A C^{0} counterexample to the Arnold conjecture

A counterexample to the Arnold conjecture

Lev Buhovsky, Vincent Humilière, Sobhan Seyfaddini
July 13, 2019
Abstract

The Arnold conjecture states that a Hamiltonian diffeomorphism of a closed and connected symplectic manifold must have at least as many fixed points as the minimal number of critical points of a smooth function on .

It is well known that the Arnold conjecture holds for Hamiltonian homeomorphisms of closed symplectic surfaces. The goal of this paper is to provide a counterexample to the Arnold conjecture for Hamiltonian homeomorphisms in dimensions four and higher.

More precisely, we prove that every closed and connected symplectic manifold of dimension at least four admits a Hamiltonian homeomorphism with a single fixed point.

11footnotetext: This author also uses the spelling “Buhovski” for his family name.

1 Introduction and main results

1.1 The Arnold conjecture

Let denote a closed and connected symplectic manifold. This paper is concerned with the celebrated conjecture of Arnold on fixed points of Hamiltonian diffeomorphisms.

Conjecture (Arnold).

A Hamiltonian diffeomorphism of must have at least as many fixed points as the minimal number of critical points of a smooth function on .

What makes this conjecture so remarkable is the large number of fixed points predicted by it. This is often interpreted as a manifestation of symplectic rigidity. In contrast to Arnold’s conjecture, the classical Lefschetz fixed-point theorem cannot predict the existence of more than one fixed point for a general diffeomorphism. Ever since its inception, this simple and beautiful conjecture has been a powerful driving force in the development of symplectic topology. The most important breakthrough towards a solution of this conjecture came with Floer’s invention of what is now called Hamiltonian Floer homology which established a variant of the Arnold conjecture on a large class of symplectic manifolds [floer86, floer88, floer89]. The above version of the Arnold conjecture has been established on symplectically aspherical111 is said to be symplectically aspherical if and , the first Chern class of , both vanish on . manifolds by Rudyak and Oprea in [rudyak-oprea] who built on earlier works of Floer [floer89b] and Hofer [hofer]. We should mention that prior to the discovery of Floer homology, the Arnold conjecture was proven by Eliashberg [eliashberg] on closed surfaces, by Conley and Zehnder [conley-zehnder] on higher dimensional tori, and by Fortune and Weinstein [fortune, FW] on complex projective spaces.

1.2 The Arnold conjecture and Hamiltonian homeomorphisms

Throughout this paper we will denote by and the groups of symplectic and Hamiltonian diffeomorphisms of , respectively. As is nowadays standard, we call symplectic homeomorphism any homeomorphism which can be written as a uniform limit of symplectic diffeomorphisms; the set of all symplectic homeomorphisms is denoted by ; see Section 2.1.

As a first attempt at defining Hamiltonian homeomorphisms, we will say that a homeomorphism of is a Hamiltonian homeomorphism if it can be written as a uniform limit of Hamiltonian diffeomorphisms. This class of homeomorphisms has been studied very extensively, from a dynamical point of view, in the case of closed surfaces222This is precisely the class of area preserving homeomorphisms with vanishing mean rotation vector.. For example, Matsumoto [matsumoto], building on an earlier paper of Franks [franks], has proven that Hamiltonian homeomorphisms of surfaces satisfy the Arnold conjecture. An important development in the study of Hamiltonian homeomorphisms of surfaces has been Le Calvez’s theory of transverse foliations [lecalvez05] which has not only proven the Arnold conjecture but also the Conley conjecture on periodic points of these homeomorphisms [lecalvez06].

In striking contrast to the rich theory in dimension two, there are virtually no results on fixed point theory of Hamiltonian homeomorphisms in higher dimensions. Indeed, none of the powerful tools of surface dynamics seem to generalize in an obvious manner to dimensions higher than two. Our first theorem proves that in fact one can not hope to prove the Arnold conjecture in higher dimensions.

Theorem 1.

