A $C^0$ counterexample to the Arnold conjecture

Abstract

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

1 Introduction and main results

1.1 The Arnold conjecture

Let $(M, \omega )$ 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 M must have at least as many fixed points as the minimal number of critical points of a smooth function on M.

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 [9, 10, 12]. The above version of the Arnold conjecture has been established on symplectically aspherical¹ manifolds by Rudyak and Oprea [33] who built on earlier works of Floer [11] and Hofer [20]. We should mention that prior to the discovery of Floer homology, the Arnold conjecture was proven by Eliashberg [7] on closed surfaces (see also Sikorav [35]), by Conley and Zehnder [6] on higher dimensional tori, and by Fortune and Weinstein [13, 14] on complex projective spaces.

1.2 The Arnold conjecture and Hamiltonian homeomorphisms

Throughout this paper we will denote by $\mathrm {Symp}(M, \omega )$ and $\mathrm {Ham}(M, \omega )$ the groups of symplectic and Hamiltonian diffeomorphisms of $(M, \omega )$, 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 $\mathrm {Sympeo}(M, \omega )$; see Sect. 2.1.

As a first attempt at defining Hamiltonian homeomorphisms, we will say that a homeomorphism $\phi$ of M 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 surfaces.² For example, Matsumoto [28], building on an earlier paper of Franks [15], 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 [25] which has not only proven the Arnold conjecture but also the Conley conjecture on periodic points of these homeomorphisms [26].

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 $C^0$, 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 Sect. 2.1 for the precise definition. From this point onward by Hamiltonian homeomorphisms we will mean those homeomorphisms of M prescribed by Definition 7. We denote the set of all Hamiltonian homeomorphisms by $\mathrm {Hameo}(M, \omega )$.

The group $\mathrm {Hameo}(M, \omega )$ has met some success. Indeed, recent results in $C^0$-symplectic topology [22,23,24] 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 $C^0$-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 $(M, \omega )$ denote a closed and connected symplectic manifold of dimension at least 4. There exists $f \in \mathrm {Hameo}(M, \omega )$ with a single fixed point. Furthermore, f can be chosen to satisfy either of the following additional properties.

• 1. Let $\mathcal {H}$ be a normal subgroup of $\mathrm {Sympeo}(M, \omega )$ which contains $\mathrm {Ham}(M, \omega )$ as a proper subset. Then, $f \in \mathcal {H}$.

• 2. Let p denote the unique fixed point of f. Then, f is a symplectic diffeomorphism of $M {\setminus } \{p\}$.

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 point³.

With regards to the second property, we point out that it is natural to expect f 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 f must necessarily be non-smooth on any symplectic manifold $(M, \omega )$ with the property⁴ that $\mathrm {Hameo}(M,\omega ) \cap \mathrm {Diff}(M) = \mathrm {Ham}(M,\omega )$.

Lastly, we remark that it is well known that $\mathrm {Ham}(M, \omega )$ is a normal subgroup of $\mathrm {Symp}(M, \omega )$. Hence, it is reasonable to expect that any alternative candidate, say $\mathcal {H}$, for the group of Hamiltonian homeomorphisms should contain $\mathrm {Ham}(M, \omega )$ and be a normal subgroup of $\mathrm {Sympeo}(M, \omega )$. It is indeed the case that $\mathrm {Hameo}(M, \omega ) \unlhd \mathrm {Sympeo}(M, \omega )$. 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.

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

In Gromov’s view [18], symplectic topology is enriched by a beautiful interplay between rigidity and flexibility. Recent results, such as [3, 22, 31], have demonstrated that this contrast between rigidity and flexibility permeates, in a surprising fashion, to $C^0$ symplectic topology as well. Symplectic rigidity manifests itself when symplectic phenomena survive under $C^0$ limits; see [2, 5, 8, 22, 31] for some examples. On the other hand, there exist instances where passage to $C^0$ limits results in spectacular loss of rigidity and prevalence of flexibility; see [3] for an example.

The main theorem of our paper tells us that fixed points of Hamiltonian diffeomorphisms become completely flexible under $C^0$ limits. It is interesting to contrast this prevalence of flexibility with the strong rigidity results of Franks [15], Matsumoto [28], and Le Calvez [25, 26] 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 [19] on tori and Ginzburg [16] in the more general setting. As mentioned earlier, the Conley conjecture has been proven for Hamiltonian homeomorphisms of surfaces by Le Calvez [25, 26]. 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 [29, 34, 37], associates to each smooth Hamiltonian H, a collection of real numbers $\{c(a, H) \in \mathbb {R}: a \in H_*(M) {\setminus } \{0\} \},$ where $H_*(M)$ denotes the singular homology of M. These numbers are referred to as the spectral invariants of H and they correspond to critical values of the associated action functional. Hence, the number of distinct spectral invariants of a Hamiltonian H gives a lower bound for the number of fixed points of the time-1 map $\phi _H^1$.

Recall that the cup length of M is defined by $cl(M):= \max \{k+1: \exists \, a_1, \ldots , a_k \in H^*(M): \forall i, \deg (a_i)\ne 0 \text { and } a_1 \cup \cdots \cup a_k \ne 0\}$. Combining techniques from Hamiltonian Floer theory and Lusternik–Shnirelman theory, Floer [11] and Hofer [20] proved that if a Hamiltonian diffeomorphism, of an aspherical symplectic manifold M, has fewer spectral invariants than the cup length of M, then it must have infinitely many fixed points; see also [21].

It is well-known that one can associate spectral invariants to any continuous Hamiltonian function; see for example [30]. 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 cl(M). Hence, we see that the correspondence between spectral invariants and fixed points breaks down in the continuous setting. See Remark 20. This leads us to the following question:

Question 3

Suppose that H is a continuous Hamiltonian with fewer spectral invariants than the cup length of M. Does $\phi ^1_H$, the time-1 map of the flow of H, have infinitely many fixed points?

A positive answer to this question could be interpreted as a $C^0$ 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 [32], Hamiltonian Floer theory allows one to associate a so-called barcode to any smooth Hamiltonian; see also [1, 27, 36]. 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 $C^2$-small Morse function. See Remark 21.

1.4 A brief outline of the construction

Construction of the homeomorphism f, 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 $(M, \omega )$ denote a closed and connected symplectic manifold of dimension at least 4. There exists $\psi \in \mathrm {Hameo}(M, \omega )$ and an embedded tree $T \subset M$ such that

• 1. T is invariant under $\psi$, i.e. $\psi (T) = T$,

• 2. All of the fixed points of $\psi$ are contained in T,

• 3. $\psi$ is smooth in the complement of T.

For the proof of Theorem 2, we will in fact need a refined version of the above result; see Theorem 13. The proof of this theorem forms the technical heart of our paper. An important ingredient used in the construction of the invariant tree T is a quantitative h-principle for curves. Quantitative h-principles have recently been introduced to $C^0$ symplectic topology by Buhovsky and Opshtein and have had numerous fascinating applications; see [3]. We should point out that M having dimension at least four is used in a crucial way in the proof of this theorem.

The second major step of our construction consists of “collapsing” the invariant tree T to a single point which will be the fixed point of our homeomorphism f. Here is a brief outline of how this is done. Fix a point $p \in M$. We construct a sequence $\varphi _i \in \mathrm {Symp}(M, \omega )$ such that $\varphi _i$ converges uniformly to a map $\varphi : M \rightarrow M$ with the following two properties:

• 1. $\varphi (T) = {p}$,

• 2. $\varphi$ is a symplectic diffeomorphism from $M {\setminus }{T}$ to $M {\setminus }\{p\}.$

Note that the first property implies that $\varphi$ is not a 1-1 map and hence, the sequence $\varphi _i^{-1}$ is not convergent. Define $f: M \rightarrow M$ as follows: $f(p) =p$ and

$$\begin{aligned}\forall x \in M{\setminus }\{p\},\ f(x)=\varphi \circ \psi \circ \varphi ^{-1}(x). \end{aligned}$$

It is not difficult to see that p is the unique fixed point of f. Indeed, on $M{\setminus }\{p\}$, the map f is conjugate to $\psi : M {\setminus }{T} \rightarrow M{\setminus }{T}$ which is fixed point free by construction.

By picking the above sequence of symplectomorphisms $\varphi _i$ carefully, it is possible to ensure that the sequence of conjugations $\varphi _i\psi \varphi _i^{-1}$ converges uniformly to f. The uniform convergence of $\varphi _i\psi \varphi _i^{-1}$ to f relies heavily on the invariance of the tree T and it occurs despite the fact that the sequence $\varphi _i^{-1}$ diverges. The details of this are carried out in Sect. 3.1. It follows that f can be written as the uniform limit of a sequence of Hamiltonian diffeomorphisms.

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

1.5 Organization of the paper

In Sect. 2, we recall some preliminary results from $C^0$ symplectic geometry. Symplectic and Hamiltonian homeomorphisms are introduced in Sect. 2.1. In Sect. 2.2, we introduce a quantitative h-principle for curves which plays an important role in our construction.

In Sect. 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 Sect. 3.2, we prove the existence of a Hamiltonian homeomorphism as described in Theorem 4, assuming a technical and important result: Theorem 25. Section 3.3, which occupies the rest of the paper, is dedicated to the proof of Theorem 25. This section contains the technical heart of the paper.

2 Preliminaries from $C^0$-symplectic topology

In this section we introduce some of our notation and recall some of the basic notions of $C^0$-symplectic geometry. In Sect. 2.1 we give precise definitions for symplectic and Hamiltonian homeomorphisms. In Sect. 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, $(M, \omega )$ will denote a closed and connected symplectic manifold whose dimension is at least 4. We equip M with a Riemannian distance d. Given two maps $\phi , \psi : M \rightarrow M,$ we denote

$$\begin{aligned} d_{C^0}(\phi ,\psi )=\sup _{x\in M}d(\phi (x),\psi (x)). \end{aligned}$$

We will say that a sequence of maps $\phi _i : M \rightarrow M$, converges uniformly, or $C^0$-converges, to $\phi$, if $d_{C^0}(\phi _i, \phi ) \rightarrow 0$ as $i \rightarrow \infty$. Of course, the notion of $C^0$-convergence does not depend on the choice of the Riemannian metric.

Recall that a symplectic diffeomorphism is a diffeomorphism $\theta : M \rightarrow M$ such that $\theta ^* \omega = \omega$. The set of all symplectic diffeomorphisms of M is denoted by $\mathrm {Symp}(M, \omega )$.

Definition 5

A homeomorphism $\theta : M \rightarrow M$ is said to be symplectic if it is the $C^0$-limit of a sequence of symplectic diffeomorphisms. We will denote the set of all symplectic homeomorphisms by $\mathrm {Sympeo}(M, \omega )$.

The Eliashberg–Gromov theorem states that a symplectic homeomorphism which is smooth is itself a symplectic diffeomorphism. We remark that if $\theta$ is a symplectic homeomorphism, then so is $\theta ^{-1}$. In fact, it is easy to see that $\mathrm {Sympeo}(M, \omega )$ forms a group.

Remark 6

More generally, one can define a symplectic homeomorphism to be a homeomorphism which is locally a $C^0$-limit of symplectic diffeomorphisms; see [3] for further details.

Recall that a smooth Hamiltonian $H: [0,1] \times M \rightarrow \mathbb {R}$ gives rise to a Hamiltonian flow $\phi ^t_H$. 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 $\mathrm {Ham}(M, \omega )$; this is a normal subgroup of $\mathrm {Symp}(M, \omega )$. We next define Hamiltonian homeomorphisms as introduced by Müller and Oh [30].

Definition 7

(Hamiltonian homeomorphisms). Denote by B an open (possibly not proper) subset of M. Let $(\phi ^t)_{t\in [0,1]}$ be an isotopy of M which is compactly supported in B. We say that $\phi ^t$ is a hameotopy, or a continuous Hamiltonian flow, of B if there exists a sequence of smooth and compactly supported Hamiltonians $H_i:[0,1] \times B \rightarrow \mathbb {R}$ such that:

• 1. The sequence of flows $\phi _{H_i}^t$ $C^0$-converges to $\phi ^t$, uniformly in t, i.e. $\displaystyle \max _{t \in [0,1]} d_{C^0}(\phi ^t_{H_i}, \phi ^t) \rightarrow 0$ as $i \rightarrow \infty$.

• 2. The sequence of Hamiltonians $H_i$ converges uniformly to a continuous function $H: [0,1] \times M \rightarrow \mathbb {R}$, i.e. $\Vert H_i - H \Vert _{\infty } \rightarrow 0$ as $i \rightarrow \infty$, where $\Vert \cdot \Vert _{\infty }$ denotes the sup norm. Furthermore,

We say that H generates $\phi ^t$, denote $\phi ^t=\phi _H^t$, and call H a continuous Hamiltonian.

A homeomorphism is called a Hamiltonian homeomorphism if it is the time-1 map of a continuous Hamiltonian flow. We will denote the set of all Hamiltonian homeomorphisms by $\mathrm {Hameo}(B, \omega )$.

It is not difficult to check that $\mathrm {Hameo}(M, \omega )$ is a normal subgroup of $\mathrm {Sympeo}(M, \omega )$.

A continuous Hamiltonian H generates a unique continuous Hamiltonian flow; see [30]. Conversely, Viterbo [38] and Buhovsky–Seyfaddini [4] (see also [23]) 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 $\phi _H^t$ is a continuous Hamiltonian flow, then $(\phi _H^t)^{-1}$ is a continuous Hamiltonian flow generated by $\bar{H}(t,x) = -H(t,\phi ^t_H(x))$; given another continuous Hamiltonian flow $\phi _K^t$, the isotopy $\phi _H^t \phi _K^t$ is also a continuous Hamiltonian flow, generated by $H \# K(t,x) := H(t,x)+K(t,(\phi ^t_H)^{-1}(x))$.

We will finish this section by recalling an important dynamical property of continuous Hamiltonian flows. Recall that a submanifold C of a symplectic manifold $(M, \omega )$ is called coisotropic if for all $p\in C$, $(T_pC)^\omega \subset T_pC$ where $(T_pC)^\omega$ denotes the symplectic orthogonal of $T_pC$. For instance, hypersurfaces and Lagrangians are coisotropic. A coisotropic submanifold carries a natural foliation $\mathcal {F}$ which integrates the distribution $(TC)^\omega$; $\mathcal {F}$ is called the characteristic foliation of C.

Assume that C is a closed and connected coisotropic submanifold of M and suppose that H is a smooth Hamiltonian. The following is a standard and important fact which relates Hamiltonian flows to coisotropic submanifolds: $H|_C$ is a function of time if and only if $\phi ^t_H$ (preserves C and) flows along the characteristic foliation of C. By flowing along characteristics we mean that for any point $p\in C$ and any time $t\geqslant 0$, $\phi _H^t(p)\in \mathcal {F}(p)$, where $\mathcal {F}(p)$ stands for the characteristic leaf through p.

The following theorem, which was proven in [22], establishes the aforementioned property for continuous Hamiltonian flows.

Theorem 8

Denote by C a closed and connected coisotropic submanifold and suppose that $\phi ^t_H$ is a continuous Hamiltonian flow. The restriction of H to C is a function of time if and only if $\phi ^t_H$ preserves C 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 seem 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 h-principle for curves

Quantitative h-principles were introduced in [3], where they were used to construct interesting examples of symplectic homeomorphisms. We will need the following quantitative h-principle for curves in the construction of our counterexample to the Arnold conjecture.

Proposition 9

(Quantitative h-principle for curves). Denote by $(M, \omega )$ a symplectic manifold of dimension at least 4. Let $\varepsilon >0$. Suppose that $\gamma _0, \gamma _1 :[0,1] \rightarrow M$ are two smooth embedded curves such that

• (i) $\gamma _0$ and $\gamma _1$ coincide near $t=0$ and $t=1$,

• (ii) there exists a homotopy, rel.end points, from $\gamma _0$ to $\gamma _1$ under which the trajectory of any point of $\gamma _0$ has diameter less than $\varepsilon$, and the symplectic area of the element of $\pi _2(M,\gamma _1 \sharp \overline{\gamma _0})$ defined by this homotopy has area 0.

Then, for any $\rho >0$, there exists a compactly supported Hamiltonian F, generating a Hamiltonian isotopy $\varphi ^s: M \rightarrow M, \; s \in [0,1]$ such that

• 1. F vanishes near $\gamma _0(0)$ and $\gamma _0(1)$ (in particular, $\varphi ^s$ fixes $\gamma _0$ and $\gamma _1$ near the extremities),

• 2. $\varphi ^1 \circ \gamma _0 = \gamma _1$,

• 3. $d_{C^0}(\varphi ^s, Id) < 2\varepsilon$ for each $s \in [0,1],$ and $\Vert F\Vert _{\infty } \leqslant \rho$,

• 4. F is supported in a $2 \varepsilon$-neighborhood of the image of $\gamma _0$.

The existence of a Hamiltonian F satisfying only properties 1 and 2 is well known. The aspect of the above proposition which is non-standard is the fact that F 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 (considerably more difficult) quantitative h-principle for discs which appeared in Theorem 2 of [3]. The proof we will present is an adaptation of the arguments therein and thus, it will not be a detailed proof.

In the next section of the paper, we will need the following fact, which is standard, and follows from Proposition 9, and hence will not be proven here.

Lemma 10

Denote by $(M, \omega )$ an exact symplectic manifold of dimension at least 4. Suppose that $\gamma _0, \gamma _1 :[0,1] \rightarrow M$ are two curves such that

• (i) $\gamma _0$ and $\gamma _1$ coincide near $t=0,1$,

• (ii) $\int _0^1 \gamma _0^* \lambda = \int _0^1 \gamma _1^* \lambda$ where $\lambda$ is any 1-form such that $\omega = d \lambda$,

• (iii) there exists a homotopy, rel. end points, from $\gamma _0$ to $\gamma _1$.

Then, for any $\rho >0$, there exists a compactly supported Hamiltonian F, generating a Hamiltonian isotopy $\varphi ^s: M \rightarrow M, \; s \in [0,1]$ such that

• 1. F vanishes near the extremities of $\gamma _0$ and $\gamma _1$,

• 2. $\varphi ^1 \circ \gamma _0 = \gamma _1$,

• 3. $\Vert F\Vert _{\infty } < \rho$.

