for � small.

The remainder of the analysis is a technique in singular perturbation theory fol-lowing [12]: one relates the � -dynamics of Equation (21) to those of the -dynamicson

�, whose orbits are of the form �� , where � �� satisfies the limiting equation�� . The Conley index theory is well-suited to this situation.

For any closed set � + �and � � � , let � �� , �� -

denote the “product” radius � neighborhood in �� . Denote by � � � �� themaximal value � �� . Due to the specific form of (21), we obtain thefollowing uniform squeezing lemma.

Lemma 21. If � is any invariant set of � �� contained in some � �� , then in fact � + � �� .Proof. Let �� be an orbit in � contained in some � �� . Take the inner product

of the �-equation with � :

�� ' � � �� ' �� ' �� ' ��


Hence�� , and we conclude that if �� ' �� for some �' � � ,

then �� grows unbounded for � � �' and therefore �� , a contradiction.Thus �� for all � � � .

By compactness of the proper braid class, it is clear that� �� , and thus the max-

imal isolated invariant set of � �� given by � � �� INV�� , is strictly contained

(and thus isolated) in � �� for some compact � + �and some � sufficiently large.

Fix � �� as above. Lemma 21 now implies that as � becomes small, � � issqueezed into � �� — a small neighborhood of a compact subset � of the criticalmanifold

�, as in Fig. 9.

This proximity of � � to�

allows one to compare the dynamics of the � � � and� � � flows. Let � ' + � � + �be an isolating neighborhood for the maximal -

dynamics invariant set �' �� INV�� ' � within the braid class

� �. Theorem 2.3Cof [12] (combined with the appropriate existence theorems for isolating blocks [58])states that if �� ' �� ' � is an index pair for the limiting equations

�� ,then a suitable index pair for the flow � �� of Equation (21) is given by

(22) (� ' �� ' �� )for � � � sufficiently small. Clearly, then, the homotopy index of �' is equal to thehomotopy index of INV

�� ' �� for all � sufficiently small. It remains to show thatthis captures the maximal invariant set � �.Lemma 22. For all sufficiently small �, INV

�� ' �� .Proof. Choose �� sufficiently small so that � � �� contains the

invariant set � � for every � � ��. Assume by contradiction that INV�� ' ��

for some sequence �� . Then, since � ' �� is isolating for � � ��, there ex-ist orbits �� of � �

which are not contained in INV�� ' �� , and such that�� ' �� , for some �� . Define �� , and set �� . The sequence �� satisfies the equa-

tions ��

��

� � �� By assumption �� , and �� , for all � � and all �� . AnArzela-Ascoli argument then yields the existence of an orbit �� + � � � �� , with �� ' �, satisfying the equation

�� . By definition�� INV�� INV

�� ' � + �� ' �, a contradiction.

Theorem 20 now follows. Since, by Theorem 15, the homotopy index is indepen-dent of the parabolic flow used to compute it, one may choose the parabolic flow� �� for � � � sufficiently small. The homotopy index of � �� on the maximal invari-ant set � � yields the wedge of all the connected components: H�� REL � � �. Wehave computed that this index is equal to the index of �

�on the original braid class:

H��REL � �.

The statement of Theorem 19 follows if we prove that �� is free for�

largeenough. Indeed, if ,� - has more than one component for small

�, then, depending

on which component of � �� ,� -, the index could differ. However, if for some�


large enough the number of components reduces to one, then Theorem 20 impliesthat all indices must be the same, proving Theorem 19. In Proposition 27 we provethat �� is indeed eventually free.

Remark 23. The proof of Theorem 20 implies that any component of the period-�� braid class� �� which does not intersect

�must necessarily have trivial

index.

Remark 24. The above procedure also yields a stabilization result for boundedproper classes which are not bounded as topological classes. In this case one simplyaugments the skeleton � by two constant strands as follows. Define � � ��

��

,where

(23) � �� %� � ��

� %� � �� Suppose �� REL � � + � �� REL � is bounded for some period

�' . It now holds that

� ��REL � � � � ��

REL � � �, and �� REL ,� � -� is a bounded class. It therefore followsfrom Theorem 19 that

� �� ' � (� �� REL � ) � H��

REL � � � �(24)

where H can be evaluated via any discrete representative of �� REL ,� � -� of anyadmissible period.

5.3. Stably free classes. We now complete the proof of Theorem 19 by showing thatdiscretized braid classes are stably free.

Given a braid � � � �� , consider the extension � � of period� � �

. Assume atfirst the simple case in which

� � �, so that � � is a period-2 braid. Draw the braid

diagram � �� as defined in �2 in the domain �� . Choose any 1-parameterfamily of curves � � � �� such that � ' � �� and so that � �is transverse8 to the braid diagram � �� for all �. Define the braid � � �� as follows:

(25) ��

In this definition, the point � � � �� is well-defined since � � is always transverse tothe braid strands and �' intersects each strand but once.

Lemma 25. For any such discrete homotopy � � , �� .Proof. It suffices to show that this path of braids does not intersect the singular

braids�

. Assume that for some �' , � �� is singular. Since � is assumed to bea nonsingular braid, every crossing of two strands in the braid diagram of � � is atransversal crossing on the “left” side of the braid. Thus, if � � �� for distinct strands � and � �, then

(26) !� ��' � �� ' " !� �� " � � 8At the anchor points, the transversality should be topological as opposed to smooth.


The braid � �� has a crossing of the � and � � strands at � �. Checking the

transversality of this crossing yields

(27)

!�� ' �� ' " !�� "� !�� ' �� ' " !�� "� !�� ' �� ' " !�� " � �

Thus the crossing is transverse and the braid is never singular.

Note that the proof of Lemma 25 does not require the braid � � to be a closedbraid diagram since it fixes the endpoints: the proof is equally valid for any localizedregion of a braid in which one spatial segment has crossings and the next segmenthas flat strands.

Corollary 26. The “shifted” extension operator which inserts a trivial period-1 braid at the� discretization point in a braid has the same action on components of � � as does � .

��

FIGURE 10. Relations in the braid group via discrete isotopy.

Proposition 27. Given any braid � � � �� , then the braid class �� is free for all� � �� word.

Proof. We must show that any braid � � � � �� which has the same topologicaltype as � is discretely isotopic to �. Place both � and � � in general position so asto record the sequences of crossings using the generators of the �-strand positivebraid semigroup, ,� � -, as in �2. Recall the braid group has relations � �� for� � � � �

and � �� ; closure requires making conjugacy classesequivalent.

The conjugacy relation can be realized by a discrete isotopy as follows: since� � �� word, � must possess some discretization interval on which there are no cross-ings. Lemma 25 then implies that this interval without crossings commutes with allneighboring discretization intervals via discrete isotopies. Exchanging discretizationintervals

�times shifts the entire braid by one discretization interval. This generates

the conjugacy relation.To realize the remaining braid relations in a discrete isotopy, assume first that �

and � � are of the form that there is at most one crossing per discretization interval. Itis then easy to check that the braid relations can be conducted via discrete isotopy:see Fig. 10.


In the case where � (and/or � �) exhibits multiple crossings on some discretizationintervals, it must be the case that a corresponding number of other discretizationintervals do not possess any crossings (since

� � �� word). Again, by inductivelyutilizing Lemma 25, we may redistribute the intervals-without-crossing and “comb”out he the multiple crossings via discrete isotopies so as to have at most one crossingper discretization interval.

We suspect that all braids in the image of � are free: a result which, if true, wouldsimplify index computations yet further.

6. DUALITY

For purposes of computation of the index, we will often pass to the homologicallevel. In this setting, there is a natural duality made possible by the fact that theindex pair used to compute the index of a braid class can be chosen to be a manifoldpair.

Definition 28. The duality operator on discretized braids is the map� � ��

given by

(28) �� Clearly

�induces a map on relative braid diagrams by defining

� ��REL � � to

be� �

REL�� . The topological action of

�is to insert a half-twist at each spatial

segment of the braid. This has the effect of linking unlinked strands, and, since�

isan involution, linked strands are unlinked by

�: see Fig. 11 for the topological action.

�

FIGURE 11. The topological action of�

.

For the duality statements to follow, we assume that all braids considered have evenperiods and that all of the braid classes and their duals are proper, so that the homo-topy index is well-defined.

Lemma 29. The duality map�

respects braid classes: if �� then �� .Bounded braid classes are taken to bounded braid classes by

�.