Every closed and connected symplectic manifold of dimension at least 4 admits a Hamiltonian homeomorphism with a single fixed point.

This theorem might suggest that, in dimensions higher than two, one should search for a different notion of Hamiltonian homeomorphisms. Indeed, such notion does exist within the field of continuous, or , symplectic topology. Motivated in part by developing a continuous analogue of smooth Hamiltonian dynamics, Müller and Oh have suggested an alternative, more restrictive, definition for Hamiltonian homeomorphisms; see Section 2.1 for the precise definition. From this point onward by Hamiltonian homeomorphisms we will mean those homeomorphisms of prescribed by Definition 7. We denote the set of all Hamiltonian homeomorphisms by .

The group has met some success. Indeed, recent results in -symplectic topology [HLS12, HLS13, HLS14] have demonstrated that Hamiltonian homeomorphisms inherit some of the important dynamical properties of smooth Hamiltonian diffeomorphisms; see Theorem 8. Furthermore, they have played a key role in the development of -symplectic topology over the past several years. However, our main theorem proves that the Arnold conjecture is not true for this notion of Hamiltonian homeomorphisms either. In fact, as we will explain below, it shows that there is no hope for proving the Arnold conjecture, as formulated above, for any alternate definition of Hamiltonian homeomorphisms which satisfies a minimal set of requirements.

Theorem 2 (Main Theorem).

Let denote a closed and connected symplectic manifold of dimension at least . There exists with a single fixed point. Furthermore, can be chosen to satisfy either of the following additional properties.

  1. Let be a normal subgroup of which contains as a proper subset. Then, .

  2. Let denote the unique fixed point of . Then, is a symplectic diffeomorphism of .

A few remarks are in order. First, we should point out that every Hamiltonian homeomorphism possesses at least one fixed point. This is because a Hamiltonian homeomorphism is by definition a uniform limit of Hamiltonian diffeomorphisms and it is a non-trivial fact that a Hamiltonian diffeomorphism has at least one fixed point. 333 This fact is an immediate consequence of Floer’s proof of the Arnold conjecture; see also [Grom].

Second, we remark that it is well known that is a normal subgroup of . Hence, it is reasonable to expect that any alternative candidate, say , for the group of Hamiltonian homeomorphisms should contain and be a normal subgroup of . It is indeed the case that . Therefore, the first property in the above theorem states that there is no hope of proving the Arnold conjecture for any alternate definition of Hamiltonian homeomorphisms.

Lastly, with regards to the second property, we point out that it is natural to expect to have at least one non-smooth point. Indeed, since Hamiltonian Floer homology predicts that a Hamiltonian diffeomorphism can never have as few as one fixed point, our homeomorphism must necessarily be non-smooth on any symplectic manifold with the property 444 It can be shown that this property holds for closed symplectic surfaces, as well as for the standard and monotone . that .

1.3 Does there exist a fixed point theory for Hamiltonian homeomorphisms?

In Gromov’s view [Grom2], symplectic topology is enriched by a beautiful interplay between rigidity and flexibility. Recent results, such as [ops, HLS13, BuOp], have demonstrated that this contrast between rigidity and flexibility permeates, in a surprising fashion, to symplectic topology as well. Symplectic rigidity manifests itself when symplectic phenomena survives under limits; see [cardin-viterbo, entov-polterovich, buhovsky, ops, HLS13] for some examples. On the other hand, there exist instances where passage to limits results in spectacular loss of rigidity and prevalence of flexibility; see [BuOp] for an example.

The main theorem of our paper tells us that fixed points of Hamiltonian diffeomorphisms become completely flexible under limits. It is interesting to contrast this prevalence of flexibility with the strong rigidity results of Franks [franks], Matsumoto [matsumoto], and Le Calvez [lecalvez05, lecalvez06] in the two-dimensional setting. Given the main result of this article, one might conclude that there is no hope of developing a sensible fixed point theory for any notion of Hamiltonian homeomorphisms in dimensions greater than two. However, there exist some interesting open questions which remain unanswered.