Proof of Proposition 9

First, by a slight Hamiltonian perturbation of $\gamma _0$ via a Hamiltonian diffeomorphism generated by a $C^\infty$-small Hamiltonian function which vanishes near $\{ \gamma _0(0), \gamma _0(1) \}$, we can, without loss of generality, assume that $\gamma _0 = \gamma _1$ on $[0,\delta ] \cup [1-\delta ,1]$, and that the images of $\gamma _0|_{(\delta ,1-\delta )}$ and $\gamma _1|_{(\delta ,1-\delta )}$ are disjoint, where $\delta > 0$ is a small positive real number. By assumption there exists a homotopy $h:[0,1] \times [0,1] \rightarrow M$ such that $h(0,t) = \gamma _0(t)$, $h(1,t) = \gamma _1(t)$ and for any fixed t, the path $s \mapsto h(s,t)$ is of diameter smaller than $\varepsilon .$ Since the dimension of M is at least 4, by the weak Whitney immersion theorem, we can approximate h by a smooth map $h' : [0,1] \times [0,1] \rightarrow M$ such that $h'(0,t) = \gamma _0(t)$ and $h'(1,t) = \gamma _1(t)$ for $t \in [0,1]$, $h'(s,t) = \gamma _0(t) = \gamma _1(t)$ for $(s,t) \in [0,1] \times ([0,\delta ] \cup [1-\delta ,1])$, and such that the restriction $h'|_{[0,1] \times (\delta ,1-\delta )}$ is a smooth immersion with a finite number of self-intersection points occuring inside the relative interior $h((0,1) \times (\delta , 1-\delta ))$, and whose image $h'([0,1] \times (\delta ,1-\delta ))$ does not intersect $\gamma _0([0,\delta ] \cup [1-\delta ,1])$. Furthermore, similarly as was done in Lemma A.1 from [3], one can find a smooth map $h'' : [0,1] \times [0,1] \rightarrow M$ whose image lies in an arbitrarily small neighborhood of $h' ([0,1] \times [0,1])$, such that as before we have $h''(0,t) = \gamma _0(t)$ and $h''(1,t) = \gamma _1(t)$ for $t \in [0,1]$, $h''(s,t) = \gamma _0(t) = \gamma _1(t)$ for $(s,t) \in [0,1] \times ([0,\delta ] \cup [1-\delta ,1])$, and the image $h''([0,1] \times (\delta ,1-\delta ))$ does not intersect $\gamma _0([0,\delta ] \cup [1-\delta ,1])$, but moreover such that the restriction $h''|_{[0,1] \times (\delta ,1-\delta )}$ is a smooth embedding, and such that for any fixed t the diameter of the curve $s \mapsto h(s,t)$ is less than $2\varepsilon$. Note that by construction, h, $h'$ and $h''$ give the same element of $\pi _2(M,\gamma _1 \sharp \overline{\gamma _0})$. Abusing our notation, we will denote $h''$ by h again.

Let m be a sufficiently large positive integer. Then, for each $1 \leqslant i < m-3$, the image $h([0,1] \times [\frac{i}{m}, \frac{i+3}{m}])$ has diameter less than $2 \varepsilon$, for given $0 \leqslant i,j \leqslant m-1$ we have $h([0,1] \times [\frac{i}{m}, \frac{i+1}{m}]) \cap h([0,1] \times [\frac{j}{m}, \frac{j+1}{m}]) \ne \emptyset$ only if $j \in \{ i-1, i, i+1\}$, and moreover we can find a neighborhood $U_i$ of each $h([0,1] \times [\frac{i}{m}, \frac{i+1}{m}])$ such that $U_i$ is diffeomorphic to a ball, such that we again have $U_i \cap U_j \ne \emptyset$ only if $j \in \{ i-1, i, i+1\}$, and such that the diameter of $U_{i-1} \cup U_i \cup U_{i+1}$ is less than $2\varepsilon$, for every $0< i < m-1$. Moreover, the union $U = U_0 \cup \cdots \cup U_m$ can be assumed to be diffeomorphic to a ball, as well as $U_i \cap U_{i+1}$, for each $0 \leqslant i < m-1$. Then in particular, $\omega$ is exact on U, i.e. $\omega = d \lambda$ on U, for some differential 1-form $\lambda$ on U. By our assumptions, $\int _0^1 \gamma _0^* \lambda = \int _0^1 \gamma _1^* \lambda$.

Step 1: Mapping points to points For each $1 \leqslant i \leqslant m-1$, we pick a Hamiltonian $G_i$ which is supported in $U_{i-1} \cap U_i$

such that

$$\begin{aligned} \phi ^1_{G_i}(\gamma _0(t)) = \gamma _1(t), \quad \forall t \in \left[ \tfrac{i}{m} - \kappa , \tfrac{i}{m}+ \kappa \right] , \end{aligned}$$

where $\kappa >0$ is sufficiently small. In particular, the $G_i$’s have mutually disjoint supports.

We let $G:= \sum _{i=1}^{m-1} G_i$ and let $\gamma _0' = \phi ^1_{G} \circ \gamma _0$. We also remark that each $G_i$ can be picked such that $\Vert G_i\Vert _{\infty }$ is as small as one wishes. Hence, we may assume that $\Vert G\Vert _{\infty } \leqslant \rho / 3$.

Step 2: Adjusting the actions Note that the two curves $\gamma _1|_{[\frac{i}{m}, \frac{i+1}{m}]}$ and $\gamma _0'|_{[\frac{i}{m}, \frac{i+1}{m}]}$ coincide near their end-points and are both contained in $U_i$. We would like to find a Hamiltonian diffeomorphism which is supported in $U_i$ and maps $\gamma _0'|_{[\frac{i}{m}, \frac{i+1}{m}]}$ to $\gamma _1|_{[\frac{i}{m}, \frac{i+1}{m}]}$. 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 $\gamma _0'$ to remove this obstruction. $\square$

Claim 11

There exists a Hamiltonian H with the following properties:

• 1. The support of H is contained in U, and we have $\phi _H^t(U_i) \subset U_i$ for every $t \in [0,1]$ and $0 \leqslant i \leqslant m$.

• 2. The curve $\gamma _0'' := (\phi _H^1) \circ \gamma _0'$ coincides with $\gamma _1$ near $t = \frac{i}{m}$, for $i = 0,1, \ldots , m$,

• 3. $\gamma _0 ''|_{[\frac{i}{m}, \frac{i+1}{m}]}$ has the same action as $\gamma _1|_{[\frac{i}{m}, \frac{i+1}{m}]}$, for $i = 0,1, \ldots , m-1$,

• 4. $\Vert H\Vert _{\infty } \leqslant \rho / 3$.

• 5. For each $0 \leqslant i,j \leqslant m-1$, the images of $\gamma _0''|_{(\frac{i}{m},\frac{i+1}{m})}$ and of $\gamma _1|_{(\frac{j}{m},\frac{j+1}{m})}$ intersect only if $i = j$.

Proof

Denote $\gamma _{0,1}' = \gamma _0'$. We perform $(m-1)$ steps, where at step i ($1 \leqslant i \leqslant m-1$) we construct a curve $\gamma _{0,i+1}'$, and find a Hamiltonian isotopy from $\gamma _{0, i }'$ to $\gamma _{0, i+1}'$.

Let us describe the ith step. Let $\gamma _{0,i}'$ be the curve provided by the previous step. First, perturb the curve $\gamma _{0,i}'$ in an arbitrarily small neighborhood of $t = \frac{i}{m} - \frac{\kappa }{2}$, so that the perturbed curve $\gamma _{0,i}''$ satisfies:

• $\gamma _{0,i}''$ coincides with $\gamma _{0,i}'$ on $[0, \frac{i}{m} - \frac{3\kappa }{4}] \cup [\frac{i}{m} - \frac{\kappa }{4} ,1]$,

• we have $\gamma _{0,i}''([\frac{i}{m} - \frac{3\kappa }{4} , \frac{i}{m} - \frac{\kappa }{4}]) \subset U_{i-1} \cap U_i$,

• the $\lambda$-actions of the restrictions of $\gamma _{0,i}''$ and $\gamma _1$ to $[\frac{i-1}{m},\frac{i}{m}]$ coincide.

Such perturbation can be performed similarly as in the Remark A.13 from [3].

Now, we claim that there exists a smooth Hamiltonian function $H_i$ supported in $U_{i-1} \cap U_i$, such that $\gamma _{0,i}'' = (\phi _{H_i}^1) \circ \gamma _{0,i}'$ on $[0,\frac{i}{m}+\kappa ]$, and such that $\Vert H_i \Vert _\infty \leqslant \rho /6$. For doing this, roughly speaking, it is sufficient to isotope (via the Hamiltonian flow) a small segment of $\gamma _{0,i}'$ so that it coincides with $\gamma _{0,i}''$, near $t = \frac{i}{m}$, or more precisely, for $t \in [\frac{i}{m} - \frac{3\kappa }{4}, \frac{i}{m} + \kappa ]$. And notice, that we are not restricted to keeping the ”right end-point” $\gamma _{0,i}''(\frac{i}{m} + \kappa )$ fixed along the isotopy. Therefore, for keeping the Hofer norm of the isotopy small, we can just ”shrink” the curve $\gamma _{0,i}'|_{[\frac{i}{m} - \frac{7\kappa }{8}, \frac{i}{m} + \kappa ]}$ to the small segment $\gamma _{0,i}'|_{[\frac{i}{m} - \frac{7\kappa }{8}, \frac{i}{m} - \frac{3\kappa }{4}]} = \gamma _{0,i}''|_{[\frac{i}{m} - \frac{7\kappa }{8}, \frac{i}{m} - \frac{3\kappa }{4}]}$ near the left end-point, and then ”expand this segment” to coincide with $\gamma _{0,i}''|_{[\frac{i}{m} - \frac{7\kappa }{8}, \frac{i}{m} + \kappa ] }$.

For a more precise explanation, denote $a = \frac{i}{m} - \frac{7\kappa }{8}$, $b = \frac{i}{m} + \kappa$, choose a smooth function $c : [a,b] \rightarrow [a, a + \frac{\kappa }{8}]$ such that $c(t) = t$ for $t \in [a,a+\frac{\kappa }{16}]$ and $c'(t) > 0$ on [a, b] , and consider families of curves $\alpha _s, \beta _s : [a,b] \rightarrow M$, $s \in [0,1]$, where $\alpha _s(t) = \gamma _{0,i}'(st + (1-s)c(t))$, $\beta _s(t) = \gamma _{0,i}''(st + (1-s)c(t))$. Note that $\alpha _0 \equiv \beta _0$. It is easy to see that one can find Hamiltonian functions $H_i'$, $H_i''$, supported in arbitrarily small neighborhood of $\gamma _{0,i}'([a+ \frac{\kappa }{8},b])$ and $\gamma _{0,i}''([a+ \frac{\kappa }{8},b])$ respectively, such that $\alpha _s = \phi _{H_i'}^s \circ \alpha _0$ and $\beta _s = \phi _{H_i''}^s \circ \beta _0$ for each $s \in [0,1]$, and such that $\Vert H_i' \Vert _{\infty }, \Vert H_i'' \Vert _{\infty } \leqslant \rho / 12$. Now let $H_i$ be the Hamiltonian function of the Hamiltonian flow $(\phi _{H_i''}^s \circ (\phi _{H_i'}^s)^{-1})_{s \in [0,1]}$. The function $H_i$ is supported in $U_{i-1} \cap U_i$, satisfies $\Vert H_i \Vert _{\infty } \leqslant \rho / 6$, and $\gamma _{0,i}'' = (\phi _{H_i}^1) \circ \gamma _{0,i}'$ on $[0,\frac{i}{m}+\kappa ]$. Now define the curve $\gamma _{0,i+1}' : [0,1] \rightarrow M$ by $\gamma _{0,i+1}' := \phi _{H_i}^1 \circ \gamma _{0,i}'$.

After performing all the $(m-1)$ steps, the $\lambda$-actions of $\gamma _{0,m-1}'$ and $\gamma _1$ on $[\frac{i-1}{m},\frac{i}{m}]$ coincide for any $1 \leqslant i \leqslant m-1$. But since the actions of $\gamma _{0,m-1}'$ and $\gamma _1$ coincide on the whole [0, 1] , it also follows that the actions of $\gamma _{0,m-1}'$ and $\gamma _1$ coincide on $[\frac{m-1}{m},1]$. Note that since all $H_i$ have disjoint supports (since the support of $H_i$ is contained in $U_{i-1} \cap U_i$), if we denote $H' = H_1 + \cdots + H_{m-1}$, then $\Vert H' \Vert _{\infty } \leqslant \rho / 6$ and $\gamma _{0,m-1}' = (\phi ^{1}_{H'}) \circ \gamma _{0,1}' = (\phi ^1_{H'}) \circ \gamma _0'$.

It is possible that for different $0 \leqslant i,j \leqslant m-1$, the images of $\gamma _{0,m-1}' |_{(\frac{i}{m},\frac{i+1}{m})}$ and of $\gamma _1|_{(\frac{j}{m},\frac{j+1}{m})}$ intersect. But then, one can easily find a $C^\infty$-small Hamiltonian function $H''$, supported inside an arbitrarily small neighborhood of $\bigcup _{i=0}^{m-1} \gamma _1([\frac{i}{m} + \frac{\kappa }{5}, \frac{i+1}{m} - \frac{\kappa }{5}])$, such that in particular we have $\Vert H'' \Vert _\infty \leqslant \rho / 6$, such that $\gamma _0'' := \phi ^1_{H''} \circ \gamma _{0,m-1}'$ satisfies the property 5 from the statement of the Claim, and moreover such that the Hamiltonian function $H := H'' \sharp H'$ that generates the flow $(\phi ^t_{H''} \circ \phi ^t_{H'})$ satisfies the property 1 from the statement of the Claim. $\square$

Step 3: Mapping $\gamma _0 ''|_{[\frac{i}{m}, \frac{i+1}{m}]}$ to $\gamma _1|_{[\frac{i}{m}, \frac{i+1}{m}]}$.

Claim 12

There exist Hamiltonians $K_i$ such that

• 1. $K_i$ is supported in $U_i$,

• 2. the support of $K_i$ intersects the images of $\gamma _0''|_{[\frac{j}{m}, \frac{j+1}{m}]}$ and of $\gamma _1|_{[\frac{j}{m}, \frac{j+1}{m}]}$ only for $j = i$,

• 3. $\Vert K_i \Vert _\infty \leqslant \rho / 6$,

• 4. $\phi ^1_{K_i} \circ \gamma _0''|_{[\frac{i}{m}, \frac{i+1}{m}]} = \gamma _1|_{[\frac{i}{m}, \frac{i+1}{m}]}$.

Proof

Before passing to the proof, let us remark that in general one can apply directly Lemma A.3 (a) of [3] to our situation, but we would not have obtained the estimate on $\Vert \cdot \Vert _\infty$. Therefore the proof is more subtle. Let us roughly explain the steps of the proof. The first step is to make a very small ($C^\infty$) perturbation of the curve $\gamma _0''$ (via the Hamiltonian diffeomorphism $\phi _{K_i'}^1$ below), in order to put the curves $\gamma _0'' |_{[\frac{i}{m}, \frac{i+1}{m}]}$ and $\gamma _1 |_{[\frac{i}{m}, \frac{i+1}{m}]}$ in general position. Then, we use the following idea. If two curves which coincide near the endpoints and with equal actions, are $C^\infty$-close, then clearly one can find a very small (with respect to the Hofer norm) Hamiltonian function that moves the first curve to the second. However, if such curves are not $C^\infty$-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 $C^\infty$-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 $\gamma _0 ''|_{[\frac{i}{m}, \frac{i+1}{m}]}$ and $\gamma _1|_{[\frac{i}{m}, \frac{i+1}{m}]}$ both lie in $U_i$, have the same $\lambda$-actions, and coincide near the endpoints. Let $\kappa > 0$ be such that $\gamma _0''(t) = \gamma _1(t)$ for $t \in [\frac{i}{m}, \frac{i}{m} + \kappa ] \cup [\frac{i+1}{m}-\kappa ,\frac{i+1}{m}]$ (we use the old notation $\kappa$ in a new situation, and in fact it is enough to replace the old $\kappa$ by $\frac{\kappa }{5}$). One can slightly perturb $\gamma _0''$ via a Hamiltonian diffeomorphism $\phi _{K_i'}^1$ generated by a $C^\infty$-small Hamiltonian function $K_i'$, so that $\gamma _{01}'' := \phi _{K_i'}^1 \circ \gamma _0''$ satisfies $\gamma _{01}''(t) = \gamma _1(t)$ for $t \in [\frac{i}{m}, \frac{i}{m} + \kappa ] \cup [\frac{i+1}{m}-\kappa ,\frac{i+1}{m}]$, and moreover $\gamma _{01}''((\frac{i}{m} + \kappa ,\frac{i+1}{m}-\kappa )) \cap \gamma _1((\frac{i}{m} + \kappa ,\frac{i+1}{m}-\kappa )) = \emptyset$. We may assume that $K_i'$ is supported in $U_i$, the support of $K_i'$ intersects the image of $\gamma _0''|_{[\frac{j}{m}, \frac{j+1}{m}]}$ and of $\gamma _1|_{[\frac{j}{m}, \frac{j+1}{m}]}$ only for $j = i$, and that $\Vert K_i' \Vert _{\infty } \leqslant \rho / 12$.

Now, one can clearly find a $C^\infty$-small Hamiltonian function $K_i''$ supported in $U_i$, so that the support of $K_i''$ intersects the image of $\gamma _0''|_{[\frac{j}{m}, \frac{j+1}{m}]}$ and of $\gamma _1|_{[\frac{j}{m}, \frac{j+1}{m}]}$ only for $j = i$, and such that the curve $\gamma _1' := (\phi _{K_i''}^1) \circ \gamma _1$ coincides with $\gamma _{01}''$ on $[\frac{i}{m}, \frac{i}{m} + \kappa '] \cup [\frac{i+1}{m}-\kappa ',\frac{i+1}{m}]$, and moreover $\gamma _1'((\frac{i}{m} + \kappa ',\frac{i+1}{m}-\kappa ')) \cap \gamma _1((\frac{i}{m} + \kappa ',\frac{i+1}{m}-\kappa ')) = \emptyset$, for some $\kappa ' > \kappa$. We may assume that $\Vert K_i'' \Vert \leqslant \rho / 12$.

Denote $\kappa '' = (\kappa + \kappa ') / 2$. The curves $\gamma _{01}''|_{[\frac{i}{m} + \kappa '',\frac{i+1}{m}-\kappa '']}$ and $\gamma _1'|_{[\frac{i}{m} + \kappa '',\frac{i+1}{m}-\kappa '']}$ lie in $U_i$, coincide near the endpoints, have the same $\lambda$-action, and their images do not intersect $\gamma _1 ([0,1])$. By Lemma A.3 (a) of [3], there exists a Hamiltonian function $K_i'''$ supported in $U_i$ and away from the endpoints $\gamma _{01}''(\frac{i}{m} + \kappa '')$, $\gamma _{01}''(\frac{i+1}{m}-\kappa '')$, such that $\gamma _1' = \phi _{K_i'''}^1 \circ \gamma _{01}''$ on $[\frac{i}{m} + \kappa '',\frac{i+1}{m}-\kappa '']$. By a general position argument and a cut-off argument, we may further assume that the support of $K_i'''$ does not intersect $\gamma _1([0,1])$, as well as $\gamma _0'' ([\frac{j}{m}, \frac{j+1}{m}])$ for $j \ne i$. Denote $\psi := \phi _{K_i'''}^1$.