Proof: It suffices to show that the map�

is a homeomorphism on the pair � �� .This is true on �� since

�is a smooth involution (

� � � �). If � � �

with �� and

(29) ��


then applying the operator�

yields points� �� with all terms in the above

inequality multiplied by �� [if is even] or by �� [if is odd]: in either case, thequantity is still non-negative and thus

� � � �. Boundedness is clearly preserved.

Theorem 30. (a) The effect of�

on an appropriate index pair is to reverse the directionof the parabolic flow.

(b) For �� REL � � + � �� REL � of period�� with � free strands,

(30) � � �� REL � ��

REL � � � � (c) For �� REL � � + � �� REL � of period

�� with � free strands,

(31) � � �H �� REL � �� H ��

REL � � � � Proof:For (a), let �� denote an index pair associated to a proper relative braid class�� REL � �. Dualizing sends � to a homeomorphic space

� �� . The following localargument shows that the exit set of the dual braid class is in fact the complement (inthe boundary) of the exit set of the domain braid: specifically,

�� Let � � �� REL � � � �

. At any singular anchor point of � , i.e., where � �� and the transversality condition is not satisfied, then it follows from Axiom (A2) that

(32) SIGN

� ��

��

SIGN �� (Depending on the form of (A2) employed, one might use � �� on the righthand side without loss.) Since the subscripts on the left side have the opposite parityof the subscripts on the right side, taking the dual braid (which multiplies the entriesby �� and ��

respectively) alters the sign of the terms. Thus, the operator�

reverses the direction of the parabolic flow.From this, we may compute the Conley index of the dual braid by reversing the

time-orientation of the flow. Since one can choose the index pair used to computethe index to be an oriented manifold pair (specifically, an isolating block: see, e.g.,[58]), one may then apply a Poincare-Lefschetz duality argument as in [42] and usethe fact that the dimension is

�� to obtain the duality formula for homology. Thisyields (b).

The final claim (c) follows from (b) by showing that within �� , the map�

isbijective on topological braid classes. Assume that �� REL � � and �� REL � � are distinctbraid classes in � �� of the same topological type. Since

�is a homeomorphism on� �� , the dual classes ��

REL�� and �� REL

�� are distinct. Claim (e) follows

upon showing that these duals are still topologically the same braid class.Since ,� REL � - � ,� � REL � -, it follows from Proposition 27 that��

REL��

� �� REL

�� for

�sufficiently large. By Lemma 29,

� �� REL

��

REL��

� ��

which, by Lemma 7 means that these braids are topologically the same. The topolog-ical action of dualizing the

��-stabilizations of �

REL � and � � REL � is to add�

full


twists. Since the full twist is in the center of the braid group (this element commuteswith all other elements of the braid group [9]), one can factor the dual braids withinthe topological braid group and mod out by

�full twists, yielding that,� �

REL�� - � ,� � �

REL�� -

We use the homological duality above to complete a crucial computation in theproof of the forcing theorems (e.g., Theorem 1) at the end of this paper. The followingsmall corollary uses duality to give the first step towards answering the question ofjust what the homotopy index measures topologically about a braid class.

Corollary 31. Consider the dual of any augmented proper relative braid. Adding a full twistto this dual braid shifts the homology of the index up by two dimensions.

Proof: Assume that� �� REL � � � is the dual of an augmented braid in period

��(the augmentation is required to keep the braid class bounded upon adding a fulltwist). The prior augmentation implies that the outer two strands of

�� “maximally

link” the remainder of the relative braid. The effect of adding a full twist to thisbraid can be realized by instead stabilizing �� REL � � � twice and then dualizing. Thehomological duality implies that for each connected component of the topologicalclass it holds that

(33)��

REL � � �� REL � � ��

REL � � �� REL � � ��

which gives the desired result for the index H via Theorem 30.

Remark 32. The homotopy version of (33) can be achieved by following a similarprocedure as in �5. One obtains a double-suspension of the homotopy index, asopposed to a shift in homology.

Remark 33. Given a braid class �� of odd period � � �� , the image under

�

is not necessarily a discretized braid at all: without some symmetry condition, thebraid will not “close up” at the ends. To circumvent this, define the dual of � to be thebraid

� �� — the dual of the period�� extension of � . The analogue of Theorem 30

above is that

(34) � � �H�SYM

�� REL� �� H��

REL � � � � �

where SYM denotes the subset of the braid class which consists of symmetric braids:�� for all .7. MORSE THEORY

It is clear that the Morse-theoretic content of the homotopy index on braids holdsimplications for the dynamics of parabolic flows and thus zeros of parabolic recur-rence relations. To this end, we have restricted ourselves to bounded proper braidclasses.


Recall that the characteristic polynomial of an index pair �� is the polynomial

(35) � � � �� ' � � �� The Morse relations in the setting of the Conley index (see [14]) state that, if � has aMorse decomposition into distinct isolating subsets ,� � -�� , then

(36)

��

for some polynomial � � with nonnegative integer coefficients.

7.1. The exact, nondegenerate case. For parabolic recurrence relations which satisfy(A3) (gradient type) it holds that if � ��

REL � � �� , then � has at least one fixed pointin �� REL � �, for any � � �� . Indeed, one has:

Lemma 34. For an exact nondegenerate parabolic flow on a bounded proper relative braidclass, the sum of the Betti numbers � of �, as defined in (35), is a lower bound on the numberof fixed points of the flow on that braid class.

Proof: The details of this standard Morse theory argument are provided forthe sake of completeness. Choose �

�a nondegenerate gradient parabolic flow on�� REL � � (in particular, �

�fixes � for all time). Enumerate the [finite number of]

fixed points ,� � -�� of ��

on this [bounded] braid class. By nondegeneracy, thefixed point set may be taken to be a Morse decomposition of INV

�� . The charac-

teristic polynomial of each fixed point is merely � � �� , where � � �� is the Morseco-index of � � . Substituting � �

into Equation (36) yields the lower bound

(37) Fix�� REL � � �� On the level of the topological braid invariant H, one needs to sum over all the

path components as follows. As in Theorem 19, choose period-�

representatives � �� (� from 0 to � ) for each path component of the topological class �� REL ,� -�. If weconsider fixed points in the union �� ' ��

REL � �, we obtain the following Morseinequalities from (37) and Theorem 19:

(38) Fix �� ' �� REL � � �� H� �

where � �H� is the� �

Betti number of H��REL � � �. Thus, again, the sum of the Betti

numbers is a lower bound, with the proviso that some components may not containany critical points.

If the topological class �� REL ,� -� is bounded the inequality (38) holds with theinvariant H��

REL � �.


7.2. The exact, degenerate case. Here, the situation is not as good, but a decentlower bound still exists.

Lemma 35. For an arbitrary exact parabolic flow on a bounded relative braid class, thenumber of fixed points is bounded below by the number of distinct nonzero monomials in thecharacteristic polynomial � � � �� .

Proof: Assuming that Fix is finite, all critical points are isolated and form a Morsedecomposition of INV

�� . The specific nature of parabolic recurrence relations re-veals that the dimension of the null space of the linearized matrix at an isolatedcritical point is at most 2, see e.g. [54]. Using this fact Dancer proves [15], via the de-generate version of the Morse lemma due to Gromoll and Meyer, that � �� for at most one index

� � �' . Equation (36) implies that,

(39)

� ��

on the level of polynomials. Since the above mentioned result of Dancer implies thatfor each �, � � � ��

, for some � � �, it follows that the number of critical

points needs to be at least the number of non-trivial monomials in � � � ��.As before, if we instead use the topological invariant H for �� REL ,� -� we obtain

that the number of monomials in � � � �H� is a lower bound for the total sum of fixedpoints over the topologically equivalent path-components.

More elaborate estimates in some cases can be obtained via the extension of theConley index due to Floer [17].