The most prominent open question is that of the Conley conjecture which in its simplest form states that a Hamiltonian diffeomorphism on an aspherical symplectic manifold has infinitely many periodic points. This conjecture was proven by Hingston [hingston] on tori and Ginzburg [ginzburg] in the more general setting. As mentioned earlier, the Conley conjecture has been proven for Hamiltonian homeomorphisms of surfaces by Le Calvez [lecalvez05, lecalvez06]. We have not been able to construct a counterexample to the Conley conjecture in higher dimensions.

The second question relates to the theory of spectral invariants. For the sake of simplicity, we limit this discussion to the case of symplectically aspherical manifolds. In that case, the theory of spectral invariants, which was introduced by Viterbo, Oh and Schwarz [viterbo, Oh05b, schwarz], associates to each smooth Hamiltonian , a collection of real numbers where denotes the singular homology of . These numbers are referred to as the spectral invariants of and they correspond to critical values of the associated action functional. Hence, the number of distinct spectral invariants of a Hamiltonian gives a lower bound for the number of fixed points of the time– map .

Recall that the cup length of is defined by . Combining techniques from Hamiltonian Floer theory and Lusternik-Shnirelman theory, Floer [floer89b] and Hofer [hofer] proved that if a Hamiltonian diffeomorphism, of an aspherical symplectic manifold , has fewer spectral invariants than the cup length of , then it must have infinitely many fixed points; see also [howard].

It is well-known that one can associate spectral invariants to any continuous Hamiltonian function; see for example [muller-oh]. In an interesting twist, it turns out that the Hamiltonian homeomorphism that we construct in the proof of Theorem 2 is generated by a continuous Hamiltonian which has at least as many distinct spectral invariants as . Hence, we see that the correspondence between spectral invariants and fixed points breaks down in the continuous setting. See Remark 3.1. This leads us to the following question:

Question 3.

Suppose that is a continuous Hamiltonian with fewer spectral invariants than the cup length of . Does , the time–1 map of the flow of , have infinitely many fixed points?

A positive answer to this question could be interpreted as a version of the Arnold conjecture.

We end this section with a brief discussion which will add to the importance of the above question. This concerns the theory of barcodes, or persistence modules. As pointed out in [PS14], Hamiltonian Floer theory allows one to associate a so-called barcode to any smooth Hamiltonian; see also [Barann, LNV, UZ]. Barcodes can be viewed as generalizations of spectral invariants. The barcode of a smooth Hamiltonian encodes all the information contained in the filtered Floer homology of that Hamiltonian. In the same way that one can associate spectral invariants to a continuous function, one can also associate a barcode to a continuous Hamiltonian function. In yet another interesting twist, it turns out that the Hamiltonian homeomorphism of Theorem 2 can be generated by a continuous Hamiltonian which has the same barcode as a -small Morse function. See Remark 3.1.

1.4 A brief outline of the construction

Construction of the homeomorphism , as prescribed in Theorem 2, takes place in two major steps. The first step, which is the more difficult of the two, can be summarized in the following theorem.

Theorem 4.

Let denote a closed and connected symplectic manifold of dimension at least . There exists and an embedded tree such that

  1. is invariant under , i.e. ,

  2. All of the fixed points of are contained in ,

  3. is smooth in the complement of .

For the proof of Theorem 2, we will in fact need a refined version of the above result; see Theorem 12. The proof of this theorem forms the technical heart of our paper. An important ingredient used in the construction of the invariant tree is a quantitative -principle for curves. Quantitative -principles have recently been introduced to symplectic topology by Buhovsky and Opshtein and have had numerous fascinating applications; see [BuOp]. We should point out that having dimension at least four is used in a crucial way in the proof of this theorem. In fact, the second step of the construction, which is outlined below, can be carried out on surfaces as well.