To finish the proof, let $K_i$ be the Hamiltonian function that generates the flow $(\psi ^{-1} \circ (\phi _{K_i''}^s)^{-1} \circ \psi \circ \phi _{K_i'}^s)_{s \in [0,1]}$. Then $K_i$ has all the desired properties. For instance, for the property 4 of the statement of the Claim, we first of all have $\gamma _1 = \psi ^{-1} \circ \gamma _1$ since the support of $\psi$ does not intersect the curve $\gamma _1$, and then we get

$$\begin{aligned} \gamma _1= & {} \psi ^{-1} \circ \gamma _1 = \psi ^{-1} \circ \left( \phi ^1_{K_i''}\right) ^{-1} \circ \gamma _1' = \psi ^{-1} \circ \left( \phi ^1_{K_i''}\right) ^{-1} \circ \psi \circ \gamma _{01}'' \\= & {} \psi ^{-1} \circ \left( \phi ^1_{K_i''}\right) ^{-1} \circ \psi \circ \phi ^1_{K_i'} \circ \gamma _{0}'' = \phi ^1_{K_i} \circ \gamma _0'' \end{aligned}$$

on $[\frac{i}{m}, \frac{i+1}{m}]$. $\square$

Let $K_{\mathrm {odd}} = K_1 + K_3 + \cdots$ and $K_{\mathrm {even}} = K_2 + K_4 + \cdots ,$ where $K_i$’s are provided to us by the above claim. We let K be a Hamiltonian such that $\phi ^s_K = \phi ^s_{K_{\mathrm {odd}}} \circ \phi ^s_{K_{\mathrm {even}}}$.

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

• 1. $\phi ^s_K(U_i) \subset U_{i-1} \cup U_i \cup U_{i+1}$ for each $0 \leqslant i \leqslant m$ and $s \in [0,1]$ (where we put $U_{-1} = U_{m+1} = \emptyset$),

• 2. $\phi ^1_K \circ \gamma _0'' = \gamma _1$,

• 3. $\Vert K\Vert _{\infty } \leqslant \rho / 3$.

We now let F be a Hamiltonian such that $\phi ^s_F = \phi ^s_K \circ \phi ^s_H \circ \phi ^s_G$. Examining the properties of K, H,  and G we see that

• 1. F vanishes near the extremities of $\gamma _0$ (and hence $\gamma _1$), in particular $\phi ^s_F$ fixes the extremities,

• 2. $\phi ^1_F \circ \gamma _0 = \gamma _1$,

• 3. $d_{C^0}(\phi ^s_F, Id) < 2 \varepsilon$ for each $s \in [0,1]$, and $\Vert F\Vert _{\infty } \leqslant \rho$,

• 4. F is supported in a $2 \varepsilon$-neighborhood of $\gamma _0$.

$\square$

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 $(M, \omega )$ will denote a closed and connected symplectic manifold whose dimension is at least four.

Theorem 13

Let H be a Morse function on M. If H is sufficiently $C^2$-small, then for every $\varepsilon ,\rho >0$, there exists $\psi \in \mathrm {Hameo}(M,\omega )$ and an embedded tree T (see Definition 14) such that:

• T is $\psi$-invariant, i.e. $\psi (T)=T$,

• T contains all the fixed points of $\psi$,

• $d_{C^0}(\phi _H^1,\psi )<\varepsilon$ and $\psi \circ \phi _H^{-1}$ is generated by a continuous Hamiltonian F such that $\Vert F\Vert _{\infty }<\rho$,

• $\psi$ coincides with a Hamiltonian diffeomorphism in the complement of any neighborhood of T.

The notion of an embedded tree which appears in the above theorem is defined as follows.

Definition 14

We will say that a compact subset T of a smooth manifold M is an embedded tree if there exists a finite tree $T_0$ and an injective continuous map $\chi :T_0\rightarrow M$, such that $T=\chi (T_0)$ and the map $\chi$ is a smooth embedding of the interior of each edge of $T_0$.

Note that we do not ask more than continuity at the vertices of $T_0$. Note also that the restriction of $\chi$ to any compact interval included in the interior of any edge is a smooth embedding.

In Sect. 3.1 below, we explain why Theorem 13 implies Theorem 2. The proof of Theorem 13 then occupies Sects. 3.2 and 3.3.

3.1 From an invariant tree to a single fixed point

In this section, we explain how one can build a Hamiltonian homeomorphism with a single fixed point from a Hamiltonian homeomorphism preserving an embedded tree (see Definition 14) that contains all of its fixed points; in other terms, we prove that Theorem 13 implies Theorem 2. This will rely on the following proposition.

Proposition 15

Let $\psi$ be a symplectic homeomorphism of $(M,\omega )$, and $T\subset M$ be an embedded tree which is invariant under $\psi$, that is $\psi (T)=T$. Assume that all the fixed points of $\psi$ are contained in T. Then, there exists $f \in \mathrm {Sympeo}(M, \omega )$, with only one fixed point p, and such that $f\circ \psi ^{-1}\in \mathrm {Hameo}(M,\omega )$.

Moreover, if $\psi$ is smooth on $M{\setminus } T$, then f can be chosen to be smooth on $M{\setminus }\{p\}$. If $\psi$ coincides with a Hamiltonian diffeomorphism in the complement of any neighborhood of T, then f can be chosen in any normal subgroup of $\mathrm {Sympeo}(M,\omega )$ which contains $\mathrm {Ham}(M,\omega )$.

Note that in the last sentence of the above proposition, we do not claim that f can be chosen to be smooth on $M{\setminus } \{p\}$ and simultaneously be contained in any normal subgroup of $\mathrm {Sympeo}(M, \omega )$ which contains $\mathrm {Ham}(M, \omega )$. As will be clear from the proof, our method of building f in the normal closure of $\mathrm {Ham}(M,\omega )$ has the effect of creating a second non-smooth point. We do not know if both properties can be satisfied at the same time.

Proof of Theorem 2

Theorem 13 provides us with a Hamiltonian homeomorphism $\psi$ which satisfies all the requirements of Proposition 15. Thus, there exists a symplectic homeomorphism f with only one fixed point p and such that $f\circ \psi ^{-1}\in \mathrm {Hameo}(M,\omega )$. Since $\psi \in \mathrm {Hameo}(M,\omega )$ we deduce that $f\in \mathrm {Hameo}(M,\omega )$ as well. Moreover, $\psi$ coincides with a Hamiltonian diffeomorphism in the complement of any neighborhood of T. This implies that f can be chosen in any normal subgroup of $\mathrm {Sympeo}(M,\omega )$ containing $\mathrm {Ham}(M,\omega )$. It also implies that $\psi$ is smooth in the complement of T and hence that f can be chosen to be smooth in the complement of p. $\square$

The following lemma will be useful for the proof Proposition 15.

Lemma 16

Let $(M, \omega )$ be a symplectic manifold of dimension at least 4 and let $T\subset M$ be an embedded tree. Let d be a Riemannian distance on M. Then, for every $\varepsilon >0$ and every open neigborhood U of T, there exists a Hamiltonian function H supported in U such that $\mathrm {diam}(\phi _H^1(T))\leqslant \varepsilon$ and $\Vert H\Vert _{\infty }\leqslant \varepsilon$.

Proof

Let $T_0$ be a finite tree and $\chi :T_0\rightarrow T$ a map as in Definition 14. Let U be an open set containing T. Without loss of generality, by replacing U by a smaller open set if needed, we may assume that U is simply connected. We first pick a Hamiltonian function $H_0$ supported in U, such that $\phi _{H_0}^1$ maps all the vertices of T inside a ball B included in U and of diameter less than $\varepsilon$. By continuity of $\chi$, there exist an open neighborhood W of the set of vertices of T and a closed subinterval $J_i\subset \mathrm {Int}(I_i)$, for every edge $I_i$, $i=1,\dots ,r$ of $T_0$, such that

• $T=(W\cap T)\cup \chi (J_1)\cup \dots \cup \chi (J_r)$,

• $\phi _{H_0}^1(W)\subset B$.

To achieve the proof, we only need to find a Hamiltonian isotopy which moves the pieces of curves $\chi (J_1), \ldots , \chi (J_r)$ into B with the endpoints kept in B along the isotopy. This can be achieved by successively choosing Hamiltonian functions $H_i$’s (for $i=1,\dots ,r$) such that for all i, $\phi _{H_i}^1(\phi _{H_0}^1(\chi (J_i)))\subset B$ and the support of $H_i$ meets neither $\phi _{H_0}^1(W\cap T)$ nor any of the curves $\phi _{H_k}^1(\chi (J_k))$, for $k=1,\dots ,i-1$ and $\chi (J_k)$, for $k=i+1,\dots ,r$. Then, the Hamiltonian diffeomorphism $h=\phi _{H_r}^1\circ \dots \circ \phi _{H_1}^1\circ \phi _{H_0}^1$ sends T into B, hence $\mathrm {diam}(h(T))\leqslant \varepsilon$.

Moreover, it is a standard fact that the above Hamiltonian functions $H_0, \dots ,H_r$ can be chosen arbitrarily small in the $\Vert \cdot \Vert _\infty$ norm. Thus, h can be generated by a Hamiltonian H satisfying $\Vert H\Vert _{\infty }\leqslant \varepsilon$. $\square$

Proof of Proposition 15

Let $\psi$ and T be as in the statement of the proposition. We will build the symplectic homeomorphism f as a $C^0$-limit of conjugates of $\psi$.

For that purpose, let $W_0\supset W_1\supset \dots \supset W_k\supset \dots \supset T$ be a sequence of nested open neighborhoods of T such that $\bigcap _{k\geqslant 0}W_k=T$. $\square$

Claim 17

There exists a sequence of open sets $(U_i)_{i\in \mathbb {N}}$, a sequence of Hamiltonian diffeomorphisms $(h_i=\phi _{H_i}^1)_{i\in \mathbb {N}}$ and a subsequence $(W_{k_i})_{i\in \mathbb {N}}$ with the following properties: For all $i\in \mathbb {N}$,

• $U_i=\varphi _i(W_{k_i})$, where $\varphi _i=h_i\circ \dots \circ h_1\circ h_0$,

• $U_{i+1}\subset U_i$,

• $H_{i+1}$ is supported in $U_i$,

• $\mathrm {diam}(\overline{U_{i+1}})\leqslant \frac{1}{3^i}$,

• $\Vert H_i\Vert _\infty \leqslant \frac{1}{3^i}$.

Proof

We will construct these sequences by induction. First set $U_0=W_0$, $H_0=0$ and $k_0=0$. Then, assume that we have constructed sequences $(U_i)_{i\in \{0,\ldots ,j\}}$, $(H_i)_{i\in \{0,\ldots ,j\}}$, $(W_{k_i})_{i\in \{0,\ldots ,j\}}$, with the desired properties.

According to Lemma 16, applied to the tree $\varphi _j(T)$, we can pick a Hamiltonian function $H_{j+1}$ supported in $U_j$, such that $\Vert H_{j+1}\Vert _\infty \leqslant \frac{1}{3^{j+1}}$ and, denoting $h_{j+1}=\phi _{H_{j+1}}^1$, $h_{j+1}(\varphi _j(T))$ is included in a ball $B_j\subset U_j$, of diameter less than $\frac{1}{3^{j}}$. Then pick $k_{j+1}>k_j$ sufficiently large so that

$$\begin{aligned} h_{j+1}\left( \varphi _j(W_{k_{j+1}})\right) \subset B_j. \end{aligned}$$

Then, $\mathrm {diam}(h_{j+1}(\varphi _j(W_{k_{j+1}})))\leqslant \tfrac{1}{3^j}$ and we can set $U_{j+1}=h_{j+1}(\varphi _j(W_{k_{j+1}}))$. Since $B_j\subset U_j$, we have $U_{j+1}\subset U_j$ and the three sequences $(U_i)_{i\in \{0,\ldots ,j+1\}}$, $(H_i)_{i\in \{0,\ldots ,j+1\}}$, $(W_{k_i})_{i\in \{0,\ldots ,j+1\}}$ still have the required properties. By induction, we obtain the claimed infinite sequences. $\square$

Since $\mathrm {diam}(\overline{U_i})$ converges to 0 when i goes to infinity and since the $U_i$’s are nested, the intersection of the closures $\bigcap _{i\in \mathbb {N}}\overline{U_i}$ is a single point. Let p denote this point.

Consider the sequence of maps $\varphi _i=h_i\circ \dots \circ h_1$. By construction, if $x\in T$ then $\varphi _i(x)\in U_i$, hence it converges to p. Moreover, for every neighborhood U of T, the restrictions $\varphi _i|_{M{\setminus } U}$ stabilize for i large. For every $x\notin T$, denote by $\varphi (x)$ the point $\varphi _i(x)$ for i large enough. The map $\varphi$ is a diffeomorphism from $M{\setminus } T$ to $M{\setminus } \{p\}$.

We define for all $x\in M{\setminus }\{p\}$,

$$\begin{aligned} f(x)=\varphi \circ \psi \circ \varphi ^{-1}(x), \end{aligned}$$

and $f(p)=p$. We see that p is the unique fixed point of f. Indeed, if we assume that f admits another fixed point $q\ne p$ then, $\varphi (q)$ would be a fixed point of $\psi$ which is not contained in T and this would be a contradiction.

The first part of Proposition 15 thus follows if we prove that f is a symplectic homeomorphism. This will be a consequence of the next claim, which requires us to introduce additional notations.

Without loss of generality, we may assume that for all i, the function $H_i(t,\cdot )$ vanishes for t in the complement of an open subinterval of [0, 1]. Let $\tau _0=0$ and for every $i=1,2,\dots$, let $\tau _i=\sum _{k=1}^i\frac{1}{2^{k}}$. We consider the sequence of smooth Hamiltonian functions $K_i$ defined as concatenations of time-reparametrizations of the $H_i$’s as follows:

$$\begin{aligned} K_i(t,x) ={\left\{ \begin{array}{ll} 2^{k+1}H_k\left( 2^{k+1}(t-\tau _k),x\right) , &{}\quad \forall x\in M,\ k=0,1,\dots ,i,\ t\in [\tau _k,\tau _{k+1}] \\ 0 &{}\quad \forall x\in M,\ t\in [\tau _{i+1},1]. \end{array}\right. } \end{aligned}$$

Each Hamiltonian $K_i$ generates the smooth isotopy $\varphi _i^t$ given by:

$$\begin{aligned} \varphi _i^t = {\left\{ \begin{array}{ll} \phi _{H_k}^{2^{k+1}(t-\tau _k)}\circ h_{k-1}\circ \dots \circ h_0, &{}\quad \forall k=0,1,\dots ,i,\ t\in [\tau _k,\tau _{k+1}] \\ h_i\circ \dots \circ h_0, &{}\quad \forall t\in [\tau _{i+1},1]. \end{array}\right. } \end{aligned}$$

Note that by construction, for all indices $i<j$, one has $K_i=K_j$ and $\varphi _i^t=\varphi _j^t$ on the time interval $[0,\tau _{i+1}]$. Also, note that for $t=1$, we have $\varphi _i^1=\varphi _i$.

Claim 18

The isotopies $\varphi _i^t\circ \psi \circ (\varphi _i^t)^{-1}$ $C^0$-converges to an isotopy of homeomorphisms $f^t$ as $i\rightarrow \infty$, satisfying $f^1=f$. In particular, f is a symplectic homeomorphism.

Proof of Claim 18

We denote $f_i^t=\varphi _i^t\circ \psi \circ (\varphi _i^t)^{-1}$ and let, for $t\in [0,1)$ $f^t$ be the homeomorphism defined by $f^t = f_i^t$ for any $i\in \mathbb {N}$ such that $\tau _{i+1}\geqslant t$. We set $f^1=f$.

First, note that by construction f is a bijection. Note also that on any time interval, of the form $[0,1-\delta ]$, with $\delta >0$, $f^t$ coincides with $f_i^t$ for i large enough. We will prove that the sequences $f_i^t$ and $(f_i^t)^{-1}$ respectively $C^0$-converge to $f^t$ and $(f^t)^{-1}$. This will imply that $f^t$ is a homeomorphism for all t (including $t=1$) and prove the claim.

Let B be a ball around p. For i large enough, $U_i\subset B$, hence $\varphi _{j}^t(W_{k_i})\subset B$ for all $j\geqslant i$ and $t\in [\tau _{j+1},1]$. The uniform continuity of $\psi$ and the invariance of T implies that for $j\geqslant i$ large enough, $\psi (W_{k_j})\subset W_{k_i}$. Thus,

$$\begin{aligned} f_{j}^t(U_j)=\varphi _{j}^t\circ \psi \circ \left( \varphi _{j}^t\right) ^{-1}(U_j)\subset B \end{aligned}$$

(1)

for j large enough and $t\in [\tau _{j+1},1]$. Let $j_0$ be such a large j.

For all j, it can be easily checked that

$$\begin{aligned} f_{j+1}^t=\rho _j^t\circ f_j^t\circ \left( \rho _j^t\right) ^{-1}, \end{aligned}$$

where

$$\begin{aligned} \rho _j^t= {\left\{ \begin{array}{ll} \mathrm {Id}, &{}\quad \forall t\in [0,\tau _{j+1}]\\ \phi _{H_{j+1}}^{2^{j+2}(t-\tau _{j+1})}, &{}\quad \forall t\in [\tau _{j+1},\tau _{j+2}]\\ h_{j+1}, &{}\quad \forall t\in [\tau _{j+2},1]. \end{array}\right. } \end{aligned}$$

It follows that if $t\in [0,\tau _{j+1}]$, then $(f_{j+1}^t)^{-1}(x)=(f_j^t)^{-1}(x)$ for all $x\in M$. If $t\in [\tau _{j+1}, 1]$, then for all $j\geqslant j_0$, $\rho _j^t$ is supported in $U_{j}\subset U_{j_0}\subset B$, thus for all $x\notin B$, (1) yields $(f_j^t)^{-1}\circ (\rho _j^t)^{-1}(x)=(f_j^t)^{-1}(x)\notin U_j$. This implies that $(f_{j+1}^t)^{-1}(x)=(f_j^t)^{-1}(x)$ for all $x\notin B$. Thus, the sequence $(f_i^t)^{-1}$ stabilizes in the complement of B, independently of t. Moreover, by construction the limit point of $(f_i^t)^{-1}(x)$ for $x\notin B$ is nothing but $(f^t)^{-1}(x)$.

It follows that for j large enough, $f_j^t\circ (f^t)^{-1}$ is supported in B, hence is $C^0$-close to $\mathrm {Id}$. This shows that $f_i^t$ $C^0$-converges to $f^t$. The same argument shows that $(f_j^t)^{-1}$ converges to $(f^t)^{-1}$. $\square$

We will now prove that the homeomorphism $f\circ \psi ^{-1}$ is a Hamiltonian homeomorphism.