7.3. The non-exact case. If we consider parabolic recurrence relations that are notnecessarily exact, the homotopy index may still provide information about solutionsof � � �. This is more delicate because of the possibility of periodic solutions forthe flow � �� . For example, if � � � �� , the indexdoes not provide information about additional solutions for � � �, as a simple coun-terexample shows. However, if � � � �� then there exists at least onesolution of � � � with the specified relative braid class. Specifically,

Lemma 36. An arbitrary parabolic flow on a bounded relative braid class is forced to have afixed point if � �� is nonzero. If the flow is nondegenerate, then the number offixed points is bounded below by the quantity

(40) (� � � �� ) �� Proof: Set � � �� REL � ��. As the vector field � has no zeros at �� , the Brouwer

degree, �� , may be computed via a small perturbation�� and is given by9

�� % �� ' ��

For a generic perturbation�� the associated parabolic flow

��

is a Morse-Smale flow

[21]. The (finite) collection of rest points ,� � - and periodic orbits ,� � - of��

then

9We choose to define the degree via � � in order to simplify the formulae.


yields a Morse decomposition of �� , and the Morse inequalities are

��

� � � ��

� � � �� The indices of the fixed points are given by � � � �� , where � � is the num-

ber of eigenvalues of� �� with positive real part, and the indices of periodic orbits

are given by � � � �� . Upon substitution of � �we obtain

��

��

��

Thus, if the Euler characteristic of � is non-trivial, then � has at least one zero in � .In the generic case the Morse relations give even more information. One has� � � �� , with � �� , and � � � �� .

It then follows that� � � � � �� , proving the stated lower

bound.The obvious extension of these results to the topological index H is left to the

reader.

8. SECOND ORDER LAGRANGIAN SYSTEMS

In this final third of the paper, we apply the developed machinery to the problemof forcing closed characteristics in second order Lagrangian systems of twist type.The vast literature on fourth order differential equations coming from second orderLagrangians includes many physical models in nonlinear elasticity, nonlinear op-tics, physics of solids, Ginzburg-Landau equations, etc. (see �1). In this context wemention the work of [1, 36, 47, 48].

8.1. Twist systems. We recall from �1 that closed characteristics at an energy level #are concatenations of monotone laps between its minima and maxima �� , whichare periodic sequences with even period

�� . The extrema are restricted to the set . / ,whose connected components are denoted by 0/ : interval components (see �1.2 forthe precise definition). The problem of finding closed characteristics can, in mostcases, be formulated as a finite dimensional variational problem on the extrema �� .The following twist hypothesis, introduced in [56], is key:

(T): �� ,�/ �� ' (� �� # )�� - has a minimizer� �� for all �� ,0/ � 0/ � � �� -, and � and � are �

-smoothfunctions of �� .

Here � � � � � �� ,� � � � �� ' %� � � � Hypothesis (T) is a weaker version of the hypothesis that assumes that the mono-

tone laps between extrema are unique (see, e.g., [32, 33, 56]). Hypothesis (T) is validfor large classes of Lagrangians � . For example, if � �� , thefollowing two inequalities ensure the validity of (T):


(a) �� # �, and

(b) ��

�� # � � � for all � � 0/ and � � � .

Many physical models, such as the Swift-Hohenberg equation (3), meet these re-quirements, although these conditions are not always met. In those cases numeri-cal calculations still predict the validity of (T), which leaves the impression that theresults obtained for twist systems carry over to much more systems for which Hy-

pothesis (T) is hard to check.10 For these reasons twist systems play a important rolein understanding second order Lagrangian systems.

Existence of minimizing laps is valid under very mild hypotheses on�

(see [29]).In that case (b) above is enough to guarantee the validity of (T). An example of aLagrangian that satisfies (T), but not (a) is given by the Erickson beam-model [28, 47,

53] � �� .

8.2. Discretization of the variational principle. We commence by repeating the un-derlying variational principle for obtaining closed characteristics as decribed in [56].In the present context a broken geodesic is a � �

-concatenation of monotone laps (alter-nating between increasing and decreasing laps) given by Hypothesis (T). A closedcharacteristic � at energy level # is a (� �

-smooth) function � � �� , � � � � � ,which is stationary for the action �/ �� with respect to variations �� , and�� , and as such is a ‘smooth broken geodesic’.

The following result, a translation of results implicit in [56], is the motivation andbasis for the applications of the machinery in the first two-thirds of this paper.

Theorem 37. Extremal points ,� � - for bounded solutions of second order Lagrangian twistsystems are solutions of an exact parabolic recurrence relation with the constraints that (i)�� ; and (ii) the recurrence relation blows up along any sequence satisfy-ing � � � � �� .

Proof: For simplicity, we restrict to the case of a nonsingular energy level # : forsingular energy levels, a slightly more involved argument is required. Denote by 0the interior of 0/ , and by � �0 � � � �� , �� 0 � 0 � � � �� - the diagonal.Then define the generating function

(41) � � �0 � 0 � � � � � �� ' (� �� # )�� the action of the minimizing lap from � to �� . That � is a well-defined function isthe content of Hypothesis (T). The action functional associated to � for a period

��system is the function

� �� ' � ��

Several properties of � follow from [56]:

(a) [smoothness] � � � � �0 � 0 �� .(b) [monotonicity] � �� for all � �� 0 .

10Another method to implement the ideas used in this paper is to set up a curve-shortening flow for

second order Lagrangian systems. See �11.


(c) [diagonal singularity] ��$ �� $� � � �� $�� $ � ��

��$ �� $� � � �� $�� $ � �� .

In general the function � � �� is strictly increasing in �� for all � �� 0/ ,and similarly �� is strictly increasing in �. The function � also has theadditional property that � �� .

Critical points of � �� satisfy the [exact] recurrence relation

� � �� (42)

where � � �� is both well-defined and � on the domains

� � � , �� 0 � � �� - �by Property (a). The recurrence function � satisfies (A2) with

� � �and

� �� (non-autonomous).11 Property (b) implies that Axiom (A1) is satisfied. Indeed,� � � � � �� , and �� .

Property (c) provides information about the behavior of � at the diagonal bound-aries of

� � , namely,

(43)��

The parabolic recurrence relations generated by second order Lagrangians are nat-urally defined on the polygonal domains

� �.Definition 38. A parabolic recurrence relation is said to be of up-down type if (43) is satis-fied.

In the next subsection we demonstrate that the up-down recurrence relations canbe embedded into the standard theory as developed in �2-�7.

8.3. Up-down restriction. The variational set-up for second order Lagrangians in-troduces a few complications into the scheme of parabolic recurrence relations asdiscussed in �2-�7. The problem of boundary conditions will be considered in thefollowing section. Here, we retool the machinery to deal with the fact that maxima

and minima are forced to alternate. Such braids we call up-down braids.12

8.3.1. The space � .

Definition 39. The spaces of general/nonsingular/singular up-down braid diagrams aredefined respectively as:

�� (�� ) � � � � � � �

� �� (�� ) � � � � � � ��

11We could also work with sequences � that satisfy �� .12The more natural term alternating has an entirely different meaning in knot theory.


Path components of � �� comprise the up-down braid types �� , and path components in� �� REL � comprise the relative up-down braid types �� REL � �� .

The set �� has a boundary in �� (44) � �� ! �� (�� ) � � � � � �"Such braids, called horizontal singularities, are not included in the definition of �� since the recurrence relation (42) does not induce a well-defined flow on the bound-ary � �� .

Lemma 40. For any parabolic flow of up-down type on �� , the flow blows up in a neighbor-hood of � �� in such a manner that the vector field points into �� . All of the conclusions ofTheorem 15 hold upon considering the �-closure of braid classes �� REL � �� in �� , denoted

�� %� �� REL � �� REL � � �� REL � �� (�� ) � � � � � � �

for all � � � sufficiently small.

Proof: The proof that any parabolic flow ��

of up-down type acts here so as tostrictly decrease the word metric at singular braids is the same proof as used inProposition 11. The only difficulty arises in what happens at the boundary of �� :we must show that �

�respects the up-down restriction in forward time.

Let � � � �� be a horizontal singularity. Then ��

is, strictly speaking, not defined

at � . Let� � ,�� - be the set of all pairs of indices for which �� . Consider

the 1-parameter family (�� )� �� defined by:

(45) ��

�� odd�� even

Defining �� , we have that �� . We now compute the vector field

component at � �� in the direction of �� :

� ��

� �� ' � �� %��

Using (42) and property (c) for � , take the limit � � �:

(46) �� ' � ��

� �� ' � ��

This implies that ��

is transverse-inwards to � �� with infinite repulsion. Due to

the boundedness of �� REL � �� and the infinite repulsion at � �� , we can choose auniform � ��

REL � � � � so that �� %� �� REL � �� is an isolating neighborhood for all� � � � ��REL � �.