The second major step of our construction consists of “collapsing” the invariant tree to a single point which will be the fixed point of our homeomorphism . Here is a brief outline of how this is done. Fix a point . We construct a sequence such that converges uniformly to a map with the following two properties:

  1. ,

  2. is a symplectic diffeomorphism from to

Note that the first property implies that is not a 1-1 map and hence, the sequence is not convergent. Define as follows: and

It is not difficult to see that is the unique fixed point of . Indeed, on , the map is conjugate to which is fixed point free by construction.

By picking the above sequence of symplectomorphisms carefully, it is possible to ensure that the sequence of conjugations converges uniformly to . The uniform convergence of to relies heavily on the invariance of the tree and it occurs despite the fact that the sequence diverges. The details of this are carried out in Section 3.1. It follows that can be written as the uniform limit of a sequence of Hamiltonian diffeomorphisms.

It is not difficult to see that is smooth on the complement of its unique fixed point. However, proving that is a Hamiltonian homeomorphism and that it satisfies the first property listed in Theorem 2 requires some more work; see Section 3.1.

1.5 Organization of the paper

In Section 2, we recall some preliminary results from symplectic geometry. Symplectic and Hamiltonian homeomorphisms are introduced in Section 2.1. In Section 2.2, we introduce a quantitative -principle for curves which plays an important role in our construction.

In Section 3.1, we prove that the existence of a Hamiltonian homeomorphism with an invariant tree, as described in Theorem 4, implies the main theorem of the paper. In Section 3.2, we prove the existence of a Hamiltonian homeomorphism as described in Theorem 4, assuming a technical and important result: Theorem 24. Section 3.3, which occupies the rest of the paper, is dedicated to the proof of Theorem 24. This section contains the technical heart of the paper.

1.6 Acknowledgments

We would like to thank Yasha Eliashberg, Viktor Ginzburg, Helmut Hofer, Rémi Leclercq, Frédéric Le Roux, Patrice Le Calvez, Emmanuel Opshtein, Leonid Polterovich, and Claude Viterbo for fruitful discussions.

LB: The research leading to this project began while I was a Professeur Invité at the Université Pierre et Marie Curie. I would like to express my deep gratitude to the members of the Institut Mathématique de Jussieu - Paris Rive Gauche, especially the team Analyse Algébrique, for their warm hospitality.

SS: I would like to thank the members of the Department of Mathematics at MIT, where a part of this project was carried out, for providing a stimulating research environment.

LB was partially supported by the Israel Science Foundation grant 1380/13, by the Alon Fellowship, and by the Raymond and Beverly Sackler Career Development Chair. VH was partially supported by the Agence Nationale de la Recherche, projets ANR-11-JS01-010-01 and ANR-12-BS020-0020. SS was partially supported by the NSF Postdoctoral Fellowship Grant No. DMS-1401569.

2 Preliminaries from -symplectic topology

In this section we introduce some of our notation and recall some of the basic notions of –symplectic geometry. In Section 2.1 we give precise definitions for symplectic and Hamiltonian homeomorphisms. In Section 2.2 we state a quantitative h-principle for curves which will play a crucial role in the proof of Theorem 2.

2.1 Symplectic and Hamiltonian homeomorphisms

Throughout the rest of this paper, will denote a closed and connected symplectic manifold whose dimension is at least . We equip with a Riemannian distance . Given two maps we denote

We will say that a sequence of maps , converges uniformly, or –converges, to , if as . Of course, the notion of –convergence does not depend on the choice of the Riemannian metric.

Recall that a symplectic diffeomorphism is a diffeomorphism such that . The set of all symplectic diffeomorphisms of is denoted by .

Definition 5.

A homeomorphism is said to be symplectic if it is the –limit of a sequence of symplectic diffeomorphisms. We will denote the set of all symplectic homeomorphisms by .

The Eliashberg–Gromov theorem states that a symplectic homeomorphism which is smooth is itself a symplectic diffeomorphism. We remark that if is a symplectic homeomorphism, then so is . In fact, it is easy to see that forms a group.