The homeomorphism $f\circ \psi ^{-1}$ is the time one map of the isotopy $f^t\circ \psi ^{-1}$. According to Claim 18, this isotopy is the $C^0$-limit of the isotopies $\varphi _i^t\circ \psi \circ (\varphi _i^t)^{-1}\circ \psi ^{-1}$. Since these isotopies are generated by the continuous Hamiltonians

$$\begin{aligned} F_i(t,x)=K_i(t,x)-K_i\left( t,\varphi _i^t\circ \psi ^{-1}\circ \left( \varphi _i^t\right) ^{-1}(x)\right) , \end{aligned}$$

the following claim implies that $f^t\circ \psi ^{-1}$ is a hameotopy and hence that $f\circ \psi ^{-1}$ is a Hamiltonian homeomorphism.

Claim 19

The continuous Hamiltonians $F_i$ converge uniformly as $i\rightarrow \infty$.

Proof

The condition $\Vert H_i\Vert _{\infty }\leqslant \frac{1}{3^i}$ implies that the sequence of Hamiltonians $K_i$ converges uniformly to the continuous function K given for all $x\in M$ and all $t\in [0,1)$ by $K(t,x) = 2^{k+1}H_k(2^{k+1}(t-\tau _k),x)$ with $k\in \mathbb {N}$ such that $t\in [\tau _k,\tau _{k+1}]$, and $K(1,x)=0$. Now according to Claim 18, $\varphi _i^t\circ \psi ^{-1}\circ (\varphi _i^t)^{-1}$ converges uniformly to $(f^t)^{-1}$. Therefore, $F_i(t,x)$ converges uniformly to $K(t,x)-K(t,(f^t)^{-1}(x))$.

We now pursue the proof of Proposition 15. Since $\varphi$ is a smooth from $M{\setminus } T$ to $M{\setminus }\{p\}$, it is obvious that if $\psi$ is smooth in the complement of T, then the above construction provides a symplectic homeomorphism f which is smooth in the complement of p.

Let us now assume that $\psi$ coincides with a Hamiltonian diffeomorphism on the complement of any neighborhood of T. Let f be constructed as above. We will modify f to define a new symplectic homeomorphism $\tilde{f}$ which is in any normal subgroup of $\mathrm {Sympeo}(M,\omega )$ containing $\mathrm {Ham}(M,\omega )$.

Let K be a smooth Hamiltonian whose time-one map does not fix the point p. Then $\phi _K^1(p)\ne f\circ \phi _K^1(p)$ and we can find a small enough ball B around p, such that $\phi _K^1(B)\cap f\circ \phi _K^1(B)=\emptyset$.

For i large enough, f coincides with $\varphi _i \circ \psi \circ \varphi _i^{-1}$ on $M{\setminus } B$. Since by assumption $\psi$ coincides with some Hamiltonian diffeomorphism on $M{\setminus } \varphi _i^{-1}(B)$, we deduce that we can write $f=h\circ g$, where h is a Hamiltonian diffeomorphism and g is a symplectic homeomorphism which is the identity on the complement of B. Now set

$$\begin{aligned} \tilde{f}=h\circ g\circ \phi _K^{1}\circ g^{-1}\circ \left( \phi _K^1\right) ^{-1}. \end{aligned}$$

We see that $\tilde{f}$ belongs to any normal subgroup of $\mathrm {Sympeo}(M,\omega )$ containing $\mathrm {Ham}(M,\omega )$. We claim that p is the only fixed point of $\tilde{f}$. To see this, first note that $\phi _K^{1}\circ g^{-1}\circ (\phi _K^1)^{-1}$ is the identity on the complement of $\phi _K^1(B)$. It follows in particular that $f=h\circ g$ and $\tilde{f}$ coincide on $M{\setminus } \phi _K^1(B)$, hence that p is the only fixed point of $\tilde{f}$ in $M{\setminus } \phi _K^1(B)$. But if $x\in \phi _K^1(B)$, then $\tilde{f}(x)$ belongs to $f\circ \phi _K^1(B)$, hence is distinct from x.

We point out that since g has only one non-smooth point, we see that $\tilde{f}$ has two non-smooth points, p and $\phi _K^1(p)$. $\square$

Remark 20

We can now justify what we claimed in the discussion of Sect. 1.3, namely the fact that our Hamiltonian homeomorphism f with a unique fixed point can be generated by a continuous Hamiltonian admiting cl(M) distinct spectral invariants. As in Sect. 1.3, we restrict our discussion to the case of an aspherical symplectic manifold.

We begin by recalling that spectral invariants of a smooth Hamiltonian depend on the Hamiltonian Lipschitz continuously; see [34]. It follows that one can define spectral invariants for any continuous function.

The argument showing that f can be generated by a continuous Hamiltonian with, at least, cl(M) distinct spectral invariants requires four steps:

• (i) The initial Hamiltonian H of the construction (see Theorem 13) is a $C^2$-small Morse function. For such a Hamiltonian, Floer theory is nothing but Morse theory and it follows from the classical Lusternik–Schnirelman theory that H must have at least cl(M) distinct spectral invariants.

• (ii) The Hamiltonian homeomorphism $\psi$ obtained in Theorem 13 can be chosen so that $\psi \circ \phi _H^{-1}$ is generated by a Hamiltonian arbitrarily small in $\Vert \cdot \Vert _\infty$ norm. By continuity of spectral invariants, this implies that $\psi$ can be generated by a (continuous) Hamiltonian, which will be denoted by G, with spectral invariants close to those of H.

• (iii) In the above proof of Proposition 15, the Hamiltonian homeomorphisms $f_i=\varphi _i\circ \psi \circ \varphi _i^{-1}$ are conjugate to $\psi$. Hence, they can be generated by the Hamiltonians $G \circ \varphi _i^{-1}$ which have the same spectral invariants as G.

• (iv) Our Hamiltonian homeomorphism f is constructed so that for i large, $f_i^{-1} \circ f$ is generated by a uniformly small Hamiltonian function $\widetilde{F_i}$. Thus, f is generated by the Hamiltonian $(G \circ \varphi _i^{-1}) \# \widetilde{F_i}$ whose spectral invariants are close to those of $G \circ \varphi _i^{-1}$ and hence to those of H. By choosing small enough perturbations, we can ensure that at least cl(M) of these spectral invariants are distinct.

Remark 21

We should point out the argument presented in the above remark can be modified to prove the following (stronger) statement on closed symplectic manifolds which are not necessarily aspherical: the Hamiltonian homeomorphism f can be generated by a continuous Hamiltonian, say G, whose spectral invariants are exactly the same as the spectral invariants of the initial $C^2$-small Morse function H.

In fact, one could go even further: it is possible to show that the continuous Hamiltonian G has the exact same barcode as the initial $C^2$-small Morse function H. Hence, despite the fact that the time-1 map of G has only one fixed point, from a Floer theoretic point of view G can not be distinguished from H. For further information on the theory of barcodes see [1, 27, 32, 36].

Although these claims, and their proofs, are very interesting, in the interest of not lengthening the paper we do not present them here.

3.2 Building the tree from a Morse function

In this section we begin the proof of Theorem 13. Our first step will be to perturb our initial $C^2$-small Morse function H so that it satisfies a number of additional properties. This is the content of the following lemma. In order to simplify our presentation, throughout the rest of this section, we will refer to local maxima/minima of a function as maxima/minima. An extremum point of a function will be a point which is either a local maximum or a local minimum.

Lemma 22

On every closed symplectic manifold $(M,\omega )$ and for every Morse function $\tilde{H}$ on M, there exists a Morse function H on M, arbitrarily $C^1$-close to $\tilde{H}$, with the following set of properties:

• 1. The function H takes distinct values at distinct critical points,

• 2. Every critical point p of H which is an extremum admits a neighborhood with Darboux coordinates $(x_1,\ldots ,x_n,y_1,\ldots ,y_n)$ in which H is of the form $H(p) + c\sum (x_i^2+y_i^2)$, where $c\in \mathbb {R}{\setminus }\{0\}$ is a constant which can be chosen to have arbitrarily small magnitude,

• 3. For every critical point of H which is not extremal, there exist local Darboux coordinates $(x_1,\ldots ,x_n,y_1,\ldots ,y_n)$ in which $H = c(x_1^2 -y_1^2) +Q$, where c is some non-zero constant, and Q is a quadratic form in the variables $(x_2, \ldots , x_n, y_2, \ldots , y_n)$.

The relevant consequence of the third condition is that locally near p, the Hamiltonian flow of H preserves the symplectic 2-plane $P=\{(x_1,0,\dots ,0,y_1, 0,\dots ,0)\}$ and acts as a linear hyperbolic flow on it. Indeed, $X_H = X_{ c(x_1^2 - y_1^2)} + X_{Q}$, where $X_H, X_{c(x_1^2 - y_1^2)}, X_{Q}$ denote the associated Hamiltonian vector fields. It can easily be checked that $X_{Q}|_P = 0$. Hence, the restriction of the Hamiltonian flow of H to the plane P coincides with the Hamiltonian flow of $c(x_1^2 -y_1^2)$.

Proof

We can assume without loss of generality that $\tilde{H}$ is a Morse function on M which has pairwise distinct critical values. We then modify $\tilde{H}$ near every critical point as follows. Let p be a critical point of $\tilde{H}$. Up to addition of a constant, we may assume that $\tilde{H}(p)=0$.

Assume that p is a local minimum and let $(x_1,\ldots ,x_n, y_1,\ldots ,y_n)$ be Darboux coordinates near p. We first make a $C^2$-small perturbation of $\tilde{H}$ so that in a small neighborhood of p, it coincides with its Hessian near p. Then, note that given two positive definite quadratic forms $Q_1\leqslant Q_2$ and two open subsets U, V such that $0\in U\subset \overline{U}\subset V$, there always exists a smooth function h which coincides with $Q_1$ on U and with $Q_2$ on the complement of V, and having 0 as its only critical point. Now define H by replacing $\tilde{H}$ by such a function h obtained in a neighborhood of p from the quadratic forms $Q_1=c\sum x_i^2+y_i^2$ for some small $c>0$ and for $Q_2$ the Hessian of H. This perturbation can be made arbitrarily $C^1$-small by using a small neighborhood of p. Local maxima are worked out similarly.

Now assume that p is not an extremum. By the Morse lemma, and since p is not an extremum, there exists a local chart on a neighborhood of p, parametrised via coordinates $(v_1,\dots ,v_n,w_1,\dots , w_n)$ by a small open ball W centered at the origin in the Euclidean space $\mathbb {R}^{2n}$, such that $p = (0,0, \ldots , 0)$ in these coordinates, and moreover $\tilde{H}$ has the form $\tilde{H}= \tilde{H}(p) + v_1^2-w_1^2+\sum _{i=2}^n\pm v_i^2\pm w_i^2$. Now, let $(x_1, \ldots , x_n , y_1, \ldots , y_n)$ be Darboux coordinates in a neighborhood of p, such that $p = (0,0, \ldots , 0)$ in these coordinates as well. Then, choose a diffeomorphism $\phi$ of M, supported in a very small ball around p, such that near p, $\phi$ carries the coordinate system $(v_1,\dots ,v_n,w_1,\dots , w_n)$ to the coordinate system $(x_1, \ldots , x_n , y_1, \ldots , y_n)$. Consider its “rescalings” $\phi _\lambda : M \rightarrow M$, for $\lambda \in (0,1)$, defined by

$$\begin{aligned} \phi _\lambda (v_1,\dots ,v_n, w_1,\dots , w_n) = \lambda \phi \left( \frac{v_1}{\lambda },\dots ,\frac{v_n}{\lambda },\frac{w_1}{\lambda },\dots , \frac{w_n}{\lambda }\right) ,\end{aligned}$$

for $(v_1,\dots ,v_n,w_1,\dots , w_n) \in \lambda W \subset \mathbb {R}^{2n}$, and by the identity on the complement of $\lambda W$. Now, replacing $\tilde{H}$ by the pushforward $H := (\phi _{\lambda })_* \tilde{H} = \tilde{H} \circ (\phi _{\lambda })^{-1}$, we get that for small enough $\lambda$, H is a smooth function on M which is $C^1$-close to $\tilde{H}$, and which has the form $H = H(p) + x_1^2-y_1^2+\sum _{i=2}^n\pm x_i^2\pm y_i^2$ in a small neighborhood of p. $\square$

In the above lemma, the fact that H and $\tilde{H}$ are $C^1$-close implies that $d_{C^0}(\phi _{\tilde{H}}^1,\phi _H^1)$ is small and $\phi _H^1\circ \phi _{\tilde{H}}^{-1}$ is generated by a uniformly small Hamiltonian. Hence, Theorem 13 now amounts to the following proposition.

Proposition 23

Let H be a Morse function on M satisfying the properties of Lemma 22. If H is sufficiently $C^2$-small, then for every $\varepsilon ,\rho >0$, there exists $\psi \in \mathrm {Hameo}(M,\omega )$ and an embedded tree T (see Definition 14) such that:

• T is $\psi$-invariant, i.e. $\psi (T)=T$,

• T contains all the fixed points of $\psi$,

• $d_{C^0}(\phi _H^1,\psi )<\varepsilon$ and $\psi \circ \phi _H^{-1}$ is generated by a continuous Hamiltonian F such that $\Vert F\Vert _{\infty }<\rho$,

• $\psi$ coincides with a Hamiltonian diffeomorphism in the complement of any neighborhood of T.

Our proof of Proposition 23 will make use of the notion of Reeb graph whose definition we now recall.

Definition 24

We assume that a function $H:M\rightarrow \mathbb {R}$ is given. For every point $x\in M$ we define $\mathcal {C}(x)$ as the connected component of x in the level set $H^{-1}(H(x))$. For a subset $X\subset M$, we define $\mathcal {C}(X)=\bigcup _{x\in X} \mathcal {C}(x)$.

The Reeb graph is the quotient space $\mathcal {R}=M/\sim$, where $\sim$ is the equivalence relation given by $x\sim y$ if and only if $\mathcal {C}(x)=\mathcal {C}(y)$.

It follows from basic Morse theory that if H is a Morse function whose critical values are pairwise distinct, then the space $\mathcal {R}$ actually carries the structure of a graph whose vertices correspond to the critical points of H and such that an edge $e_{p,q}$ between two critical points p and q corresponds to a connected open subset $U_{p,q}\subset M$ such that:

• $U_{p,q}$ contains no critical point of H,

• The canonical projection $U_{p,q}\rightarrow U_{p,q}/\sim$ is a fibration onto an interval (this interval can be seen as parametrizing the edge),

• The closure $\overline{U_{p,q}}$ has two boundary components, one containing p and the other containing q.

Fig. 1

The Reeb graph of a height function H on a 2-torus, the bi-degrees of the different vertices (Note that the bi-degree (1, 1) cannot happen on a surface) and an example of an edge $e_{p,q}$ with the corresponding open set $U_{p,q}$

By construction, H descends to a well defined function on $\mathcal {R}$, which we still denote H. This function is monotone on each edge. We assign orientations to the edges so that H is decreasing on each of them. As for any oriented graph, any vertex p admits a bi-degree which is a couple $(d_-(p),d_+(p))$, where $d_-(p)$ is the number of edges with head end at p and $d_+(p)$ is the number of edges with tail end at p. This bi-degree is related to the Morse index of p. Indeed, local maxima have bi-degree (0, 1), local minima have bi-degree (1, 0), and all the other points have bi-degree (1, 1), (2, 1) or (1, 2) (but (2, 1) is only possible for points of Morse index $\dim (M)-1$ and (1, 2) is only possible for points of Morse index 1).

Proof of Proposition 23

Let $\varepsilon ,\rho >0$. Since M is connected, so is the Reeb graph $\mathcal {R}$ of H, and there exists a subgraph $\mathcal {T}$ of $\mathcal {R}$ which is a tree and contains all the vertices of $\mathcal {R}$. To each vertex of $\mathcal {T}$ we associate the corresponding critical point of H. This gives an embedding of the vertices of $\mathcal {T}$ into M. The complicated part of the proof will be to embed the edges.

By assumption, near every critical point p of H which is not extremal, there exist Darboux coordinates $(x_1,\ldots ,x_n,y_1,\ldots ,y_n)$ such that $H= c(x_1^2 -y_1^2) + Q'$, where $c\ne 0$ and $Q'$ is a quadratic form whose kernel is the plane $P=\{(x_1,0,\dots ,0,y_1,0,\dots ,0)\}$. As mentioned after the statement of Lemma 22, this implies in particular that locally near p, the flow of H preserves P and acts on it as the flow of the quadratic form $c(x_1^2-y_1^2)$, which is linear hyperbolic. Thus the flow of H admits in P two orbits converging to p when time goes to $+\infty$ and two orbits converging to p when time goes to $-\infty$. Moreover, in the plane P and still locally near the point p, these four orbits are the frontiers of four regions; two of them correspond to $H < H(p)$ and the two others to $H > H(p)$.

To each edge $e_{p,q}$ with tail end at p, we associate one of the two orbits converging to p in the past. The local picture above shows that this orbit belongs to the closure $\overline{U_{p,q}}$. Whenever there are two such edges (i.e. $d_+(p)=2$) we demand that the two associated orbits are distinct. Again, the local picture shows that this is possible. We let $x_{p,q}$ be a point located near p on the orbit associated to $e_{p,q}$. To summarize the situation, we have:

$$\begin{aligned} \overline{U_{p,q}}\ni \phi _H^t(x_{p,q}){\mathop {\longrightarrow }\limits ^{t\rightarrow -\infty }} p. \end{aligned}$$

We associate in a similar way to each edge $e_{q,p}$ with head end at p a point $y_{q,p}$ such that:

$$\begin{aligned} \overline{U_{q,p}}\ni \phi _H^t(y_{q,p}){\mathop {\longrightarrow }\limits ^{t\rightarrow +\infty }}p. \end{aligned}$$

For each oriented edge $e_{p,q}$ we now choose a path $\gamma _{p,q}:[0,1]\rightarrow M$ satisfying the following properties (Fig. 2):

• For all $t\in [0,1]$, $\frac{d}{dt}(H\circ \gamma _{p,q}(t))<0$,

• $\gamma _{p,q}(0)=p$ if p is a local maximum and $\gamma _{p,q}(0)=x_{p,q}$ otherwise,

• $\gamma _{p,q}(1)=q$ if q is a local minimum and $\gamma _{p,q}(1)=y_{p,q}$ otherwise.

Note that it follows from the above properties that $U_{p,q}=\mathcal {C}(\gamma ((0,1)))$.