8.3.2. Universality for up-down braids. We now show that the topological informationcontained in up-down braid classes can be continued to the canonical case describedin �2. As always, we restrict attention to proper, bounded braid classes, proper be-ing defined as in Definition 13, and bounded meaning that the set �� REL � �� isbounded in �� . Note that an up-down braid class �� REL � �� can sometimes bebounded while �� REL � � is not. To bounded proper up-down braids we assigna homotopy index. From Lemma 40 it follows that for � sufficiently small the set� � %� �� %� �� REL � �� is an isolating neighborhood in �� whose Conley index,

� ��REL � � � � �� %��

is well-defined with respect to any parabolic flow ��

generated by a parabolic recur-rence relation of up-down type, and is independent of �. As before, non-triviality of� �� %� � implies existence of a non-trivial invariant set inside � � %� (see �8.3.3).

The obvious question is what relationship holds between the homotopy index� ��

REL � � � � and that of a braid class without the up-down restriction. To answerthis, augment the skeleton � as follows: define � � � � � �

� � ��

, where� �� %� � ��

� %� � �� The topological braid class ,� REL � � - is bounded and proper by construction.

Theorem 41. For any bounded proper up-down braid class �� REL � �� in � �� REL � ,

� ��REL � � � � � � ��

REL � � � Proof. From Lemma 55 in Appendix A we obtain a parabolic recurrence relation

� 'for which � � is a solution. We denote the associated parabolic flow by �

�' . Definetwo functions

� and�� in � �� , with

� � � � � � �� , and� �� for � �� ,� ��

, and�� for � � �� ,

�� , for some � � � and

� � � to bespecified later. Introduce a new recurrence function � � �� '� �� for odd, and � � �� '� �� for even. Theassociated parabolic flow will be denoted by �

�, and �� by construction.

Since the braid class �� REL � � � is proper, � � �� REL � � � is an isolating neigh-borhood with invariant set INV

�� . If we choose�

large enough, and � sufficientlysmall, then the invariant set INV

�� lies entirely in �� %� �� REL � � �� %�. Indeed,

for large�

we have that for each , � � �� has a fixed sign on the complementof � � %�. Therefore, � ��

REL � � � � � �� %� �. Now restrict the flow �� to

� � %� + �� REL � . We may now construct a homotopy between �� and �

�, via

�� (see Appendix A), where � and the associated flow �

�are defined

by (42). The braid � � is stationary along the homotopy and therefore

� �� %� �� %� ��

which proves the theorem.

We point out that similar results can be proved for other domains� � with various

boundary conditions. The key observation is that the up-down restriction is reallyjust an addition to the braid skeleton.


8.3.3. Morse theory. For bounded proper up-down braid classes �� REL � �� the Morsetheory of �7 applies. Combining this with Lemma 40 and Theorem 41, the topologicalinformation is given by the invariant H of the topological braid type �� REL ,� � -�.

Corollary 42. On bounded proper up-down braid classes, the total number of fixed points ofan exact parabolic up-down recurrence relation is bounded below by the number of monomialsin the critical polynomial � � � �H� of the homotopy index.

Proof. Since all critical point are contained in � � %� the corollary follows from theLemmas 35, 40 and Theorem 41.

9. MULTIPLICITY OF CLOSED CHARACTERISTICS

We now have assembled the tools necessary to prove Theorem 1, the general forc-ing theorem for closed characteristics in terms of braids, and Theorems 2 and 3, theapplication to singular and near-singular energy levels. Given one or more closedcharacteristics, we keep track of the braiding of the associated strands, including atwill any period-two shifts. Fixing these strands as a skeleton, we add hypotheticalfree strands and compute the homotopy index. If nonzero, this index then forcesthe existence of the free strand as an existing solution, which, when added to theskeleton, allows one to iterate the argument and produce an infinite family of forcedclosed characteristics.

The following lemma (whose proof is straightforward and thus omitted) will beused repeatedly for proving existence of closed characteristics.

Lemma 43. Assume that � is a parabolic recurrence relation on � � with � a solution.Then, for each integer � � �

, there exists a lifted parabolic recurrence relation on �� forwhich every lift of � is a solution. Furthermore, any solution to the lifted dynamics on �� projectes to some period-

�solution. 13

The primary difficulties in the proof of the forcing theorems are (i) computingthe index (we will use all features of the machinery developed thus far, includingstabilization and duality); and (ii) asymptotics/boundary conditions related to thethree types of closed interval components 0/ : a compact interval, the entire real line,and the semi-infinite ray.

All of the forcing theorems are couched in a little braid-theoretic language:

Definition 44. The intersection number of two strands �� , �� of a braid � is the numberof crossings in the braid diagram, denoted

� �� of crossings of strands

The trivial braid on � strands is any braid (topological or discrete) whose braid diagram

has no crossings whatsoever,14 i.e., � �� , for all � � � �. The full-twist braid on �strands, is the braid of � connected components, each of which has exactly two crossings with

every other strand, i.e., �� for all � �� .

13This does not imply a �-periodic solution, but merely a braid diagram � of period �.14Recall, all our braids are positive, and crossings cannot be “undone”.


Among discrete braids of period two, the trivial braid and the full twist are dualsin the sense of �6.

9.1. Compact interval components. Let # be a regular energy level for which theset . / contains a compact interval component 0/ .

Theorem 45. Suppose that a twist system with compact 0/ possesses one or more closedcharacteristics which, as a discrete braid diagram, form a nontrivial braid. Then there existsan infinity of non-simple, geometrically distinct closed characteristics in 0/ .

In preparation for the proof of Theorem 45 we state a technical lemma, whose[short] proof may be found in [56].

Lemma 46. Let 0/ � �� , then there exists a �' � � such that

(1) � �� , � �� , and(2) � � �� , � � �� ,

for any � � � �' .

Proof of Theorem 45. Via Theorem 37, finding closed characteristics is equivalent tosolving the recurrence relation given by (42). Define the domains

� �� , �� 0 �/ � �� - � �� , �� 0 �/ � �� - � ��

Denote by� �� the set of

�� -periodic sequences ,� � - for which �� .By Lemma 46, choosing � � � � �' small enough forces the vector field � � �� tobe everywhere transverse to �� , making

� �� positively invariant for the inducedparabolic flow �

�.

By Lemma 43, one can lift the assumed solution(s) to a pair of period�� single-

stranded solutions to (42), �

and ��, satisfying ��

�� . Define the cones�� ,� � � �� - � �� ,� � � �� -

The combination of the facts ��

� � � � � �, Axiom (A1), and the behavior of� on �� implies that on the boundaries of the cones �� and �� the vector field� is everywhere transverse and pointing inward. Therefore, �� and �� are alsopositively invariant with respect to the parabolic flow �

�. Consequently, � �� has

global maxima ��

and ��

on �� and �� respectively. The maxima ��

and ��

have the property that � �� , � � � � �. As a braid diagram, � �,� ��

� �� - is a stationary skeleton for the induced parabolic flow �

�.

Having found the solutions ��

and ��

we now choose a compact interval 0 � 0/ ,such that the skeletal strands are all contained in 0 . In this way we obtain a properparabolic flow (circumventing boundary singularities) which can be extended to aparabolic flow on ��

�� REL � . Let �� REL � �� be the relative braid class with a period�� free strand � � ,� � - which links the strands �

and ��

with linking number �while satisfying � �� : see Fig. 12 below.

As an up-down braid class, �� REL � �� is a bounded proper braid class provided� � �� , and the Morse theory discussed in �7 and �8.3.3 then requires

the evaluation of the invariant H of the topological class �� REL ,� � -�. In this case,


��

��

��FIGURE 12. A representative braid class for the compact case: � �� .

since � ��REL � � � � � � ��

REL � � � � � ��REL � �, augmentation is not needed, and

H��REL � � � � H ��

REL � �. The nontriviality of the homotopy index H is given bythe following lemma, whose proof we delay until �10.

Lemma 47. The Conley homology of H��REL � � is given by:

(47) � �H� ��

� � � �� else.