Remark 6. More generally, one can define a symplectic homeomorphism to be a homeomorphism which is locally a –limit of symplectic diffeomorphisms; see [BuOp] for further details.

Recall that a smooth Hamiltonian gives rise to a Hamiltonian flow . A Hamiltonian diffeomorphism is a diffeomorphism which arises as the time-one map of a Hamiltonian flow. The set of all Hamiltonian diffeomorphisms is denoted by ; this is a normal subgroup of . We next define Hamiltonian homeomorphisms as introduced by Müller and Oh [muller-oh].

Definition 7 (Hamiltonian homeomorphisms).

Denote by an open (possibly not proper) subset of . Let be an isotopy of which is compactly supported in . We say that is a hameotopy, or a continuous Hamiltonian flow, of if there exists a sequence of smooth and compactly supported Hamiltonians such that:

  1. The sequence of flows –converges to , uniformly in , i.e. as .

  2. The sequence of Hamiltonians converges uniformly to a continuous function , i.e. as , where denotes the sup norm. Furthermore,

We say that generates , denote , and call a continuous Hamiltonian.

A homeomorphism is called a Hamiltonian homeomorphism if it is the time– map of a continuous Hamiltonian flow. We will denote the set of all Hamiltonian homeomorphisms by .

It is not difficult to check that is a normal subgroup of .

A continuous Hamiltonian generates a unique continuous Hamiltonian flow; see [muller-oh]. Conversely, Viterbo [viterbo06] and Buhovsky–Seyfaddini [buhovsky-seyfaddini] (see also [HLS12]) proved that a continuous Hamiltonian flow has a unique (up to addition of a function of time) continuous generator.

One can easily check that generators of continuous Hamiltonian flows satisfy the same composition formulas as their smooth counterparts. Namely, if is a continuous Hamiltonian flow, then is a continuous Hamiltonian flow generated by ; given another continuous Hamiltonian flow , the isotopy is also a continuous Hamiltonian flow, generated by .

We will finish this section by recalling an important dynamical property of continuous Hamiltonian flows. Recall that a submanifold of a symplectic manifold is called coisotropic if for all , where denotes the symplectic orthogonal of . For instance, hypersurfaces and Lagrangians are coisotropic. A coisotropic submanifold carries a natural foliation which integrates the distribution ; is called the characteristic foliation of .

Assume that is a closed and connected coisotropic submanifold of and suppose that is a smooth Hamiltonian. The following is a standard and important fact which relates Hamiltonian flows to coisotropic submanifolds: is a function of time if and only if (preserves and) flows along the characteristic foliation of . By flowing along characteristics we mean that for any point and any time , , where stands for the characteristic leaf through .

The following theorem, which was proven in [HLS13], establishes the aforementioned property for continuous Hamiltonian flows.

Theorem 8.

Denote by a closed and connected coisotropic submanifold and suppose that is a continuous Hamiltonian flow. The restriction of to is a function of time if and only if preserves and flows along the leaves of its characteristic foliation.

The above theorem indicates that continuous Hamiltonian flows inherit some of the fundamental dynamical properties of their smooth counterparts. In light of this, it would some reasonable to expect the Arnold conjecture to hold for Hamiltonian homeomorphisms. But of course, Theorem 2 tells us that this is quite far from reality.

2.2 A quantitative -principle for curves

Quantitative –principles were introduced in [BuOp], where they were used to construct interesting examples of symplectic homeomorphisms. We will need the following quantitative –principle for curves in the construction of our counterexample to the Arnold conjecture.

Proposition 9 (Quantitative -principle for curves).

Denote by a symplectic manifold of dimension at least 4. Let . Suppose that are two smooth embedded curves such that

  1. and coincide near and ,

  2. there exists a homotopy, rel.end points, from to under which the trajectory of any point of has diameter less than , and the symplectic area of the element of defined by this homotopy has area .