For each non extremal critical point p, and for each piece of orbit $\beta$ which is either of the form $\{\phi _H^t(x_{p,q}):t\in (-\infty ,1]\}$ or $\{\phi _H^t(y_{q,p}):t\in [-1,+\infty )\}$, we choose some open sets $V_p(\beta )$, such that

$$\begin{aligned} \mathrm {image}(\beta )\subset V_p(\beta )\subset M{\setminus }\{p\}. \end{aligned}$$

We choose these open sets sufficiently small such that all the $V_p(\beta )$’s, for p ranging over all non-extremal critical points of H and $\beta$ over all pieces of orbits as above, are pairwise disjoint.

Fig. 2

The right-hand side represents the tree $\mathcal {T}$ obtained from the Reeb graph of Fig. 1. On the left hand side, the arrows correspond to the four orbits converging to non extremal points, and the blue curves represent possible choices for the curves $\gamma _{p,q}$

We are now ready to construct the perturbation of H and the embedding of $\mathcal {T}$. We will proceed by induction on the edges of $\mathcal {T}$. For that purpose we number the edges $e_1,e_2,\dots ,e_N$, and for each index i, we let $p_i$ and $q_i$ be respectively the tail end and head end of $e_i$. With the previous notations, this means that $e_i=e_{p_i,q_i}$. We also rename the points $x_i=x_{p_i,q_i}$ and $y_i=y_{p_i,q_i}$, when they are defined and the paths $\gamma _i=\gamma _{p_i,q_i}$. For the sake of brevity, we omit the piece of orbit from the notation and denote $V_{p_i}$ and $V_{q_i}$ the corresponding open subsets of the form $V_p(\beta )$ when they are defined; it will always be clear from the context which orbit is considered. If $p_i$ is a local maximum, we set $V_{p_i}=\emptyset$; if $q_i$ is a local minimum, we set $V_{q_i}=\emptyset$.

Assume that for some $j\geqslant 0$, we have realized the edges $e_1,\dots ,e_j$ in M, i.e, we have built smooth embeddings $\alpha _1,\dots ,\alpha _j:\mathbb {R}\rightarrow M$ and Hamiltonian homeomorphisms $\theta _1,\dots ,\theta _j$, such that the following holds:

• 1. $\lim _{t\rightarrow -\infty }\alpha _i(t)=p_i$ and $\lim _{t\rightarrow +\infty }\alpha _i(t)=q_i$,

• 2. For all $i\in \{1,\dots ,j\}$ and all $t\in \mathbb {R}$, $\theta _i\circ \phi _H^1(\alpha _i(t))=\alpha _i(t+1)$,

• 3. $\theta _i\circ \phi _H^1$ has the same fixed points as $\phi _H^1$,

• 4. For all $i\in \{1,\dots ,j\}$, $\alpha _i$ takes values in $V_{p_i}\cup \mathcal {C}(\gamma _i((0,1)))\cup V_{q_i}$, and the images of the $\alpha _i$’s are pairwise disjoint,

• 5. For all $i\in \{1,\dots ,j\}$, $\theta _i$ is generated by a continuous Hamiltonian $F_i$ supported in $\{p_i\}\cup V_{p_i}\cup \mathcal {C}(\gamma _i((0,1)))\cup V_{q_i}\cup \{q_i\}$, and the interiors of the supports of the $\theta _i$’s are pairwise disjoint,

• 6. $d_{C^0}(\theta _i,\mathrm {Id})<\varepsilon$ and $\Vert F_i\Vert _{\infty }<\rho$.

• 7. For any neighborhoods of $p_i,q_i$, the homeomorphism $\theta _i$ coincides with a Hamiltonian diffeomorphism in the complement of the union of those neighborhoods.

We want to show that it is possible to build $\alpha _{j+1}$ and $\theta _{j+1}$ such that the above Properties 1–7 still hold. After shrinking the open sets $V_{p_{j+1}}$, $V_{q_{j+1}}$ if needed so that they intersect neither the supports of $F_1,\dots ,F_j$ nor the images of the curves $\alpha _1,\dots ,\alpha _j$, we construct $\alpha _{j+1}$ and $\theta _{j+1}$ by applying Theorem 25 below to the critical points $p_{j+1}$, $q_{j+1}$, to the curve $\gamma _{j+1}$ and to the open subsets $V_{p_{j+1}}$, $V_{q_{j+1}}$.

Following this process by induction, we get curves $\alpha _1,\dots ,\alpha _N$ and Hamiltonian homeomorphisms $\theta _1,\dots ,\theta _N$ satisfying Properties 1–7. The union of the images $\alpha _i$, denoted by T, is an embedding of the tree $\mathcal {T}$ in M in the sense of Definition 14. Moreover, it is easy to check that the Hamiltonian homeomorphism $\psi =\theta _N\circ \dots \circ \theta _1\circ \phi _H^1$ meets the requirements of Proposition 23. $\square$

3.3 Connecting two critical points

The goal of this section is to explain the construction of the individual edges of the invariant tree of Proposition 23. This is achieved in Theorem 25.

3.3.1 Statement of the result

Theorem 25

Let H be a Morse function as described in Lemma 22 and denote by p, q two critical points of H, at most one of them being an extremum. We assume that there exists a smooth embedded curve $\gamma :[0,1]\rightarrow M$ such that

• $\gamma (0)=p$ if p is a maximum of H, and $\phi _H^t(\gamma (0))\rightarrow p$ when t goes to $-\infty$ if p is not a maximum,

• $\gamma (1)=q$ if q is a minimum of H, and $\phi _H^t(\gamma (1))\rightarrow q$ when t goes to $+\infty$ if q is not a minimum,

• for all $t\in [0,1]$, $\frac{d}{dt}(H(\gamma (t))) <0$.

Let $\varepsilon ,\rho >0$ be positive real numbers, let $V_p,V_q$ be open sets such that

$$\begin{aligned} {\left\{ \begin{array}{ll} V_p =\emptyset , \quad &{}\text { if }p\text { is a maximum,}\\ V_p \supset \{\phi _H^t(\gamma (0)):-\infty <t\leqslant 1\}, \quad &{}\text { if }p\text { is not a maximum,} \end{array}\right. } \end{aligned}$$

and

$$\begin{aligned} {\left\{ \begin{array}{ll} V_q =\emptyset , \quad &{}\text { if }q\text { is a minimum,} \\ V_q \supset \left\{ \phi _H^t(\gamma (1)):-1\leqslant t<+\infty \right\} , \quad &{}\text { if }q\text { is not a minimum.} \end{array}\right. } \end{aligned}$$

Then, there exist $\theta \in \mathrm {Hameo}(M,\omega )$ and a smooth embedded curve $\alpha : \mathbb {R}\rightarrow M$, with the following properties:

• 1. $\lim _{t \rightarrow -\infty } \alpha (t) = p$ and $\lim _{t \rightarrow +\infty } \alpha (t) = q$,

• 2. $\theta \circ \phi ^1_H (\alpha (t)) = \alpha (t+1)$ for all $t \in \mathbb {R}$,

• 3. $\theta \circ \phi ^1_H$ has the same set of fixed points as $\phi _H^1$,

• 4. Fot all $t\in \mathbb {R}$, $\alpha (t)$ belongs to the open set $V_p\cup \mathcal {C}(\gamma ((0,1)))\cup V_q$,

• 5. $\theta$ is generated by a continuous Hamiltonian F which is supported in $\{p\}\cup V_p\cup \mathcal {C}(\gamma ((0,1)))\cup V_q\cup \{q\}$.

• 6. $d_{C^0}(\mathrm {Id}, \theta ) < \varepsilon$ and $\Vert F\Vert _{\infty }<\rho$.

• 7. For any neighborhoods of p, q, the homeomorphism $\theta$ coincides with a Hamiltonian diffeomorphism in the complement of the union of those neighborhoods.

One of the main difficulties we will have to face in proving Theorem 25 is to perform perturbations of $\phi ^1_H$ without creating new fixed points. Away from the fixed points of $\phi _H^1$, i.e. critical points of H, a $C^0$-small perturbation will not create such fixed points. However, a $C^0$-small perturbation near a critical point can create new fixed points, and for this reason neighborhoods of critical points will require special treatment. In order to surmount these difficulties, we will build the perturbation and the invariant curve of Theorem 25 in three steps: First, we build one end of the curve near one of the two critical points (this construction is achieved in Sect. 3.3.2 in the case of a local minimum/maximum). We then extend the invariant curve such that it reaches a sufficiently small neighborhood of the second critical point (Sect. 3.3.3). Finally, we finish the construction of the invariant curve in the neighborhood of the second critical point (Sect. 3.3.4).

3.3.2 Connecting a max/min to a nearby point

We denote by $\mathbb {R}^{2n}$ the Euclidean space of dimension 2n equipped with the standard symplectic structure $\omega _0 = \sum _{i=1}^{n} dx_i\wedge dy_i$. In the following theorem, $\Vert x\Vert$ denotes the Euclidean norm of a point $x \in \mathbb {R}^{2n}$.

Theorem 26

Let $H: \mathbb {R}^{2n} \rightarrow \mathbb {R}$ be a Hamiltonian of the form $H = c \sum _{i=1}^{n}(x_i^2 +y_i^2)$, where c is a non-zero constant. There exists $A >0$ such that if $|c| < A$, then the following statement holds for any point $x \in \mathbb {R}^{2n}$ and any $\varepsilon , \delta >0$.

Denote by $B \subset \mathbb {R}^{2n}$ an open ball which is centered at the origin and contains the point x. There exists a Hamiltonian homeomorphism $\theta$, whose support is compactly contained in B, and a smooth injective immersion $\alpha : [-1, \infty ) \rightarrow B {\setminus }\{0\}$ with the following properties:

• 1. $\alpha (t) =\phi ^t_H(x)$ for $t \in [-1, 0]$ and $\alpha (t) \rightarrow 0$ as $t\rightarrow \infty$, where 0 denotes the origin,

• 2. $\theta \circ \phi ^1_H (\alpha (t)) = \alpha (t+1)$ for all $t \in [-1, \infty )$,

• 3. $\theta \circ \phi ^1_H$ has only one fixed point and that is the origin,

• 4. The support of $\theta$ intersects the complement of the ball $B(0,\Vert x\Vert )$ of radius $\Vert x\Vert$ only in an $\varepsilon$-neighborhood of $\{\phi _H^t(x):t\in [1,2]\}$,

• 5. $d_{C^0}(Id, \theta ) < \varepsilon$ and $\Vert F\Vert _{\infty } \leqslant \delta$, where F denotes a continuous Hamiltonian such that $\phi ^1_F = \theta$,

• 6. For any neighborhood of 0, there exists a compactly supported Hamiltonian diffeomorphism of B which coincides with $\theta$ in the complement of that neighborhood.

Proof of Theorem 26

Without loss of generality we may assume that $\Vert x\Vert =1$. We remark here that throughout the proof we will use the fact that $\phi ^t_H$, for each t, is a linear isometry of $\mathbb {R}^{2n}$, without explicitly mentioning it. We will assume that $\varepsilon >0$ is very small in comparison to $\Vert \phi ^1_H(x) - x\Vert$.

We will now pick a sequence of curves $\alpha _0, \alpha _1, \ldots ,$ which will be joined together, at a later stage, to form the invariant curve $\alpha$. Let $x_0 = x$ and define $\alpha _0 :[0,1] \rightarrow B$, $t\mapsto \phi ^t_H(x_0)$. Let $0< \rho <1$ be a constant such that $1- \rho$ is very small in comparison to $\varepsilon$. Now, let $x_1 = \rho \, \phi ^1_H(x_0)$ and more generally, for each $i\geqslant 1, x_i = \rho ^i \phi ^i_H(x_0)$ and let $\alpha _i:[i, i+1] \rightarrow B$ denote the curve $\alpha _i(t) = \rho ^i \phi ^t_H(x_0)$ for each $t\in [i,i+1].$ Denote $y_i := \phi ^1_H(x_i)$. Note that $\alpha _i$ satisfies the following identity:

$$\begin{aligned} \alpha _i(t) = \phi ^{t-i}_H(x_i) = \phi ^{t-i-1}_H(y_i), \quad \forall t\in [i,i+1]. \end{aligned}$$

(2)

Fig. 3

Settings for the beginning of the proof of Theorem 26

Step 1: Preliminary preparations for the construction of the invariant curve. For each non-negative integer i, let $a_i:[0,1] \rightarrow B$ be the curve $a_i(t) := (1-t)y_i + t x_{i+1}$. Observe that, the length of $a_i$ is $\rho ^i - \rho ^{i+1}.$ Note that $\rho ^i - \rho ^{i+1} < \rho ^i \varepsilon .$ Furthermore, the image of $a_i$ is disjoint from the remaining $a_j$’s and is contained in the shell $\{z \in \mathbb {R}^{2n}: \rho ^{i+1} \leqslant \Vert z\Vert \leqslant \rho ^i\}$. See Fig. 3.

We will now introduce some of the notation which will be used throughout the proof. Let $U_0$ be a shell which is a slight enlargement of the shell $\{z \in \mathbb {R}^{2n}: \rho \leqslant \Vert z\Vert \leqslant 1\}$. For $i >0$, we set $U_i:= \rho ^i U_0$. Note that $U_i$ is a shell which is a slight enlargement of the shell $\{z \in \mathbb {R}^{2n}: \rho ^{i+1} \leqslant \Vert z\Vert \leqslant \rho ^i\}$ and furthermore, $U_i \cap U_j = \emptyset$ if $j \notin \{i-1, i, i+1\}$.

Next, let $W_0$ be a small neighborhood of the image of the curve $a_0$ which is contained in $U_0$ and has the following properties: the diameter of $W_0$ is less than $\varepsilon$, $W_0$ is convex, $W_0 \cup \rho W_0$ is also convex, and there exists a small positive number $\kappa$ such that $\phi ^t_H(y_0), \phi ^t_H(x_{1}) \in W_0$ if and only if $t \in (-2 \kappa , 2 \kappa )$. Lastly, we require that $W_0 \cap \rho \phi ^1_H(W_0) = \emptyset$. Observe that, for this last requirement it is sufficient to take $A < \pi$. For $i >0$, we define $W_i := \rho ^i \phi ^i_H(W_0)$, or equivalently, $W_i = \rho \phi ^1_H(W_{i-1})$. What is important to observe about the sets $W_i$ is that they have the following properties: $W_i$ is a small neighborhood of the image of $a_i$ which is contained in $U_i$ and the diameter of $W_i$ is less than $\rho ^i \varepsilon$. The sets $W_i$ are convex and so are $\phi ^1_H(W_i) \cup W_{i+1}$. The $W_i$’s are pairwise disjoint and furthermore $W_j \cap \phi ^1_H(W_i) \ne \emptyset$ only if $j=i+1$. Also, note that for $t\in [-1,1]$ we have $\phi ^t_H(y_i), \phi ^t_H(x_{i+1}) \in W_i$ if and only if $t \in (-2 \kappa , 2 \kappa )$. Finally, we remark that among the sets $W_j$ the only one which intersects the image of $\phi ^1_H\circ \alpha _i$ is $W_i$. Our set up is summarized in Fig. 3.

Next, we find Hamiltonians, say $G_i$, such that $G_i$ is supported in $W_i$, and

$$\begin{aligned} \phi ^1_{G_i}\left( \phi _H^t(y_i)\right) = \phi ^t_H(x_{i+1}), \quad \forall \;t \in [-\kappa , \kappa ]. \end{aligned}$$

(3)

Let $G: = \sum _{i=0}^{\infty }G_i$; we remark that by Lemma 10, we can pick $G_i$ such that $\Vert G_i\Vert _{\infty }$ is arbitrarily small. Note that the $G_i$’s are supported in $W_i$’s which are pairwise disjoint. Therefore, G is well-defined and in fact $\Vert G\Vert _{\infty }$ can be made arbitrarily small. Furthermore, G generates a hameotopy whose flow is the (infinite) composition $\Pi _{i=1}^{\infty } \phi ^t_{G_i}$: to see this observe that the flows $\phi ^t_{G_i}$ are supported in $W_i$’s which are disjoint and whose diameters are smaller than $\rho ^i \varepsilon$. We point out that $\phi ^1_G$ coincides with a Hamiltonian diffeomorphism of B in the complement of any neighborhood of 0.

Let $\psi _1$ denote the Hamiltonian homeomorphism $\phi ^1_G \circ \phi ^1_H$.

Step 2: A first approximation to the invariant curve $\alpha$. Consider the smooth curve $\alpha '(t) : [-1,\infty ) \rightarrow B$ given by the following formula:

$$\begin{aligned} \alpha '(t):= {\left\{ \begin{array}{ll}\phi _H^{t}(x) &{} \quad t\in [-1,0], \\ \phi ^1_{G_{i}} \circ \alpha _i(t) &{} \quad t\in [i,i+1], \quad \forall i \geqslant 0.\end{array}\right. } \end{aligned}$$

To see that the above formula defines a smooth curve one can check, using Eqs. (2) and (3), $\alpha '(t) = \phi ^{t-i}_H(x_{i})$ for $t\in [i -\kappa , i + \kappa ]$ for each $i\geqslant 1$. One can ensure that $\alpha '$ is injective, by perturbing the Hamiltonians $G_i$, if necessary, and by picking the constant A, from the statement of the theorem, to be sufficiently small⁵; we leave it to the reader to check the details of this. The curve $\alpha '$ and its image under $\psi _1$ are drawn in Fig. 4.

Fig. 4

The curve $\alpha '$ in blue and its image by $\psi _1$ in green

It is evident that $\psi _1( \alpha '(t)) \ne \alpha '(t+1)$ for $t>0$. In Steps 3 and 4, we will be modifying $\alpha '$ and $\psi _1$ to establish the invariance stated in the second property from the statement of the theorem. The next few claims record some properties of these two curves which will be used later in Steps 3 and 4. $\square$

Claim 27

For A sufficiently small, $\psi _1( \alpha '(t)) = \alpha '(t+1)$ for $t\in [-1, 0]$.

Proof

If $t \in [-1,0]$, then one can easily conclude from the definition of $\alpha '$ that we have $\psi _1(\alpha '(t)) = \phi ^1_{G_0}\phi ^{1}_H \phi ^{t}_H(x)= \phi ^1_{G_0} \phi ^{t+1}_H(x) = \alpha '(t+1)$. $\square$

Claim 28

For each $i \geqslant 0$, the two curves $\psi _1 ( \alpha '|_{[i,i+1]})$ and $\alpha '|_{[i+1,i+2]}$ coincide near their endpoints. More precisely, $\psi _1 ( \alpha '(t)) = \alpha '(t+1)$ for $t\in [i, i+\kappa ] \cup [i+1 - \kappa , i+1]$.