In particular � � � �H� � �� .From the Morse theory of Corollary 42 we derive that for each � satisfying� � �

� � � �� there exist at least two distinct period-�� solutions of (42), which

generically are of index�� and

��

. In this manner, the number of solutionsdepends on � and � . To construct infinitely many, we consider � -fold coveringsof the skeleton � , i.e., one periodically extends � to a skeleton contained in � ��

� ,� � �

. Now � must satisfy � � �� . By choosing triples �� such

that �� are relative prime, we obtain the same Conley homology as above, andtherefore an infinity of pairs of geometrically distinct solutions of (42), which, viaLemma 43 and Theorem 37 yield an infinity of closed characteristics.

Note that if we set �� and �

�� , then the admissible ratios

�� for finding

closed characteristics are determined by the relation� � ��

� �� (48)

Thus if �

and ��

are maximally link, i.e. � � �� , closed characteristics exist, for allratios in � �� .9.2. Non-compact interval components: 0/ � � . On non-compact interval com-ponents, closed characteristics need not exist. An easy example of such a systemis given by the quadratic Lagrangian � � � �� , with � � �

.Clearly 0/ � � for all # � �, and the Lagrangian system has no closed charac-teristics for those energy levels. For � � �

the existence of closed characteristicsstrongly depends on the eigenvalues of the linearization around �. To treat non-compact interval components, some prior knowledge about asymptotic behavior ofthe system is needed. We adopt an asymptotic condition shared by most physicalLagrangians: dissipativity.


Definition 48. A second order Lagrangian system is dissipative on an interval component0/ � � if there exist pairs � � � � �� , with � � and � �� arbitrarily large, such that

(49) �� where ��

�$ � � �

�$� �� .

Dissipative Lagrangians admit a strong forcing theorem:

Theorem 49. Suppose that a dissipative twist system with 0/ � � possesses one or moreclosed characteristic(s) which, as discrete braid diagram in the period-two projection, formsa link which is not a full-twist (Definition 44). Then there exists an infinity of non-simple,geometrically distinct closed characteristics in 0/ .

Proof. After taking the � -fold covering of the period-two projection, the hypothe-ses imply the existence two sequences �

and �

�that form a braid diagram in �

��whose intersection number is not maximal, i.e. � ��

�� . Following

Definition 48, choose 0 � �� , with � � � � �� such that � � � � � � � �� for all ,and let

� �� and� �� be as in the proof of Theorem 45. Furthermore define the cone� �� ,� � � ��

� � � �� -

Since � � ��

� � � �� , the vector field � given by (42) is transverse to � � . More-over, the set � is contractible, compact, and � is pointing outward at the boundary� � . The set � is therefore negatively invariant for the induced parabolic flow �

�.

Consequently, there exists a global minimum � � in the interior of � . Define theskeleton � to be � �� ,� ��

� �� -.Consider the up-down relative braid class �� REL � �� described as follows: choose� to be a

�� -periodic strand with �� , such that � has intersectionnumber

�� with each of the strands �

� ��, � � � �

� � �� , as in Fig. 13. For � � �,�� REL � �� is a bounded proper up-down braid class. As before, in order to apply

the Morse theory of Corollary 42, it suffices to compute the homology index of thetopological braid class �� REL ,� � -�:

Lemma 50. The Conley homology of H��REL � � � is given by:

(50) � �H� ��

� � �� else

In particular � � � �H� � �� .By the same covering/projection argument as in the proof of Theorem 45, infin-

itely many solutions are constructed within the admissible ratios��

�� (51)

Theorem 49 also implies that the existence of a single non-simple closed character-istic yields infinitely many other closed characteristics. In the case of two unlinkedclosed characteristics all possible ratios in � �� can be realized.


��

��

��

FIGURE 13. A representative braid class for the case 0/ � � : � �� .

9.3. Half spaces 0/ � �� . The case 0/ � �� (or 0/ � �� ) shares much with

both the compact case and the the case 0/ � � . Since these 0/ are non-compact weagain impose a dissipativity condition.

Definition 51. A second order Lagrangian system is dissipative on an interval component0/ � �� if there exist arbitrarily large points � � �� such that

��

��

For dissipative Lagrangians we obtain the same general result as Theorem 45.

Theorem 52. Suppose that a dissipative twist system with 0/ � �� possesses one or moreclosed characteristics which, as a discrete braid diagram, form a nontrivial braid. Then thereexists an infinity of non-simple, geometrically distinct closed characteristics in 0/ .

Proof. We will give an outline of the proof since the arguments are more-or-lessthe same as in the proofs of Theorems 45 and 49. Assume without loss of generalitythat 0/ � �� . By assumption there exist two sequences �

and �

�which form

a nontrivial braid in �� , and thus � � � � � ��

�� . Defining the cone ��

as in the proof of Theorem 45 yields a global maximum ��

which contributes to theskeleton �� ,� ��

� �� -. Consider the braid class �� REL

�� defined by addingthe strand � such that � � � � �� and � links with the strands �

and �

�with linking

number � , � � �� .

Notice, in contrast to our previous examples, that �� REL�� is not bounded. In

order to incorporate the dissipative boundary condition that � � � � � is attracting,we add one additional strand �

�. Set �

��

�� for even, and �

�� , for odd.

As in the proof of Theorem 49 choose � � large enough such that � � � � �� . Let� � be a parabolic recurrence relation such that � � ��

� � � �. Using � � one can con-struct yet another recurrence relation � �� which coincides with � on �� REL

�� andwhich has �

�as a fixed point (use cut-off functions). By definition the skeleton

� � ,� � ��

� �� - is stationary with respect to the recurrence relation � �� .

Now let �� REL � �� be as before, with the additional requirement that �� . This defines a bounded proper up-down braid class. The homologyindex of the topological class �� REL ,� � -� is given by the following lemma (see�10).


Lemma 53. The Conley homology of H��REL � � � is given by

(52) � �H� ��

� � � �� else

In particular � � � �H� � �� .For the remainder of the proof we refer to that of Theorem 45.

��

��

��

��

FIGURE 14. A representative braid class for the case 0/ � �� : � �� .

9.4. A general multiplicity result and singular energy levels.

9.4.1. Proof of Theorem 1. Lagrangians for which the above mentioned dissipativityconditions are satisfied for all (non-compact) interval components at energy # , are

called dissipative at # .15 For such Lagrangians the results for the three different typesof interval components are summarized in Theorem 1 in �1. The fact that the pres-ence of a non-simple closed characteristic, when represented as a braid, yields a non-trivial, non-maximally linked braid diagram, allows us to apply all three Theorems45, 49, and 52, proving Theorem 1.

9.4.2. Singular energy levels. The forcing theorems in �9.1 - �9.3 are applicable forall regular energy levels provided the correct configuration of closed characteristicscan be found a priori. In this section we will discuss the role of singular energylevels; they may create configurations which force the existence of (infinitely) manyperiodic orbits. The equilibrium points in these singular energy levels act as seedsfor the infinite family of closed characteristics.

For singular energy levels the set . / is the union of several interval compo-nents, for which at least one interval component contains an equilibrium point. If� �$� �� at an equilibrium point � �, then such a point is called non-degenerateand is contained in the interior of an interval component. For applying our resultsof the previous section the nature of the equilibrium points may play a role.

15One class of Lagrangians that is dissipative on all its regular energy levels is described by

��

� � � �� pointwise in � � � � .


��

�

� � � ��

��

��

�

�

�

��

��

�

� � � ��

��

��

�

�

FIGURE 15. The gradient of � � for the case with two equilibria anddissipative boundary conditions. On the left, for # � �, the regions�� with the maxima and minima �� are depicted, as well as the

superlevel set ��

. On the right, for # � �� ' �, the region � , con-taining an index 1 point, is indicated.

9.4.3. 0/ � � . We examine the case of a singular energy level # � � such that0/ � � and 0/ contains at least two equilibrium points. At first glance one observesthat if the equilibria can be regarded as periodic orbits then Theorem 49 would apply:a regularisation argument makes this rigorous. Let # � � be the energy level inwhich ./ is the concatenation of three interval components �� ,i.e., the equilibria are � and �. We remark that the nature of the equilibrium pointsis irrelevant; there is a global reason for the existence of two unlinked periodic orbitin the energy levels # � �� ' � for some small

' � �. In these regular energy levelswe can apply Theorem 49, and a limit procedure ensures that the periodic solutionspersist to the degenerate energy level # � �, proving Theorem 2.

Recall from [56] that two equilibrium points imply the existence of maximum ��and minimum �� , both simple closed characteristics, see Fig. 15. Define the regions�� , �� -, and � � � , �� -, where �� is the point where the dissipativity condition (seeDefinition 48) is satisfied. Then �� and �� .