Then, for any , there exists a compactly supported Hamiltonian , generating a Hamiltonian isotopy such that

  1. vanishes near and (in particular, fixes and near the extremities),

  2. ,

  3. for each and ,

  4. is supported in a -neighborhood of the image of .

The existence of a Hamiltonian satisfying only properties 1 and 2 is well known. The aspect of the above proposition which is non-standard is the fact that can be picked such that properties 3 and 4 are satisfied as well. We should point out that the above proposition is a variation of a quantitative –principle for discs which appeared in Theorem 2 of [BuOp]. The proof we will present is an adaptation of the arguments therein. In fact, the quantitative –principle of [BuOp] is considerably more difficult to prove than our proposition and for this reason we will only sketch an overview of the proof of the above proposition.

Proof of Proposition 9.

First, by a slight Hamiltonian perturbation of via a Hamiltonian diffeomorphism generated by a -small Hamiltonian function which vanishes near , we can, without loss of generality, assume that on , and that the images of and are disjoint, where is a small positive real number. By assumption there exists a homotopy such that , and for any fixed , the path is of diameter smaller than Since the dimension of is at least , by the weak Whitney immersion theorem, we can approximate by a smooth map such that and for , for , and such that the restriction is a smooth immersion with a finite number of self-intersection points occuring inside the relative interior , and whose image does not intersect . Furthermore, similarly as was done in Lemma A.1 from [BuOp], one can find a smooth map whose image lies in an arbitrarily small neighbourhood of , such that as before we have and for , for , and the image does not intersect , but moreover such that the restriction is a smooth embedding, and such that for any fixed the diameter of the curve is less than . Note that by construction, , and give the same element of . Abusing our notation, we will denote by again.

Let be a sufficiently large positive integer. Then, for each , the image has diameter less than , for given we have only if , and moreover we can find a neighborhood of each such that is diffeomorphic to a ball, such that we again have only if , and such that the diameter of is less than , for every . Moreover, the union can be assumed to be diffeomorphic to a ball, as well as , for each . Then in particular, is exact on , i.e. on , for some differential -form on . By our assumptions, .

Step 1: Mapping points to points. For each , we pick a Hamiltonian which is supported in such that

where is sufficiently small. In particular, the ’s have mutually disjoint supports.

We let and let . We also remark that each can be picked such that is as small as one wishes. Hence, we may assume that .

Step 2: Adjusting the actions. Note that the two curves and coincide near their end-points and are both contained in . We would like to find a Hamiltonian diffeomorphism which is supported in and maps to . However, there is an obstruction to finding such a Hamiltonian diffeomorphism. These two curves do not necessarily have the same action. The goal of this step is to modify to remove this obstruction.

Claim 10.

There exists a Hamiltonian with the following properties:

  1. The support of is contained in , and we have for every and .

  2. The curve coincides with near , for ,

  3. has the same action as , for ,

  4. .

  5. For each , the images of and of intersect only if .

Proof.

Denote . We perform steps, where at step () we construct a curve , and find a Hamiltonian isotopy from to .

Let us describe the th step. Let be the curve provided by the previous step. First, perturb the curve in an arbitrarily small neighbourhood of , so that the perturbed curve satisfies:

  • coincides with on ,

  • we have ,

  • the -actions of the restrictions of and to coincide.

Such perturbation can be performed similarly as in the Remark A.13 from [BuOp].

Now, we claim that there exists a smooth Hamiltonian function supported in , such that on , and such that . For doing this, roughly speaking, it is sufficient to isotope (via the Hamiltonian flow) a small segment of so that it coincides with , near , or more precisely, for . And notice, that we are not restricted to keeping the "right end-point" fixed along the isotopy. Therefore, for keeping the Hofer’s norm of the isotopy small, we can just "shrink" the curve to the small segment near the left end-point, and then "expand this segment" to coincide with .