Proof

We will check that if $t\in [i -\kappa , i + \kappa ]$, then $\psi _1 \circ \alpha '(t) = \alpha '(t+1)$. To see this write $t = i+s$, where $s\in [-\kappa , \kappa ]$, and note that $\psi _1 \circ \alpha '(t) = \phi ^1_G \phi ^1_H \phi ^{s}_H(x_{i})= \phi ^1_G \phi ^{s}_H(y_{i}) = \phi ^1_{G_{i}} \phi ^s_H(y_i) = \phi ^s_H(x_{i+1}) = \alpha '(t+1).$ The last equality follows from the fact that for $i \geqslant 1$, we have $\alpha '(t) = \phi _H^{t-i}(x_i)$ when $t \in [i- \kappa , i+\kappa ]$. $\square$

For each $i \geqslant 0$, let $V_i$ be the shell $U_i \cup U_{i+1}$. Note that $V_i \cap V_j = \emptyset$ if $j \notin \{i-2, i-1, i, i+1, i+2 \}$.

Claim 29

For each $i\geqslant 0$, there exists a homotopy, which is constant near endpoints, from $\psi _1 \circ \alpha '|_{[i,i+1]}$ to $\alpha '|_{[i+1,i+2]}$ which is supported in $V_i$ and furthermore, under this homotopy the trajectory of any point of $\psi _1 \circ \alpha '|_{[i,i+1]}$ has diameter less than $2 \rho ^i \varepsilon$.

Proof

We point out that, as a consequence of Claim 28, the above two curves coincide near their endpoints. The idea of the proof of this claim is very simple: the straight-line homotopy $F(s,t) = (1-s) \psi _1 \circ \alpha '|_{[i, i+1]} + s \alpha '|_{[i+1, i+2]}$ satisfies all the required properties. The fact that $W_i \cup \rho W_i$ is convex, and of diameter less that $2 \rho ^i \varepsilon$, is used here to ensure that the trajectories of the points $\psi _1 \circ \alpha '|_{[i,i+1](t)}$ for t near $i, i+1$ are of diameter less than $2 \rho ^i \varepsilon$. Checking the proof of this claim in detail, although quite straight forward, is rather tedious. Hence, we will omit the proof. $\square$

Step 3: Constructing the curve $\alpha$ Let $\lambda$ be the 1-form $\sum _{i=1}^n x_i dy_i$. We define the action of any curve $a: [0,1] \rightarrow \mathbb {R}^{2n}$ to be the integral $\int _0^1 a^* \lambda .$ We will now finish the construction of the curve $\alpha$.

We begin by defining $\alpha |_{[-1,1]} = \alpha '|_{[-1,1]}$ and proceed to inductively construct the curve $\alpha$ such that it satisfies the following properties: for each $i\geqslant -1$, the two curves $\psi _1 \circ \alpha |_{[i,i+1]}$ and $\alpha |_{[i+1,i+2]}$ have the same action, and $\alpha$ coincides with $\alpha '$ for all values of t except near $t= i+ \frac{1}{2}$, where $i \geqslant 1$. Suppose that, for some $k\geqslant 1$, we have constructed such $\alpha :[-1,k] \rightarrow \mathbb {R}^{2n}$. Next, we make a $C^0$ small perturbation of the curve $\alpha '|_{[k,k+1]}$, in an arbitrarily small neighborhood of the point $\alpha '(k+ \frac{1}{2})$ to obtain a new curve, which we will call $\alpha |_{[k,k+1]}$, such that $\psi _1 \circ \alpha |_{[k-1,k]}$ and $\alpha |_{[k,k+1]}$ have the same action. It is evident from the construction that $\alpha$ coincides with $\alpha '$ for all t except for values of t near $i+ \frac{1}{2}$, where $i\geqslant 1$. It is also clear that $\alpha$ be can be picked to be arbitrarily $C^0$ close to $\alpha '$. We should add that it is a well-known fact that one can make an arbitrary adjustment to the action of a curve by performing a $C^0$-small perturbation; see for example Remark A.13 of [3].

We finish this step by pointing out that the curve $\alpha$ satisfies the first of the four properties listed in the statement of Theorem 26.

Step 4: Turning $\alpha$ into an invariant curve In this final step of the proof, we will perturb $\psi _1$ to a Hamiltonian diffeomorphism $\psi$ such that $\psi \circ \alpha (t) = \alpha (t+1)$.

Claim 30

For each $i\geqslant 0$, there exists a Hamiltonian $K_i$ supported in $V_i = U_i\cup U_{i+1}$ such that

• 1. $\phi _{K_i}^1 \circ \psi _1 \circ \alpha (t) = \alpha (t+1)$ for each $t\in [i,i+1]$,

• 2. $d_{C^0}(Id, \phi _{K_i}^1) < 4\rho ^i \varepsilon$ and $K_i$ can be picked such that $\Vert K_i\Vert _{\infty }$ is as small as one wishes,

• 3. $\mathrm {supp}(K_i)$ is contained within the $4\varepsilon$ neighborhood of $\alpha ([i+1,i+2])$,

• 4. $\alpha ([j, j+1]) \cap \mathrm {supp}(K_i) = \emptyset$ if $j \ne i+1$. Similarly, $\psi _1 \circ \alpha ([j,j+1]) \cap \mathrm {supp}(K_i)= \emptyset$ for $j\ne i$.

Proof

In the previous step we constructed $\alpha$ such that $\psi _1\circ \alpha |_{[i,i+1]}$ and $\alpha |_{[i+1, i+2]}$ have the same action. Furthermore, since we obtained $\alpha |_{[i,i+1]}$ from $\alpha '|_{[i, i+1]}$ by making a $C^0$-small perturbation of $\alpha '|_{[i, i+1]}$ near $t= i+ \frac{1}{2}$ we conclude, by Claim 29, that there exists a homotopy, which is constant near endpoints, from $\psi _1 \circ \alpha |_{[i,i+1]}$ to $\alpha |_{[i+1,i+2]}$ which is supported in $V_i$ and furthermore, under this homotopy the trajectory of any point of $\psi _1 \circ \alpha |_{[i,i+1]}$ has diameter less than $2 \rho ^i \varepsilon$. The first three properties of the claim then follow immediately from Proposition 9. As for the fourth property, since the dimension of M is at least four, by making a small perturbation of the homotopy from $\psi _1 \circ \alpha |_{[i,i+1]}$ to $\alpha |_{[i+1,i+2]}$, we may assume the image of the homotopy does not intersect $\alpha ([j, j+1])$, for $j \ne i+1$, or $\psi _1 \circ \alpha ([j,j+1])$, for $j \ne i$. Fix small $\kappa _i > 0$ such that the homotopy is constant on $\psi _1 \circ \alpha |_{[i,i+\kappa _i] \cup [i+1-\kappa _i]}$. Let W be a small neighborhood of the image of the restricted homotopy from $\psi _1 \circ \alpha |_{[i+\kappa _i,i+1-\kappa _i]}$ to $\alpha |_{[i+1+\kappa _i,i+2-\kappa _i]}$ such that $\alpha ([j, j+1]) \cap W = \emptyset$, if $j \ne i+1$ and $\psi _1 \circ \alpha ([j,j+1]) \cap W = \emptyset$ for $j\ne i$. Now, it is easy to see that $K_i$ can be picked to have its support contained in W. This implies the fourth property of the claim. $\square$

Note that for each i, $V_i \cap V_j \ne \emptyset$ only if $j\in \{i-2, i-1, i, i+1, i+2\}$. Hence, we see that the sets $V_0, V_3, V_6, \ldots$ are mutually disjoint and therefore the supports of $K_0, K_3, K_6, \ldots$ are mutually disjoint as well. This combined with the fact that $\Vert K_i\Vert _{\infty }$ can be picked to be as small as we wish, implies that the sum $F_0:= K_0+ K_3 + K_6 + \cdots$ defines a continuous function. Furthermore, as a consequence of the disjointness of the supports of these functions, we see that $F_0$ is a continuous Hamiltonian whose flow is $\phi ^t_{F_0}= \phi ^t_{K_0} \circ \phi ^t_{K_3} \circ \phi ^t_{K_6} \cdots$. Similarly, we define $F_1 = K_1+K_4 +K_7 + \cdots$ and $F_2 = K_2 + K_5 +K_8 + \cdots$. These functions generate the hameotopies $\phi ^t_{F_1}= \phi ^t_{K_1} \circ \phi ^t_{K_4} \circ \phi ^t_{K_7} \cdots$ and $\phi ^t_{F_2}= \phi ^t_{K_2} \circ \phi ^t_{K_5} \circ \phi ^t_{K_8} \cdots$, respectively. Observe that, as a consequence of the second item in Claim 30, $F_0, F_1, F_2$ can be picked such that their norms are as small as one wishes.

We define $\theta := \phi ^1_{F_2}\circ \phi ^1_{F_1}\circ \phi ^1_{F_0}\circ \phi ^1_G$ and $\psi := \theta \circ \phi ^1_H = \phi ^1_{F_2} \circ \phi ^1_{F_1} \circ \phi ^1_{F_0}\circ \psi _1$. Clearly, $\theta$ is a Hamiltonian homeomorphism which is compactly supported in B.

We will now check that $\theta$ and the curve $\alpha$ satisfy the properties listed in the statement of Theorem 26. We have already checked the fact that the first property is satisfied.

First, we will show that $\psi ( \alpha (t)) = \alpha (t+1)$, for all $t\in [-1, \infty )$. If $t \in [-1,0]$, then this follows from Claim 27. Indeed, for $t \in [-1,1]$ we have $\alpha (t) = \alpha '(t)$. Moreover, for $t \in [-1,0]$ we have $\psi (\alpha '(t)) = \phi ^1_{F_2} \circ \phi ^1_{F_1} \circ \phi ^1_{F_0} (\psi _1( \alpha '(t))) = \phi ^1_{F_2} \circ \phi ^1_{F_1} \circ \phi ^1_{F_0} (\alpha '(t+1)) = \alpha '(t+1)$. The reason the last equality holds is that, by the 4th item of Claim 30, $\alpha '(t+1) = \alpha (t+1)$ is not contained in the support of any of $F_0, F_1, F_2$. Next, consider $t \in [0, \infty )$. Fix i and $t\in [i,i+1]$. According to Claim 30, $\alpha (t+1), \psi (\alpha (t)) \notin \mathrm {supp}(K_j)$ for any j other than $j=i$. Hence, $\psi ( \alpha (t)) = \phi ^1_{F_2} \circ \phi ^1_{F_1}\circ \phi ^1_{F_0}\circ \psi _1( \alpha (t)) = \phi ^1_{K_i} \circ \psi _1( \alpha (t)) = \alpha (t+1)$.

To establish the remaining properties we will need the following claim. We define a nested sequence of balls $B_i \supset B_{i+1}$ as follows: For $i\geqslant 0$ we set $B_i = \{0\} \cup (\cup _{k\geqslant i} V_k)$. Note that $B_0$ is a slight enlargement of the unit ball and furthermore $B_0$ contains the supports of all $F_0, F_1, F_2$, and G.

Claim 31

Suppose that $p\in B_0 - \{ 0\}$. Let i denote the smallest integer such that $p \in U_i$. Then, for each $j\in \{0,1,2\}$,

• if $i>0$, then $\phi ^1_{F_j}(p) \in B_{i-1}$ and $\Vert \phi ^1_{F_j}(p) - p \Vert \leqslant 4\rho ^{i-1} \varepsilon$ .

• if $i=0$, then $\phi ^1_{F_j}(p) \in B_{0}$ and $\Vert \phi ^1_{F_j}(p) - p\Vert \leqslant 4 \varepsilon$.

• $\phi ^1_G(p) \in B_{i-1}$ and $\Vert \phi ^1_G(p) -p\Vert \leqslant \rho ^{i-1} \varepsilon$.

Proof

We begin with the statement about $\phi ^1_{F_j}(p)$. We will only prove this for $j=0, i>0$ and leave the remaining cases to the reader. Recall that $F_0 = K_0 + K_3 + K_6 + \cdots$ and that $\mathrm {supp}(K_m) \subset V_m = U_m \cup U_{m+1}$ for all m. Hence, the point p can only be in the support of $K_{i-1}, K_i, K_{i+1}$. Now, only one of these three Hamiltonians enters the definition of $F_0$ and so we see that $\phi ^1_{F_0}(p) \in \{\phi ^1_{K_{i-1}}(p), \phi ^1_{K_{i}}(p), \phi ^1_{K_{i+1}}(p)\}$. The result then follows from Claim 30. The statement about $\phi ^1_G(p)$ is proven similarly. $\square$

We will now prove that $\psi$ has no fixed points other than 0. Recall that $\psi = \theta \circ \phi ^1_H.$ Using the above claim, and the fact that $\theta = \phi ^1_{F_2} \circ \phi ^1_{F_1} \circ \phi ^1_{F_0} \circ \phi ^1_G$, we see that

• 1. $\Vert \theta \circ \phi ^1_H(p) -\phi ^1_H(p)\Vert \leqslant 13 \rho ^{i-4} \varepsilon$, when $p \in B_i$ and $i\geqslant 4$

• 2. $\Vert \theta \circ \phi ^1_H(p) -\phi ^1_H(p)\Vert \leqslant 13\varepsilon$, when $p \in B_i$ and $i\leqslant 3.$

Suppose that $p \in U_i$ where $i \geqslant 4$. We will show that p can not be a fixed point of $\theta \circ \phi ^1_H$. First, note that since $U_i$ is a slight enlargement of the shell $\{z: \rho ^{i+1} \leqslant \Vert z\Vert \leqslant \rho ^{i} \}$, we can assume that $|p| \geqslant \rho ^{i+2}.$ Recall that, we picked $\varepsilon$ to be very small in comparison to the number $C= \Vert \phi ^1_H(z) -z\Vert$, where z is any point such that $\Vert z\Vert =1$. It follows that $\Vert \phi ^1_H(p) -p \Vert \geqslant C \rho ^{i+2}$. Hence, we see that if $C \rho ^{i+2} > 13 \rho ^{i-4} \varepsilon$, then $\theta \circ \phi ^1_H(p) \ne p$. Of course, by picking $\varepsilon$ to be sufficiently small we can make sure that the inequality $C \rho ^{i+2} > 13 \rho ^{i-4} \varepsilon$ holds. (Recall that we picked $\rho$ such that $1-\rho$ is small in comparison to $\varepsilon$.) We leave the case where $p\in U_i$ and $i \leqslant 3$ to the reader.

Next, we will show that the support of $\theta$ satisfies the fourth property from the statement of the theorem. Indeed, examining the proof we can see that the Hamiltonians $G_i, K_i$ were all picked to be supported within at most a $4\varepsilon$ neighborhood of $\alpha ([i+1, i+2])$. Of these Hamiltonians it is only $G_0, K_0$ whose supports intersect the complement of $B(0, \Vert x\Vert )$. This implies that the support of $\theta$ intersects the complement of $B(0, \Vert x\Vert )$ only in a $4\varepsilon$-neighborhood of $\alpha ([1,2])$. Now, we leave it to the reader to check that $\alpha ([1,2])$ is contained within an $\varepsilon$-neighborhood of $\{\phi ^t_H(x): t \in [1,2] \}$. Replacing $\varepsilon$ by $\frac{\varepsilon }{5}$ throughout the proof yields the result.

We now check the 5th property from the statement of the theorem. Since $\theta = \phi ^1_{F_2} \circ \phi ^1_{F_1} \circ \phi ^1_{F_0} \circ \phi ^1_G$, it is an immediate consequence of Claim 31 that $d_{C^0}(\theta , Id) \leqslant 13 \varepsilon$. Hence, if we replace $\varepsilon$ by $\frac{\varepsilon }{13}$ then we obtain $d_{C^0}(Id, \theta ) \leqslant \varepsilon .$ Next, let F denote the generating Hamiltonian of the continuous Hamiltonian flow $\phi ^t_{F_2}\circ \phi ^t_{F_1}\circ \phi ^t_{F_0}\circ \phi ^t_G$. It follows from the composition formulas mentioned in Sect. 2.1 that $\Vert F\Vert _{\infty } \leqslant \Vert F_0\Vert _{\infty } + \Vert F_1\Vert _{\infty } + \Vert F_2\Vert _{\infty } + \Vert G \Vert _{\infty }$. As we have already mentioned, $F_0, F_1, F_2, G$ can be picked to have norms as small one wishes. Hence, the same is true for $\Vert F\Vert _{\infty }$.

Finally, we check the 6th and final property. Recall the nested sequence of balls $B_i \supset B_{i+1}$ introduced before Claim 31. Let $B_i^c$ denote the complement of $B_i$. Observe that in $B_i^c$, for any i, each of $\phi ^1_{F_0}, \phi ^1_{F_1}, \phi ^1_{F_2}$, and $\phi ^1_G$ coincides with a Hamiltonian diffeomorphism. Furthermore, using Claim 31, it can easily be checked that each of these homeomorphisms maps $B_i^c$ into $B_{i+1}^c$. Combining these facts together we see that $\theta$ coincides with a Hamiltonian diffeomorphism in the complement of each $B_i$. $\square$

3.3.3 Connecting two non critical points by an invariant curve

The main goal of this section is to prove the following technical result which is needed for the construction of the invariant curve of Theorem 25.

Theorem 32

Denote by $H: M \rightarrow \mathbb {R}$ a smooth autonomous Hamiltonian on a closed symplectic manifold $(M, \omega )$ of dimension at least 4. Let z, w be two points in M. Suppose that there exists a smooth embedded curve $\gamma : [0,1] \rightarrow M$ such that $\gamma (0) = z, \gamma (1) = w$, such that H is not constant on $\gamma$, and the map $\Gamma : [0,1] \times \mathbb {R} \rightarrow M$ defined by $\Gamma (s,t) = \phi _{H}^t(\gamma (s))$ is a smooth embedding on $(\{0\}\times [-2,1])\cup ((0,1)\times [0,1]) \cup (\{1\}\times [-1,2])$.

Then, for any constants $\varepsilon , \rho > 0$ and any neighborhood U of $\Gamma ([0,1] \times [0,1])$, there exist

• a Hamiltonian diffeomorphism $\psi : M \rightarrow M$,

• and a smooth embedded curve $\alpha : [0,k+1] \rightarrow M$, where $k \in \mathbb {N}$,

such that:

• 1. $\psi (\alpha (s)) = \alpha (s+1)$ for any $s \in [0,k]$,

• 2. There exists $t_0 \in (1,2)$ such that for all $s\in [0,t_0]$, $\alpha (s)=\phi _H^{s-2}(z)$, and $\alpha ([t_0, k+1]) \subset \displaystyle U\cup \mathcal {C}(\gamma ((0,1)))$,

• 3. Let V be any neighborhood of the point $\phi _H^{\frac{1}{2}}(w)$. Then, $\alpha$ can be chosen to satisfy the following property:

  $$\begin{aligned} \alpha (k+s){\left\{ \begin{array}{ll} =\phi _H^{s}(w), &{} \quad s\in \left[ 0,\tfrac{1}{2}-b\right] \cup \left[ \tfrac{1}{2}+b,1\right] \\ \in V, &{} \quad s\in \left[ \tfrac{1}{2}-b,\tfrac{1}{2}+b\right] , \end{array}\right. } \end{aligned}$$

  for some small $b>0$. Furthermore, $\psi \circ \alpha (k+s) = \phi _H^{s+1}(w)$ for $s \in [0,\tfrac{1}{2}-b]\cup [\tfrac{1}{2}+b,1]$,

• 4. $d_{C^0}(\psi , \phi _{H}^1) < \varepsilon$ and the Hamiltonian diffeomorphism $\theta := \psi \circ \phi _{H}^{-1}$ is generated by a Hamiltonian, say F, such that $\Vert F \Vert _{\infty }<\rho$. Furthermore, F is supported inside $\displaystyle U\cup \mathcal {C}(\gamma ((0,1)))$,

• 5. $\psi$ has the same set of fixed points as $\phi ^1_H$.