Since � � is a � �-function on int �� it follows from Sard’s theorem that there

exists a regular value ��

such that � �� . Consider

the connected component of the super-level set ,� � � �� - which contains �� . Theouter boundary of this component is a smooth circle and �� points inwards on thisboundary circle. Let �

�be the interior of the outer boundary circle in question. By

continuity it follows that there exists a positive constant ' such that �

�remains an

isolating neighborhood for # � �� ' �. In the following let # � �� ' � be arbitrary.


Define � � , �� - � �� . It follows fromthe properties of � (see �8) that � is again an isolating neighborhood, see Fig. 15. Itholds that � � � �� , and ,� ��

� � �� - forms a Morse decomposition. The Morse

relations (36) now give� � � ��

� � � � � � � � ��

where � � is a nonnegative polynomial. This implies that � ��

contains an index 1solution �

. We can now define � � � , �� - � �

�. In exactly the same way we find an index 1 solution � � � � � . Notice,

that by construction � �� . Theorem 49 now yields an infinity of closedcharacteristics for all � � # � ' . As described in �9.2 these periodic solutions canbe characterised by � and � , where �� is any pair of integers such that � � � and �

and � are relative prime (or � � � � �). Here

�� is the period of the solution �� %� and�� %� � � � � � �� %� � � � �.

In the limit # � � the solutions � and � �

may collapse onto the two equilibriumpoints (if they are centers). Nevertheless, the infinite family of solutions still exists inthe limit # � �, because the extrema of the associated closed characteristics may onlycoalesce in pairs at the equilibrium points. This follows from the uniqueness of theinitial value problem of the Hamiltonian system. Hence in the limit # � � the type�� of the periodic solution is conserved when we count extrema with multiplicityand intersections without multiplicity.

Note that when the equilibria are saddle-foci then � and � �

stay away from ��

inthe limit # � �. Extrema may still coalesce at the equilibrium points as # � �, butintersections are counted with respect to �

and � �. Finally, in the regular energy

levels # � �� ' � Theorem 49 provides at least two solutions of each type (except� � � � �

); in the limit # � � one cannot exclude the possibility that two solutionsof the same type coincide.

Remark 54. Theorem 2, proved in this subsection, is immediately applicable to theSwift-Hohenberg model (3) as described in �1. Notice that if the parameter � satsifies� � �

, then Theorem 2 yields the existence of infinitely many closed chararacteris-

tics at energy # � �� , and nearby levels. However from the physical point ofview it is also of interest to consider the case � �

. In that case there exists onlyone singular energy level, and one equilibrium point. This case can be treated withour theory, but the nature of the equilibrium point comes into play. If an equilibriumpoint is a saddle or saddle-focus no periodic solutions, it is possible that no addi-tional periodic orbits exist (see [55]). However, if the equilibrium point is a center aninitial non-simple closed characteristic can be found by analyzing an improper braidclass, which by, Theorem 1, then yields infinitely many closed characteristics. Thetechniques involved are very similar to those used in the present and subsequentsections. We will not work out the details here since this falls outside of the scope ofthis paper.

9.4.4. 0/ � �� , or 0/ � �� . The remaining cases are dealt with in Theorem 3. Wewill restrict the proof here to the case that 0/ contains an equilibrium point that is


a saddle-focus, as the center case can be treated as in [3].16 It also follows for theprevious that there is no real difference between 0/ being compact or a half-line. Forsimplicity we consider the case that 0/ is compact.

�

��

��

��

��

FIGURE 16. On the left the gradient of � � for the case of one saddle-focus equilibrium and compact boundary conditions. Clearly a sad-dle point is found in � . On the right the perturbation of one equi-librium to three equilibria.

Let us first make some preliminary observations. When � � is a saddle-focus, thenin # � � there exists a solution �

such that � � � � � � � . This follows from thefact that there is a point �� , � � � � �� , close to �� at which the vector �� points to the north-west (see Fig. 16 and [56]). This solution �

is a saddle point,its rotation number being unknown. The impression is that �� is a minimum(with � � �), and if � � were a periodic solution, then one would have a linked pair�� and �

to which one could apply Theorem 45. Since � � is a saddle-focus itdoes not perturb to a periodic solution for # � �. Hence we need to use a differentregularisation which conveys the information that � � acts as a minimum. The formof the perturbation that we have in mind is depicted in Fig. 16, where we have drawnthe “potential” � �� .

This idea can be formalized as follows. Choose a function � � ��' �� such that� � �� , � ��

for � � , � �� strictly decreases on � � � �� and � �� for� � �. Add a perturbation

� � �� $$ � ��' �� ! ��

� " ��16Indeed, for energy levels � � , sufficiently small, a small simple closed charecteristic exists due

to the center nature of the equilibrium point at � ; spectrum � �� , � � �. This small simple closed

charecteristic will have a non-trivial rotation number close to �� . The fact that the rotation number is non-

zero allows one to use the arguments in [3] to construct a non-simple closed characteristic. As a matter of

fact a linked braid diagram is created this way.


to the Lagrangian, i.e.�� , where �' � � �$� �� . The new Euler-

Lagrange equation near � � becomes� �$�� $��$� � �$�� $� �� ( �$�$ � �� )� ��

where all partial derivatives of � are evaluated at �� , and where � is the vector�� in phase space. Hence for all small � there are now two addi-tional equilibria near � �, denoted by �

� � �� and �� .Since ��

�� , the difference between

�# �� and

�# �� is� �� . To level this difference we add another small perturbation to�� of the form

� �� $$ � � �� $ � �� , i.e. �� , where where � � is chosen so

that �# �� # �� (of course �

� and �� shift slightly), and � � � � �� . Using the sameanalysis as before we conclude that a neighborhood of � � in the energy level # �

��

looks just like Fig. 15. In � � , �� - we find a

minimum. Choose an regular energy level # � slightly larger than �# �� # �� (with# � � � ��, such that the minimum in � persists. Taking this minimum and the orig-

inal � — which persists since we have only used small perturbations, preserving �

(see Fig. 16) as an isolating neighborhood — we apply Theorem 45.Finally, we take the limit � � �. The solutions now converge to solutions of the

original equation in the degenerate energy level. It follows that in the energy level# � � a solution of type �� exists, where the number of extrema has to be countedwith multiplicity since extrema can coalesce in pairs at � �.

10. COMPUTATION OF THE HOMOTOPY INDEX

Theorems 45, 49, and 52 hang on the homology computations of the homotopyinvariant for certain canonical braid classes (Lemmas 47, 50, and 53). Our strategy(as in, e.g., [3]) is to choose a sufficiently simple system (an integrable Hamiltoniansystem) which exibits the braids in question and to compute the homotopy index viaknowing the structure of an unstable manifold. By the topological invariance of thehomotopy index, any computable case suffices to give the index for any period

�.

Consider the first-order Lagrangian system given by the Lagrangian � � �� , where we choose � �� to be an even four-well potential, with� �� , and � ��

. The Lagrangian system �� defines an integrableHamiltonian system on � �

, with phase portrait given in Fig. 17.Linearization about bounded solutions � �� of the above Lagrangian system

yields the quadratic form

� �� ' ��

' � �� ' � � ��

which is strictly positive for all � � � �. For such choices of the time-1 map

defined via the induced Hamiltonian flow ��, i.e., �� $ � � ��

�� $ �, isan area preserving monotone twist map. The generating function of the twist map is


FIGURE 17. The integrable model in the �� plane; there are cen-ters at � ��

and saddles at �� .

given by the minimization problem

�� $ � %$� � ' � � ��

where � �� ,� � �� -.17 The func-tion �� is a smooth function on � �

, with � �� . The recurrence function� � �� satisfies Axioms (A1)-(A3), andthus defines an exact (autonomous) parabolic recurrence relation on � � ��. Wechoose the potential � such that the bounded solutions within the heteroclinic loopbetween � � �

and � � ��have the property that the period �� is an increasing

function of the amplitude � , and � � �� , as � � �.

This single integrable system is enough to compute the homotopy index of thethree families of braid classes in Lemmatta 47, 50, and 53 in �9.