For a more precise explanation, denote , , choose a smooth function such that for and on , and consider families of curves , , where , . Note that . It is easy to see that one can find Hamiltonian functions , , supported in arbitrarily small neighbourhood of and respectively, such that and for each , and such that . Now let be the Hamiltonian function of the Hamiltonian flow . The function is supported in , satisfies , and on . Now define the curve by .

After performing all the steps, the -actions of and on coincide for any . But since the actions of and coincide on the whole , it also follows that the actions of and coincide on . Note that since all have disjoint supports (since the support of is contained in ), if we denote , then and .

It is possible that for different , the images of and of intersect. But then, one can easily find a -small Hamiltonian function , supported inside an arbitrarily small neighbourhood of , such that in particular we have , such that satisfies the property from the statement of the Claim, and moreover such that the Hamiltonian function that generates the flow satisfies the property from the statement of the Claim.

Step 3. Mapping to .

Claim 11.

There exist Hamiltonians such that

  1. is supported in ,

  2. the support of intersects the images of and of only for ,

  3. ,

  4. .

Proof.

Before passing to the proof, let us remark that in general one can apply directly Lemma A.3 (a) of [BuOp] to our situation, but we would not have obtained the estimate on . Therefore the proof is more subtle. Let us roughly explain the steps of the proof. The first step is to make a very small () perturbation of the curve (via the Hamiltonian diffeomorphism below), in order to put the curves and in general position. Then, we use the following idea. If two curves which coincide near the endpoints and with equal actions, are -close, then clearly one can find a very small (with respect to the Hofer’s norm) Hamiltonian function that moves the first curve to the second. However, if such curves are not -close, then we can use a “conjugation trick”: Instead of moving the first curve to the second via a Hamiltonian diffeomorphism, we find a third curve which is -close to the first curve, and such that the pair “(first curve, second curve)” could be mapped to the pair “(first curve, third curve)” via a symplectomorphism. After that, it is clearly enough to move the first curve to the third curve by a Hamiltonian flow with a very small Hofer’s norm, and then conjugate the flow with the symplectomorphism. The details of this are carried out below.

The restrictions and both lie in , have the same -actions, and coincide near the endpoints. Let be such that for (we use the old notation in a new situation, and in fact it is enough to replace the old by ). One can slightly perturb via a Hamiltonian diffeomorphism generated by a -small Hamiltonian function , so that satisfies for , and moreover . We may assume that is supported in , the support of intersects the image of and of only for , and that .

Now, one can clearly find a -small Hamiltonian function supported in , so that the support of intersects the image of and of only for , and such that the curve coincides with on , and moreover , for some . We may assume that .

Denote . The curves and lie in , coincide near the endpoints, have the same -action, and their images do not intersect . By Lemma A.3 (a) of [BuOp], there exists a Hamiltonian function supported in and away from the endpoints , , such that on . By a general position argument and a cut-off argument, we may further assume that the support of does not intersect , as well as for . Denote .

To finish the proof, let be the Hamiltonian function that generates the flow . Then has all the desired properties. For instance, for the property 4 of the statement of the Claim, we first of all have since the support of does not intersect the curve , and then we get

on .

Let and where ’s are provided to us by the above claim. We let be a Hamiltonian such that .

One can deduce the following facts, without much difficulty, from the above claim:

  1. for each and (where we put ),

  2. ,

  3. .

We now let be a Hamiltonian such that . Examining the properties of and we see that

  1. vanishes near the extremities of (and hence ), in particular fixes the extremities,

  2. ,

  3. for each , and ,

  4. is supported in a –neighborhood of .

3 Proof of Theorem 2

The most important step towards the proof of Theorem 2 will be to establish the following result which is a refined version of Theorem 4. Throughout this section will denote a closed and connected symplectic manifold whose dimension is at least four.

Theorem 12.

Let be a Morse function on . If is sufficiently –small, then for every , there exists and an embedded tree (see Definition 13) such that:

  • is -invariant, i.e. ,

  • contains all the fixed points of ,

  • and