Proof of Theorem 32

By replacing U with a smaller open subset we can assume that there exists a diffeomorphism $\Phi : U \rightarrow (-c, 1+c) \times (-c, 1+c) \times (-c,c) \times \cdots \times (-c,c) \subset \mathbb {R}^{2n}$, where $c>0$ is sufficiently small. By slightly decreasing c, if necessary, we may assume that $\Phi$ is a bi-Lipschitz map, where we consider the metric g on M and the standard euclidean metric on $(-c, 1+c) \times (-c, 1+c) \times (-c,c) \times \cdots \times (-c,c)$. We can further suppose that this diffeomorphism identifies $\Gamma (s,t)$ with $(s,t, 0, \ldots , 0)$ for all $s, t \in [0,1]$. Note that here we are relying on the fact that $\Gamma$ is an embedding on $[0,1] \times [0,1]$.

Step 1: Preliminary preparations for the construction of the invariant curve We begin by picking $\delta >0$ such that $\delta <\frac{\varepsilon }{2}$ and $\forall x, y \in M$

$$\begin{aligned} \mathrm {if} \; d(x,y)< \delta , \; \mathrm {then} \quad \forall t\in [0,1],\; d\left( \phi ^t_H(x), \phi ^t_H(y)\right) < \frac{\varepsilon }{2}. \end{aligned}$$

(4)

Pick $m \in \mathbb {N}$ large enough such that $d(\gamma (\frac{i}{m}), \gamma (\frac{i+1}{m})) < \frac{\delta }{2}.$ For each $0 \leqslant i < m$, we define small neighborhoods $U_i$ of $\Gamma ([\tfrac{i}{m}, \tfrac{i+1}{m}] \times [0,1])$ by

$$\begin{aligned} U_i := \Phi ^{-1} \left( \left( \tfrac{i}{m} - a, \tfrac{i+1}{m} +a \right) \times (-c, 1+c) \times (-c,c) \times \cdots \times (-c,c)\right) , \end{aligned}$$

where $a>0$ is taken to be so small that $U_i \cap U_j \ne \emptyset$ only if $j \in \{ i-1, i, i+1\}$. Clearly, $U_i \subset U$ for all i.

Take $r >0$ satisfying the following two criteria: r is small in comparison to $\delta$ and the r-neighborhood of $\gamma (\frac{i}{m})$ is contained in $U_i$. For each $0< i < m$ choose a ball $B_i$ in the r-neighborhood of $\gamma (\frac{i}{m})$ such that $B_i \subset U_i \cap \mathcal {C}(\gamma ((0,1)))$, such that $H(B_i) \cap H(B_j) = \emptyset$ for any $0< i< j < m$, and such that moreover $H(x_0), H(x_m) \notin H(B_i)$ for each $0< i < m$. It is not hard to show that such a choice of $B_i$ is always possible; here it is important to remember that, because $\Gamma$ is an embedding, $\gamma$ does not pass through any of the critical points of H. Note that we may also assume that $B_{i+1} \subset U_i$ for $0\leqslant i < m-1$.

We can ensure that for each $x \in B_i \cup B_{i+1}$ the image of the curve $t \mapsto \phi ^t_H(x), t\in [0,1]$ is contained in $U_i$; this will be used in Step 4.

By the Poincaré recurrence theorem, we can find points $x_i \in B_i, 0< i <m$, and integers $k_{i} \geqslant 2$ with the property that $\phi _H^{k_{i}}(x_i) \in B_i$. Let $y_i := \phi _H^{k_{i}}(x_i)$. By an arbitrarily $C^2$-small perturbation of H inside the open set U away from the curves $(\phi _H^{t}(x_0))_{t \in [-2,0]}$ and $(\phi _H^{t}(x_m))_{t \in [0,2]}$, if needed, we may assume that the curve $(\phi _H^t(x_i))_{t \in [-1,k_i+1]}$ is embedded, for each i. If i is 0 or m, we set $x_0 = \phi ^{-2}_H(z), y_0 = z, k_0 =2$ and $x_m = w, y_m = \phi ^1_H(x_m), k_m =1$. We remark that for each $0 \leqslant i < m$, the points $x_i$ and $x_{i+1}$ do not belong to the same level set of H; this will be used in the next step of the proof. Figure 5, below, describes the settings from Step 1.

Fig. 5

Settings from Step 1

Step 2: Turning the sequence $x_0, \ldots , x_m$ into a genuine orbit For each $0 \leqslant i \leqslant m-1$ pick smooth embedded curves $a_i:[0,1] \rightarrow M$ such that

• $a_i(0) = y_i$ and $a_i(1) = x_{i+1}$,

• the diameter of $a_i$ is smaller than $\delta$ [note that $d(y_i, x_{i+1}) < \delta$],

• the image of $a_i$ is disjoint from the image of $a_j$, for $j\ne i$, and it is contained in the open set $U_i$,

• for each i the image of $a_i$ does not intersect any of the curves $t \mapsto \phi ^t_H(x_j), \; t \in [-1, k_{j}+1]$, for any value of j, except at the two points $y_i = \phi _H^{k_{i}}(x_i)$ and $x_{i+1} = \phi ^0_H(x_{i+1})$. Moreover, for each i the image of $a_i$ does not intersect the curves $(\phi ^t_H(x_0))_{t \in [-2, 0]}$ and $(\phi ^t_H(x_m))_{t \in [0, 2]}$, except at the point $x_0$ and $x_m$, respectively.

The above conditions, combined with the fact that the points $x_i$ were picked such that $y_i$ and $x_{i+1}$ are not on the same trajectory of $\phi ^t_H$, guarantee that for each $0 \leqslant i \leqslant m-1$ we can find $W(a_i)$, a small neighborhood of the image of $a_i$, with the following properties: $W(a_i) \subset U_i$, the diameter of $W(a_i)$ is less than $\delta$, these neighborhoods are disjoint, and each of the curves $(\phi ^t_H(y_i))_{t\in [{ -1,1}]}$ and $(\phi ^t_H(x_{i+1}))_{t\in [{ -1, 1}]}$ takes values in $W(a_i)$ if and only if $t \in (-2 \kappa _i, 2 \kappa _i)$ where $\kappa _i>0$ is small. Now, let $\kappa = \min \{\kappa _1, \ldots , \kappa _{m-1} \}$. By shrinking the neighborhoods $W(a_i)$ we may assume the following:

If $t \in [-1, 1]$, then for $0 \leqslant i \leqslant m-1$ we have

$$\begin{aligned} \phi ^t_H(y_i) \in W(a_i) \iff \phi ^t_H(x_{i+1}) \in W(a_i) \iff t \in (-2\kappa ,2\kappa ). \end{aligned}$$

(5)

If $t \in [-2, 1]$, then

$$\begin{aligned} \phi ^t_H(y_{0}) \in W(a_0) \iff t \in (-2\kappa ,2\kappa ). \end{aligned}$$

(6)

If $t \in [-1, 2]$, then

$$\begin{aligned} \phi ^t_H(x_{m}) \in W(a_m) \iff t \in (-2\kappa ,2\kappa ). \end{aligned}$$

(7)

Next, we find a Hamiltonian, say $G_1$, the time-1 map of whose flow we will denote by $\tau$, such that $G_1$ is supported in the union of $W(a_i)$’s, and

$$\begin{aligned} \tau \left( \phi _H^t(y_i)\right) = \phi ^t_H(x_{i+1}), \quad \forall \;t \in [-\kappa , \kappa ]. \end{aligned}$$

(8)

Let $\psi _1 := \tau \phi ^1_H$; observe that $\psi _1^{k_{i}}(x_i) = x_{i+1}$ for each $0 \leqslant i \leqslant m-1$. Also, note that $d_{C^0}(\tau , Id) < \delta$ and thus $d_{C^0}(\psi _1, \phi ^1_H)< \delta < \varepsilon$. Lastly, we remark that $G_1$ can be picked such that $\Vert G_1\Vert _{\infty }$ is as small as desired.

Step 3: A first approximation to the invariant curve $\alpha$ Let $k = k_0 + k_1 + \cdots + k_{m-1}$ and consider the smooth curve $\alpha _1 : [0,k+1] \rightarrow M$ given by the following formula: For $0 \leqslant i \leqslant m$, let $K_i = \sum _{j=0}^{i} k_j$ and define

$$\begin{aligned} \alpha _1(t) := {\left\{ \begin{array}{ll} \phi ^t_H(x_0) &{}\quad t \in [0,1] \\ \tau \circ \phi ^t_H(x_0) &{}\quad t \in [1,K_0] \\ \phi ^{t - K_{i}}_H(x_{i+1}) &{}\quad t \in \left[ K_{i},K_{i+1} -1 \right] , 0\leqslant i \leqslant m-2 \\ \tau \circ \phi ^{t - K_{i}}_H(x_{i+1}) &{}\quad t \in \left[ K_{i+1} -1, K_{i+1} \right] , 0\leqslant i \leqslant m-2 \\ \phi ^{t-k}_H(x_m) &{}\quad t \in [k,k+1] \end{array}\right. } \end{aligned}$$

The fact that the above formula yields a smooth curve is an immediate consequence of Eq. (8). Since the $x_i$’s belong to different level sets of H, by slightly perturbing Hamiltonian diffeomorphism $\tau$ if needed we can guarantee that this curve is embedded. $\square$

Remark 33

In this remark, we will prove that the curve $\alpha _1$ satisfies the second property from the statement of the theorem.

We take $t_0 = K_0 - 2 \kappa = 2 - 2 \kappa$. Then, it follows from the definitions of $\alpha _1$ and $\tau$, and the fact that $x_0 = \phi ^{-2}_H(z)$, that $\alpha _1(t) = \phi ^{t-2}_H(z)$ for $t \in [0, t_0]$. We remark that $\alpha _1([t_0, k+1])\subset U \cup \mathcal {C}(\gamma ((0,1)))$. This can easily be checked from the definition of $\alpha _1$ and noting the following facts: $\tau$ is supported in U, and $\{x_i: 1 \leqslant i \leqslant m\} \subset U \cap \mathcal {C}(\gamma ((0,1))$. In Step 4 we will have to modify $\alpha _1$ to obtain $\alpha$. However, we will never change $\alpha _1$ on $[0,t_0]$ and all of the modifications will take place inside U. Hence, the second property from the statement of the theorem will continue to hold for $\alpha$ as well.

Remark 34

Note that $\alpha _1(k+s) = \phi ^s_H(w)$, for $s \in [0,1]$, and $\psi _1 \circ \alpha _1(k+s) = \phi _H^{s+1}(w)$ for $s \in [0,1]$. So at this stage the third property is also satisfied for any choice of V and any sufficiently small value of b. Throughout the remainder of the proof we will have to modify the curve $\alpha _1$. However, the modifications on $[k,k+1]$ will be such that the the formula given in the third property will remain true. Furthermore, we will also modify $\psi _1$ to a different map $\psi$. Now, $\psi$ will be constructed such that the support of $\theta := \psi \circ \phi _H^{-1}$ will not intersect the curve $\phi ^1_H \circ \alpha ([k, k+1])$. It will then follow that $\psi \circ \alpha (k+s) = \phi _H^{s+1}(w)$ for $s \in [0,\tfrac{1}{2}-b]\cup [\tfrac{1}{2}+b,1]$. This establishes the third property from the statement of the theorem.

Remark 35

For $0 \leqslant l \leqslant k$, we let $\alpha _1|_{[l, l+1]}:[0,1] \rightarrow M$ denote the curve $\alpha _1|_{[l, l+1]}(t):= \alpha _1(l+t), \; \forall t \in [0,1]$. Note that $\psi _1 \circ \alpha _1 |_{[l-1,l]} = \alpha _1|_{[l, l+1]}$ for $l \notin \{K_i : 0 \leqslant i \leqslant m-1 \}.$ For the remaining values of l the two curves $\psi _1 \circ \alpha _1 |_{[l-1,l]}$ and $\alpha _1|_{[l, l+1]}$ only coincide near their endpoints. Indeed, it can easily be checked, using Eqs. (5) and (8), that both of the above curves coincide with the curve $t \mapsto \phi ^t_H(x_{i+1})$ on $[0, \kappa ] \cup [1-\kappa ,1]$.

Remark 36

Let $l=K_i, 0\leqslant i \leqslant m-1$. The two curves $\psi _1 \circ \alpha _1 |_{[l-1,l]}$ and $\alpha _1|_{[l, l+1]}$ are both contained in $U_i$. This follows from the construction of $\tau$ and the fact that the curves $t \mapsto \phi ^t_H(x_i)$ and $t\mapsto \phi ^t_H(y_i)$, $t\in [0,1]$, are both contained in $U_i$. We leave the details of this to the reader.

We will next check that for each $l \in \{K_i : 0 \leqslant i \leqslant m-1 \}$ the $C^0$ distance between the two curves $\psi _1 \circ \alpha _1 |_{[l-1,l]}$ and $\alpha _1|_{[l, l+1]}$ is small. More precisely, we will prove:

Claim 37

$d( \psi _1 \circ \alpha _1(l-1+t) , \alpha _1(l+t)) < \varepsilon ,$ for each $t \in [0,1],$ and for each $l \in \{K_i : 0 \leqslant i \leqslant m-1 \}$.

Proof of Claim

Using the definition of $\alpha _1$, it can easily be checked that $\alpha _1(K_i+t) = \phi ^t_H(x_{i+1})$ and $\alpha _1(K_i-1 +t) = \tau \phi ^{k_i-1+t}_H(x_{i}) = \tau \phi ^{-1+t}_H(y_{i})$. It follows that $\psi _1 \circ \alpha _1(K_i-1+t) = \tau \phi ^1_H \tau \phi ^{-1 + t}_H (y_{i})$.

Now, recall that the curves $t \mapsto \phi ^t_H(y_{i})$ and $t \mapsto \phi ^t_H(x_{i})$ intersect the support of $\tau$ only for $t \in (-2 \kappa , 2 \kappa )$. Using this fact it can be checked that

$$\begin{aligned} \tau \phi ^1_H \tau \phi ^{-1 + t}_H (y_{i}) = {\left\{ \begin{array}{ll} \tau \phi ^t_H(y_{i}), &{}\quad t\in [0, 2 \kappa ],\\ \phi ^t_H(y_{i}), &{}\quad t\in [2 \kappa , 1-2\kappa ],\\ \tau \phi ^1_H \left( \tau \phi ^{-1 + t}_H (y_{i})\right) , &{}\quad t\in [1-2 \kappa , 1]. \end{array}\right. } \end{aligned}$$

If $t\in [0, 2 \kappa ]$ then $\tau \phi ^t_H(y_{i}), \phi ^t_H(x_{i+1}) \in W(a_{i})$ and the set $W(a_{i})$ has diameter less than $\delta$, thus $d(\tau \phi ^t_H(y_{i}), \phi ^t_H(x_{i+1}))< \delta < \varepsilon$.

If $t \in [2 \kappa , 1-2\kappa ]$, then we must prove that $d(\phi ^t_H(y_{i}), \phi ^t_H(x_{i+1})) < \varepsilon$. This follows immediately from Eq. (4) and the fact that $d(y_{i}, x_{i+1}) < \delta$.

Finally, we consider the case $t\in [1-2 \kappa , 1]$: Write $\phi ^t_H(x_i) = \phi ^1_H (\phi ^{-1+t}_H(x_i))$. Note that in this case $\tau \phi ^{-1 + t}_H (y_{i})$ and $\phi ^{-1 +t}_H(x_{i+1})$ are both contained in the set $W(a_{i})$ which has diameter less than $\delta$. We see, via Eq. (4), that

$$\begin{aligned}&d\left( \tau \phi ^1_H\left( \tau \phi ^{-1 + t}_H (y_{i})\right) , \phi ^t_H(x_{i+1})\right) \leqslant d_{C^0}(\tau , Id) \\&\quad + \, d\left( \phi ^1_H\left( \tau \phi ^{-1 + t}_H (y_{i})\right) , \phi ^1_H\left( \phi ^{-1+t}_H(x_{i+1})\right) \right)< \delta + \frac{\varepsilon }{2} < \varepsilon , \end{aligned}$$

which proves our claim. $\square$

Step 4: Constructing the curve $\alpha$ Let $\mathcal {V}$ denote a contractible open subset of M. Since $\mathcal {V}$ is contractible there exists a 1-form $\lambda$ such that $\omega = d\lambda$ inside $\mathcal {V}$. Using the 1-form $\lambda$ we can define the action of any curve $a: [0,1] \rightarrow \mathcal {V}$ to be the integral $\int _0^1 a^* \lambda .$ Of course, the action of a curve depends on $\lambda$, however the difference of action between two curves with the same endpoints does not depend on the choice of $\lambda$ or $\mathcal {V}$.

For any $0 \leqslant l \leqslant k$, the two curves $\alpha _1|_{[l, l+1]}$ and $\psi _1 \circ \alpha _1|_{[l-1,l]}$ coincide near their endpoints. Furthermore, one can find a contractible neighborhood of $\alpha _1|_{[l, l+1]}$ which contains $\psi _1 \circ \alpha _1|_{[l-1,l]}$: Indeed, as mentioned in Remark 35, if $l \notin \{K_i : 0 \leqslant i \leqslant m-1 \}$, then $\psi _1 \circ \alpha _1 |_{[l-1,l]} = \alpha _1|_{[l, l+1]}$. Hence, we can take any sufficiently small neighborhood of $\alpha _1|_{[l,l+1]}$. If $l = K_i, \; 0 \leqslant i \leqslant m-1$, then by Remark 36 both curves are contained in $U_i$ which is contractible.

If $l=1$ the two curves $\alpha _1|_{[l, l+1]}$ and $\psi _1 \circ \alpha _1|_{[l-1,l]}$ coincide. Beginning with $l=2$, for each $2 \leqslant l \leqslant k$ we successively make $C^0$-small perturbations of $\alpha _1|_{[l,l+1]}$ near $t= l+ \tfrac{1}{2}$ to obtain an embedded curve $\alpha$ with the property that the two curves $\alpha |_{[l,l+1]}$ and $\psi _1 \circ \alpha |_{[l-1,l]}$ have the same action and are still contained in the same contractible open set mentioned in the previous paragraph. Here, we are again using the fact that one can make an arbitrary adjustment to the action of a curve by performing a $C^0$-small perturbation; see Remark A.13 of [3]. See Fig. 6.