We begin by identifying the following periodic solutions. Set � %� � ,� %�� -,� %�� , and �

� %� � ,� � %�� -, � � %�� . Let �

� �� be a solution of �� with

�� (minimum), ��

, and � �� ' � � , �' � � . For arbitrary

�this implies that

� � ��

��

��

��'�

�

where we choose so that�� . For � � �

set� �� and define � � �� ,� �� -, with� ��

�� , and � � � ,�� -, with � ��

�� ' �� , � � � � �. Clearly, � �� ,

for all�� .

Next choose �� , a solution of �� with �� (minimum), which oscil-lates around both equilibria �

and��

, and in between the equilibria � and��,

and � �� , � � � . As before

� � ��

��

17The strict positivity of the quadratic form � via the choice of

�yields a smooth family of hyperbolic

minimizers.


Let�� and choose �' � � � �

such that�'� �� ' � � �

Set � � � ,� �� -, � �� , and � � � ,� �� -, with � �� , � � � � �. For� � �� the solutions �� and �� have exactly

�� intersections. Therefore, if we choose sufficiently small, i.e.

�� is large, then it also holds that � �� %� �� %� � � �� .

Finally we choose the unique periodic solution � ��, with �� , ��

(minimum), and � �� , �� . Let � � �� , and choose �� ,

and consequently the amplitude � , so that�'�� ' � ��

��

The solution � is part of a circle hyperbolic of solutions � � �� , � � � �� . Define�� , with � �� ,� � ��-, where � � �� . As before, since

the intersection number of �� and � � is equal to

�� , it holds that � �� %� � � �

� , for sufficiently small. Moreover, � �� %� � � �� . From this point on is fixed. Wenow consider three different skeleta � .

I: � � ,� � %� � �� %� �� -. The relative braid class �� REL � �� is defined as follows:� � %�� %�� , and � links with the strands � � and � � with linking number � ,� �

� � � . The topological class �� REL ,� -� is precisely that of Lemma 47 [Fig. 12]and as such is bounded and proper.

II: � � ,� � %� �� %� � � � � � � �� -. The relative braid class �� REL � �� is defined as

follows: � � %�� %�� , � links with the strands � � and � � with linking number� , � �

� � � , and � links with � � with linking number � . The topological class�� REL ,� -� is precisely that of Lemma 53 [Fig. 14] and as such is bounded andproper.

III: � � ,� � %� �� -. The relative braid class �� REL � �� is defined as

follows: � � %�� , � links with the strands � � and � � with linking number � , and �links with � � and � � with linking number � . The topological class �� REL ,� -� is notbounded [Fig. 18[right]]. The augmentation of this braid class is bounded.

Since for I and II the topological classes are bounded and proper, the invariant H

is independent of period of the chosen representative, and can be easily computedfrom the above integrable model.

For cases I and II the closure of collection of topologically equivalent braid classesis an isolating neighborhood for the parabolic flow �

�induced by the recurrence

relation � � � � (defined via �� ). The invariant set is given by the normallyhyperbolic circle ,� ��-�� . For this reason the index H can be computed via theconnected component that contains the critical circle; we denote this neighborhoodby � . The Conley index of � can be determined via computing � $ �,� ��-�, the un-stable manifold associated to this circle. This computation is precisely that appearingin the calculations of [3, pp. 372]: � $ �,� ��-� is orientable and of dimension

�� , and

thus

(53) H��REL � � � � �� (� � � ��) � (� � ,� -) � � ��


The Conley homology is given by � �H� � � for� � �

� � � �� , and � �H� � �elsewhere. This completes the proofs of the Lemmas 47 and 53.

Consider case III. It holds that

�� REL � � � (� �� REL � ) �� The discrete class for period

�� is bounded, but for periods� � �� this is not the case.

However, by augmenting the braid, we obtain from (24) that

H��REL � � � H ��

REL � � � �where � � � � � ,� %� ��

%� -. Since the topological class �� REL ,� � -� is boundedand proper, we may use the previous calculations to conclude that

H��REL � � � � �� ,� -� � � ��

Our motivation for this computation is to complete the proof of Lemma 50. Let

�

FIGURE 18. The augmentation of the braid from Lemma 50 [left] isthe dual of the type III braid [right].�� REL � � � denote the period

�� braid class described by Fig. 13, with linking num-bers denoted by � � and � �, and let �� REL � � denote a type-III braid of period

�� . Then,it is straightforward to see (as illustrated in Fig. 18) that, for � � � � � and � � � �� ,

(54) �� REL �� ! �� REL �� "

Lemma 50 gives the index for the augemented class �� REL , � � � -�, which isbounded and proper as a topological class. The above considerations allow us tocompute the homology of H��

REL � � � � via Theorem 30:

(55)

� � (H�� REL � � � �) �� H �� REL

�� H��

REL � �� H��REL � � ��

� � � �� else�� else

The linking numbers � � and � � are exactly those of Lemma 50, completing the proofthereof.


11. POSTLUDE

11.1. Extensions and questions: dynamics. There are several ways in which thebasic machinery introduced in this paper can be generalized to other dynamical sys-tems.1. Periodicity in the range. Although we consider the anchor points to be in � , one mayjust as well constrain the anchor points to lie in �

and work in the universal cover.Such additional structure is used in the theory of annulus twist maps [3, 8, 34, 35, 40].All of our results immediately carry over to this setting. We note that compact-typeboundary conditions necessarily follow.

2. Aperiodic dynamics. In the spirit of this paper one can also extend to the theory toinclude braid diagrams with “infinite length” strands. To be more precise, considerbraid diagrams on the infinite 1-d lattice, and omit the spatial periodicity of the recur-rence function � � �� . In this context several compactness issue come into play.To name a few: (a) the parabolic flows generated by aperiodic recurrence relationsno longer live on a finite dimensional space but on the infinite dimensional space�� . See [3] for a case similar to this; (b) the Conley index needs to be replaced a

priori with an infinite dimensional analogue such as developed by Rybakowski [50].However, if one considers braid diagrams with finite word metric, the stabilizationtheory of �5 should allow one to define the necessary invariants via the finite dimen-sional theory in this paper. This is not unlike the procedure one can follow in thetreatment of parabolic PDE’s [2].

3. Fixed boundary conditions. Our decision to use closed braid diagrams is motivatedby applications in Lagrangian systems; however, one can also fix the end pointsof the braid diagrams. In this setting one can define a braid invariant in the samespirit as is done for closed braids. The proof of stabilization is not sensitive to thetype of boundary conditions used. Such an extension of the theory to include fixedendpoints is useful in applications to parabolic PDE’s [2].

4. Traveling waves and period orbits. The stationary solutions we find in this paperare but the beginnings of a dynamical skeleton for the systems considered. The nextlogical step would be to classify connecting orbits between stationary solutions: sev-eral authors (e.g., [38]) have considered these problems analytically in the context oftraveling wave phenomena in monotone lattice dynamics. There is a precedent ofusing Conley indices to prove existence theorems for connecting orbits (e.g., [45]):we anticipate that such applications are possible in our setting. One could as wellallow the skeletal strands to be part of a periodic motion (in the case of non-exact re-currence relations). In this setting one could look for both fixed points and periodicsolutions of a given braid class.

5. Long-range coupling. Assume that the recurrence relations � � are functions of theform � � �� for some � . Even if a strong monotonicity condition holds,�� for all � �� the proof of Proposition 11 still encounters a difficulty: twostrands with a simple (codimension-one) tangency can have enough local crossingsto negate the parabolic systems’ separation. Such monotone systems do exhibit an


ordering principle [4, 24] (initially nonintersecting strands will never intersect), butadditional braiding phenomena is not automatically present.

6. Higher-dimensional lattice dynamics. In parabolic PDE’s of spatial dimension greaterthan one, the straightforward generalization of the lap number (number of con-nected components of a suitable intersection) does not obey a monotonicity property.Perhaps for certain classes of recurrence relations, there is enough isolation forced toseparate a generalized form of braiding (immersion classes of graphs of lattices).This remains a challenging problem.

7. Arbitrary second-order Lagrangians. Our principal dynamical goal is to prove ex-istence theorems for periodic orbits with a minimal amount of assumptions, partic-ularly “genericity” assumptions (which are, in practice, rarely verifiable). To thisend, we have been successful for second-order Lagrangians modulo the twist as-sumption. Although this assumption is provably satisfied in numerous contexts, webelieve that it is not, strictly speaking, necessary. Its principal utility is in the reduc-tion of the problem to a finite-dimensional recurrence relation. We believe that theforcing results proved in �8 are valid for all second-order Lagrangian systems. Wepropose that a version of the curve-shortening techniques of Angenent [5] shouldyield a homotopy index for smooth curves to which our forcing theorems apply.

8. Stabilization, smooth curves, and uniform parabolic PDEs. Theorem 19 suggestsstrongly that the homotopy index for discretized braids extends to and agrees withan analogous index for parabolic dynamics on spaces of continuous (or perhapssmooth) curves via parabolic PDE’s, in which case Rybakowski’s version of the Con-ley index has to be employed. Using the standard spatial discretization of parabolicPDE’s in the spirit of Fiedler-Rocha [19] one should be able to reformulate the investi-gation of the invariant dynamics of parabolic PDE’s in terms of parabolic recurrencerelations and its associated invariants. In this sense we believe that our theory willalso allow one to formulate a Morse-type forcing theory for parabolic PDE’s. Theprincipal advantage of this approach is that the invariant for the “continuous” caseis defined, due to stabilization, via a finite dimensional configuration. Stabilizationallows one to compute the invariant via its coarsest possible representation, usingexisting computational homology algorithms [2].

11.2. Extensions and questions: topology. The homotopy index is, as a topologicalinvariant of braid pairs, utterly useless. Nevertheless, there is topological meaningintrinsic to this index, the precise topological interpretation of which is as yet un-clear. One observes that the index captures some Morse-theoretic data about howthe space of embedded monotonic closed curves in a solid torus sits inside of the im-mersed monotonic closed curves. Any topological interpretation is certainly relatedto linking data of the free strands with the skeleton, as evidenced by the examples inthis paper. Though the total amount of linking should provide some upper boundto the dimension of the homotopy index, linking numbers alone are insufficient tocharacterize the homotopy index.

We close with several related questions about the homotopy index itself.


1. Realization. It is clear that given any polynomial in , there exists a braid pairwhose homological Poincare polynomial agrees with this. [Idea: take Example 3 of�4 and stack disjoint copies of the skeleton vertically, using as many free copies andstrands as necessary to obtain the desired homology.] Can a realization theorem beproved for the homotopy index itself? As a first step to this, consider replacing thereal coefficients in the homological index with integral coefficients. Does torsion everoccur? We believe not, with the possible exception of a

� � torsion.

2. Product formulas and the braid group. Perhaps the most pressing problem for the ho-motopy index is to determine a product formula for the concatenation of two braidswith compatible skeleta. This would eliminate the need for computing the index viacontinuation to an integrable model system as in �10.

However, since we work on spaces of closed braids, a product formula is, strictlyspeaking, not well-defined. The group structure on the braid group �� does notextend naturally to a group structure on conjugacy classes: where one “cuts open”the braid to effect a gluing can change the resulting braid class dramatically. Theone instance in which a product operation is natural is a power of a closed braid.Here, splitting the closed braid to an open braid and concatenating several copiesthen reclosing yields equivalent closed braids independent of the representative ofthe conjugacy class chosen.

Such a product/power formula, in conjunction with numerical methods of indexcomputation effective in moderately low dimensions, would allow one to computemany invariants.

3. Improper and unbounded classes In certain applications one also needs to deal withimproper braid classes �� REL � �. To such classes one can also assign an index. Theinterpretation of the index as a Morse theory will not only depend on the topologicaldata, but also on the behavior of the flow �

�at

��. The simplest case is when

�� consists of finitely many points. This for example happens when � consists ofonly one strand. The homotopy index is then defined by the intrinsic definition in(14). The interpretation of the index and the associated Morse theory depends on thelinearization ��

� �� . The definition of the index in case of more complicatedsets

�� and the Morse theoretic interpretation will be subject of future study.Similar considerations hold for unbounded classes.

5. General braids. The types of braids considered in this paper are positive braids.Naturally, one wishes to extend the ideas to all braids; however, several compli-cations arise. First, passing to discretized braids is invalid — knowing the anchorpoints is insufficient data for reconstructing the braid. Second, compactness is trou-blesome — one cannot merely bound braid classes via augmentation.

We can model general braids dynamically using recurrence relations with nearestneighbor coupling allowing “positive”, or “negative” interaction. This idea appearsin the work of LeCalvez [34, 35] and can be translated to our setting via a change ofvariables — coordinate flips — of which are duality operator

�is a particular exam-

ple. However the compactness and discretization issues remain. LeCalvez works in


the setting of annulus maps, where one can circumvent these problems: the generalsetting is more problematic.

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APPENDIX A. CONSTRUCTION OF PARABOLIC FLOWS

In this appendix, we construct particular parabolic flows on braid diagrams inorder to carry out the continuation arguments for the well-definedness of the Conleyindex for proper braid diagrams. The constructions are explicit and are generated byrecurrence relations � � �� , with � �� , which are of the form:

� � �� (56)

with � � � �� , and � ��

� �� , � �

�

�� for all �� . By

definition, such recurrence relations are parabolic.

Lemma 55. For any � � �� there exists a parabolic flow ��

under which � is stationary.

Proof. In order to have �� , the sequences ,�� - need to satisfy

� � �� , for some parabolic recurrence relation. We will construct� by specifying the appropriate functions ,� � � �� - as above. In the constructionto follow, the reader should think of the anchor points ,� �� - of the fixed braid � asconstants.

For each such that the values ,� �� -� are distinct, one may choose � � �� ,�� , and

� �� to be any � function which interpolates the defined values � ��

This generates the desired parabolic flow.In the case where there are several strands � � � � � �� for which � �� are

all equal, the former construction is invalid: � is not well-defined. According to

Definition 4, we have for each ��

This implies that if we order the ,�� -� so that� � ��

then the corresponding sequence ,� �� -� satisfies� �� ¿From Lemma 56 below, there exist increasing functions � and � such that

� �� Define � � and �� in the following manner: set � � �� and �� .Thus it follows that there exists a well defined value � �� (� �� )which is independent of � . For any other strands � �, repeat the procedure, defining

the slices � � �� , �� and the points � �� , choosing new functions � and �


if necessary. To extend these functions to global functions � � �� and � � �� , sim-ply perform a �

homotopy in � without changing the monotonicity in the � and variables: e.g., on the interval �� , choose a monotonic function � �� for which

� �� and � �� . Then set

� � �� Such a procedure, performed on the appropriate �-intervals, yields a smooth �-monotonic interpolation. Repeat with � � �� . Finally, choose any function

� ��which smoothly interpolates the preassigned values. These choices of � �, �� and

�give the desired recurrence relation, and consequently the parabolic flow �

�.

Lemma 56. Given two sequences of increasing real numbers � � �� and� � �� , there exist a pair of strictly increasing functions � and � such that

(57) � �� Proof. Induct on � , noting the triviality of the case � � �

. Given increasing

sequences �� and �� , choose functions � and � which satisfy (57) for� � � � : this is a restriction on � and � only for values in �� since outsideof this domain the functions can be arbitrary as long as they are increasing. Thus,modify � and � outside this interval to satisfy

� ��

for some fixed constant � � �. These functions satisfy (57) for all � and�

.

Lemma 57. For any pair of equivalent braids �� , there exists a path � ��in � �� and a continuous family of parabolic flows �

�� , such that �� , for all

� �� .Proof. Given � any point in � �� , consider any parabolic recurrence relation � �

which fixes � and which is strictly monotonic in � and . From the proof of Lemma 55,� � exists. For every braid � � sufficiently close to � , there exists

�a near-identity

diffeomorphism of � �� which maps � to � �. The recurrence relation � � � ��fixes � �

and is still parabolic since�

cannot destroy monotonicity. Choosing a short smoothpath

� �of such diffeomorphisms to ID proves the lemma on small neighborhoods in� �� , which can be pieced together to yield arbitrary paths.

SCHOOL OF MATHEMATICS AND CDSNS, GEORIGA INSTITUTE OF TECHNOLOGY, ATLANTA GA,

30332-0160 USA

MATHEMATICS INSTITUTE, LEIDEN UNIVERSITY, 2300 RA LEIDEN, NETHERLANDS

MATHEMATICS INSTITUTE, LEIDEN UNIVERSITY, 2300 RA LEIDEN, NETHERLANDS; AND CDSNS,

GEORGIA INSTITUTE OF TECHNOLOGY, ATLANTA GA, 30332-0160 USA

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