Fig. 6

A portion of the curve $\alpha _1$ in blue, its image by $\psi _1$ in green, and the curve $\alpha$ obtained from $\alpha _1$ by local perturbations which are represented in red

We will ensure that the perturbations made in the previous paragraph satisfy the following properties. In the case where $l \in \{K_i : 0 \leqslant i \leqslant m-1 \}$, we require the $C^0$ distance between $\alpha |_{[l,l+1]}$ and $\psi _1 \circ \alpha |_{[l-1,l]}$ to be less than $\varepsilon .$ The fact that this can be achieved follows from Claim 37 and by taking the perturbations from the previous paragraph to be sufficiently small. Next consider the case where $l \notin \{K_i : 0 \leqslant i \leqslant m-1 \}$. In the previous paragraph, we obtained $\alpha$ from $\alpha _1$ by making a $C^0$ small perturbation of $\alpha _1$ in the interior of a small closed interval which contains the point $t = l + \frac{1}{2}$; let $I_l \subset [l, l+1]$ denote this interval. Since the intervals $I_l$ can be taken to be arbitrarily small and the action-adjusting perturbations from the previous step can be taken to have arbitrarily small support, we can pick contractible neighborhoods $V_l$ of $\alpha |_{I_l}$ such that these neighborhoods are pairwise disjoint and the image of $\psi _1 \circ \alpha |_{I_{l-1}}$ is contained in $V_l$ for each l. Furthermore, we require that $\phi ^1_H(V_l) \cap V_l = \emptyset$ and that $\alpha (t) \in V_l$ only if $t \in (l,l+1)$. For $l = K_i$ we of course can assume that $\psi _1 \circ \alpha |_{[l-1,l]}$ and $\alpha |_{[l,l+1]}$ are contained in $U_i$. We remark that by taking $V_k$ and $I_k$ to be sufficiently small we can ensure that the formula for $\alpha$ from the third property of the statement of the theorem continues to hold.

Lastly, we point out that since $\alpha _1([2, k+1])$ is contained in $U \cup \mathcal {C}(\gamma ((0,1)))$ (see Remark 33), by picking the sets $V_l$ to be small enough, we may assume that each $V_l$ is contained in $U \cup \mathcal {C}(\gamma ((0,1)))$.

Step 5: Turning $\alpha$ into an invariant curve In this final step of the proof, we will perturb $\psi _1$ to a Hamiltonian diffeomorphism $\psi$ such that $\psi \circ \alpha |_{[l-1,l]} = \alpha |_{[l,l+1]}$. Recall that in Step 4 we arranged that the two curves $\psi _1 \circ \alpha |_{[l-1,l]}$ and $\alpha |_{[l,l+1]}$ have the same action.

First assume that $l \notin \{ K_i : 0 \leqslant i \leqslant m-1 \}$. Then, we can find a Hamiltonian diffeomorphism $\eta _l$, generated by a Hamiltonian which is compactly supported in $V_l$, such that $\eta _l \circ \psi _1 \circ \alpha |_{I_{l-1}} = \alpha |_{I_l}$. Now, let $\eta$ be the composition of the $\eta _l$’s for $l \notin \{K_i : 0 \leqslant i \leqslant m-1 \}$ and write $\psi _2 = \eta \circ \psi _1$ and observe that

$$\begin{aligned} \psi _2 \circ \alpha |_{[l-1,l]} = \alpha |_{[l, l+1]}. \end{aligned}$$

(9)

Since the supports of $\eta _l$’s are disjoint and can be taken to be as small as one wishes, we obtain the following inequalities

$$\begin{aligned} d_{C^0}\left( \psi _2, \phi ^1_H\right) = d_{C^0}\left( \eta \circ \psi _1, \phi ^1_H\right) \leqslant d_{C^0}(\eta , Id) + d_{C^0}\left( \psi _1, \phi ^1_H\right)< \delta < \varepsilon . \end{aligned}$$

We remark that we can assume that $\eta$ is the time-1 map of a Hamiltonian, say $G_2$, whose norm $\Vert G_2\Vert _{\infty }$ can be made as small as one wishes. Existence of such Hamiltonian $G_2$ follows from Lemma 10.

We will now deal with the case $l \in \{K_i : 0 \leqslant i \leqslant m-1 \}$. Fix i and to simplify our notation denote $\gamma _0(t) := \psi _2 \circ \alpha |_{[K_i-1,K_i]}(t) = \psi _2 \circ \alpha (K_i-1+t)$ and $\gamma _1(t) := \alpha |_{[K_i,K_i+1]}(t) = \alpha (K_i+t).$ The curves $\gamma _0, \gamma _1$ coincide for $t\in [0, \kappa ] \cup [1- \kappa ,1]$: by Remark 35 this was true for $\alpha _1$, and $\alpha$ differs from $\alpha _1$ only for t near $K_i+\frac{1}{2}$. Recall from Step 4 that the $C^0$ distance between $\gamma _0$ and $\gamma _1$ is less than $\varepsilon$ and that both of these curves are contained in $U_i$.

Claim 38

There exists a homotopy $F_i(s,t): [0,1] \times [\kappa ,1-\kappa ] \rightarrow U_i$ rel. end points from $\gamma _0|_{[\kappa , 1- \kappa ]}$ to $\gamma _1|_{[\kappa , 1- \kappa ]}$ such that under this homotopy the trajectory of any point of $\gamma _0|_{[\kappa , 1- \kappa ]}$ has diameter less than $C \varepsilon$ for some constant $C>0$ that only depends on U, the diffeomorphism $\Phi$ introduced before Step 1 and the Riemannian metric.

Proof

The existence of the homotopy $F_i$ follows from the fact that the diffeomorphism $\Phi$ picked at the beginning of the proof identifies $U_i$ with the box $(\frac{i}{m} - a, \frac{i+1}{m} +a ) \times (-c, 1+c) \times (-c,c) \times \cdots \times (-c,c)$. We can take $F_i$ to be the straight-line homotopy in these coordinates. The statement about diameter of trajectories of points on $\gamma _0|_{[\kappa , 1- \kappa ]}$ follows from the fact that the $C^0$ distance between $\gamma _0$ and $\gamma _1$ is less than $\varepsilon$. The constant C is given by the ratio of the pull back, via $\Phi$, of the Riemannian metric d to $(-c, 1+c ) \times (-c, 1+c) \times (-c,c) \times \cdots \times (-c,c)$ and the usual Euclidean metric. $\square$

The dimension of M is at least 4 and thus we can perturb the homotopy $F_i$, provided to us by the above claim, to ensure that its image intersects the image of $\alpha$ only for $t \in (K_i, K_i+1)$ and the image of $\psi _2 \circ \alpha$ only for $t \in (K_i -1,K_i)$. Let $W_i \subset U_i$ be a small neighborhood of the image of the homotopy $F_i$ which intersects the images of $\alpha$ and $\psi _2 \circ \alpha$ only for $t \in (K_i, K_i+1)$ and $t \in (K_i-1, K_i)$, respectively. Lastly, recall that we picked the sets $U_i$ in Step 1 such that $U_i$ can only intersect $U_{i-1}$ and $U_{i+1}$. Hence, $W_i$ can only intersect $W_{i-1}$ and $W_{i+1}$.

Now, we apply the h-principle of Proposition 9 to obtain a Hamiltonian, say $G_i'$, generating a Hamiltonian isotopy $\varphi _i^t, t\in [0,1]$, such that

• 1. $G_i'$ is supported in $W_i$ and $\Vert G_i'\Vert _{\infty }$ is as small as one wishes,

• 2. $\varphi _i^1 ( \psi _2 \circ \alpha |_{[K_i-1,K_i]}) = \alpha |_{[K_i,K_i+1]}$,

• 3. $d_{C^0}(Id, \varphi _i^1) < 2C \varepsilon$.

Let $\phi _{\mathrm {odd}}$ denote the composition of the diffeomorphisms $\varphi _i^1$ for odd i. Note that $\phi _{\mathrm {odd}}$ is the time-1 map of the Hamiltonian $G_{\mathrm {odd}}$ which is the sum of all $G_i'$ for odd i. Define $\phi _{\mathrm {even}}$ and $G_{\mathrm {even}}$ similarly. Let $\psi := \phi _{\mathrm {even}} \phi _{\mathrm {odd}} \psi _2$.

We remark that $W_{m-1}$ is disjoint from $\phi ^1_H \circ \alpha ([k, k+1])$. As pointed out in Remark 34, this ensures that the third property from the statement of the theorem holds.

We will now check that it is indeed true that $\psi \circ \alpha (t) = \alpha (t+1)$ for $t\in [0,k]$. If $l \notin \{K_i : 0 \leqslant i \leqslant m-1 \}$, then $\psi \circ \alpha |_{[l-1,l]} = \psi _2 \circ \alpha |_{[l-1,l]}$: this is because we picked the sets $W_i$, whose union contains the supports of $\phi _{\mathrm {even}}$ and $\phi _{\mathrm {odd}}$, such that they do not intersect the image of $\psi _2 \circ \alpha |_{[l-1,l]}$. We have already checked, in Eq. (9), that $\psi _2 \circ \alpha (t) = \alpha (t+1)$ for $t \in [l-1,l]$. We must next check that $\psi \circ \alpha |_{[l-1,l]} = \alpha |_{[l,l+1]}$ for $l \in \{K_i : 0 \leqslant i \leqslant m-1 \}$. Fix i and let $l = K_i$. Then, $\phi _{\mathrm {even}} \phi _{\mathrm {odd}} \psi _2 \circ \alpha |_{[K_i-1,K_i]} = \varphi _i^1 \psi _2 \circ \alpha |_{[K_i-1,K_i]}$ because we picked the sets $W_i$ such that the image of $\psi _2 \circ \alpha |_{[K_i-1,K_i]}$ only intersects $W_i$. Now, $\varphi _i^1$ was picked such that $\varphi _i^1 (\psi _2 \circ \alpha |_{[K_i-1,K_i]}) = \alpha |_{[K_i,K_i+1]}.$

The rest of the proof is dedicated to verifying the fourth and the fifth properties from the statement of the theorem. We will first check that $d_{C^0}(\psi , \phi ^1_H) < (4C+1) \varepsilon$. First, note that $d_{C^0}( \phi _{\mathrm {odd}}, Id) < 2C \varepsilon$ because $\phi _{\mathrm {odd}}$ is the composition of the diffeomorphisms $\varphi ^1_i$, for odd i, which have disjoint supports and each of which satisfies $d_{C^0}(\varphi _i^1, Id) < 2C \varepsilon$. Similarly, $d_{C^0}( \phi _{\mathrm {even}}, Id) < 2C \varepsilon$. It follows that $d_{C^0}(\psi , \phi ^1_H) = d_{C^0}( \phi _{\mathrm {even}} \phi _{\mathrm {odd}} \psi _2, \phi ^1_H) \leqslant d_{C^0}( \phi _{\mathrm {even}} \phi _{\mathrm {odd}}, Id) + d_{C^0}(\psi _2, \phi ^1_H) < 4C \varepsilon + \varepsilon$; recall that we proved earlier that $d_{C^0}(\psi _2, \phi ^1_H) < \varepsilon$. By going back to Step 1 and replacing $\varepsilon$ with $\frac{\varepsilon }{4C+1}$ we obtain $d_{C^0}(\psi , \phi ^1_H) < \varepsilon$.

It is fairly easy to see that we obtained $\psi$ from $\phi ^1_H$ by composing it, on the left, with the Hamiltonian diffeomorphism $\theta := \phi _{even} \circ \phi _{odd} \circ \eta \circ \tau$; we denote by F the generating Hamiltonian of $\theta$. Now, the supports of the diffeomorphisms $\tau , \eta , \phi _{odd}, \phi _{even}$ are all contained in the union of U and the open sets $V_l, l \notin \{K_i : 0 \leqslant i \leqslant m-1 \}$. Recall that, as was mentioned at the end of Step 4, each of the sets $V_l$ is contained in $U \cup \mathcal {C}(\gamma ((0,1)))$. This proves the claim about the support of F.

As was remarked throughout the proof, the generating Hamiltonians of $\tau , \eta , \phi _{odd}, \phi _{even}$, which we denoted by $G_1, G_2, G_{\mathrm {odd}}, G_{\mathrm {even}}$, respectively, were picked to have norms as small as one wishes. This proves the claim about $\Vert F\Vert _{\infty }$; we have established the fourth property in the statement of the theorem.

To finish the proof of the theorem, it remains to show that $\psi$ has the same set of fixed points as $\phi ^1_H$. Now, $\psi = \theta \circ \phi ^1_H$ and $\theta$ is supported in the union of U and the open sets $V_l$. Since U and the $V_l$’s do not contain any of the fixed points of $\phi ^1_H$ we conclude that the fixed points of $\phi ^1_H$ are all fixed points of $\psi$ as well. Now, to show that $\psi$ has no additional fixed points we will check that any point x in the support of $\theta$ cannot be a fixed point of $\psi$. First, suppose that $x \in U$. Since U is a small neighborhood of the compact set $\Gamma ([0,1] \times [0,1])$, which contains no fixed points of $\phi ^1_H$, there exists a positive number, say $\varrho$, such that $d(x, \phi ^1_H(x)) > \varrho$ for all $x \in U$. Since $d_{C^0}(Id, \theta ) < \varepsilon$, picking $\varepsilon$ to be smaller than $\varrho$ guarantees that $\theta \phi ^1_H(x) \ne x$. Next, suppose that $x \notin U$. Hence, it must be the case that x is contained in one of the $V_l$’s. We leave it to the reader to check, using the condition $\phi ^1_H(V_l) \cap V_l = \emptyset$, that x can not be a fixed point of $\psi$. This completes the proof of Theorem 32. $\square$

3.3.4 Proof of Theorem 25

In this section, we prove Theorem 25 with the help of Theorems 26 and 32.

Let H be a Morse function as described in Lemma 22, and let $\varepsilon >0$. Let p, q be critical points of H, at least one of the two being non-extremal and let $\gamma :[0,1]\rightarrow M$ be a curve as in the statement of Theorem 25. We will give the proof under the assumption that q is not a minimum. The proof in the case where q is a minimum (hence p is not a maximum), can be easily adapted from this case.

Let $\varepsilon ,\rho >0$, let $V_p$ be either the empty set if p is a maximum or an open set containing $\{\phi _H^t(\gamma (0)):-\infty <t\leqslant 1\}$ is p is not a maximum, and let $V_q$ be an open set containing $\{\phi _H^t(\gamma (1)):-1\leqslant t<+\infty \}$.

Our construction will be carried out in three steps: first in the neighborhood of p, then from the neighborhood of p to that of q, and finally in the neighborhood of q. Figure 7 below might help to grasp the construction.

Step 1: Construction in the neighborhood of p We need to consider two cases.

The simpler case is where p is not a maximum. In this case, we set $z=\gamma (0)$ and $\alpha _1(t)=\phi _H^{t-2}(z)$ for all $t\in (-\infty , 1]$. At this point there is no need for a perturbation of $\phi _H^1$, thus we set $\theta _1=\mathrm {Id}$. It is clear that, $\alpha _1$ and $\theta _1$ satisfy the requirements of Theorem 25 for $\alpha$ and $\theta$.

Now consider the case when p is a maximum. In this case, $\gamma (0)=p$ and it is our assumption that in some Darboux coordinates around p, H is of the form $c\sum _{i=1}^n(x_i^2+y_i^2)$ where c is a small negative constant. We denote by $\Vert \cdot \Vert$ the standard euclidean norm in these coordinates, and by B(0, r) the euclidean ball centered at 0 and of radius r. Let z be a point on the image of $\gamma$, distinct from p, but close enough to p so that we may apply Theorem 26 to the Hamiltonian $H'=-H$ and to the point $x=\phi _H^{-2}(z)=\phi _{H'}^2(z)$. Given a ball B centered at p and containing x, and given $\varepsilon >0$, Theorem 26 provides us with a curve $\alpha ':[ - 1,+\infty )\rightarrow B {\setminus } \{p\}$ and a Hamiltonian homeomorphism $\theta '$ satisfying :

• $\alpha '(t) =\phi ^t_{H'}(x)$ for $t \in [- 1,0]$ and $\alpha '(t) \rightarrow p$ as $t\rightarrow +\infty$,

• $\theta ' \circ \phi ^1_{H'} (\alpha '(t)) = \alpha '(t+1)$ for all $t \in [-1, \infty )$,

• $\theta '(p)=p$ and $\theta ' \circ \phi ^1_{H'}$ has the same fixed points as $\phi _{H'}^1$,

• The support of $\theta '$ is contained in the union of the open ball $B(0,\Vert x\Vert )$ and the $\varepsilon$-neighborhood of $\{\phi _{H'}^t(x):t\in [1,2]\}$,

• $d_{C^0}(Id, \theta ') < \varepsilon '$ and $\Vert F'\Vert _{\infty } \leqslant \rho$, where $F'$ denotes a continuous Hamiltonian such that $\phi ^1_{F'} = \theta '$,

• For any neighborhood of p, there exists a compactly supported Hamiltonian diffeomorphism of B which coincides with $\theta '$ in the complement of that neighborhood.

Fig. 7

The case “p not a maximum” is on the left, and the case “p a maximum” is on the right. The curves $\alpha _1$, $\alpha _2$ and $\alpha _3$ are respectively in green, blue and red. The supports of the successive perturbations on $\phi _H^1$ are included in the grey regions

Now define $\alpha _1(t)=\theta '^{-1}(\alpha '( -t))$ for all $t\in (-\infty , 1]$ and $\theta _1=\theta '^{-1}$. We deduce the following from the above properties:

• 1. $\alpha _1(t) =\phi ^{t-2}_{H}(z)$ for $t \in [0,1]$ and $\alpha _1(t) \rightarrow p$ as $t\rightarrow -\infty$,

• 2. $\theta _1 \circ \phi ^1_{H} (\alpha _1(t)) = \alpha _1(t+1)$ for all $t \in (-\infty , 0]$,

• 3. $\theta _1 \circ \phi ^1_{H}$ has the same fixed points as $\phi _H^1$,

• 4. $\alpha _1$ takes values in the closed punctured ball centered at p and having z on its boundary, which is contained in $\mathcal {C}(\gamma ((0,1)))$.

• 5. The support of $\theta _1$ is contained in the union of the open ball $B(0,\Vert x\Vert )$ and the $\varepsilon$-neighborhood of $\{\phi _{H}^t(x) :t\in [-2,-1]\}$,

• 6. $d_{C^0}(Id, \theta _1) < \varepsilon$ and $\Vert F_1\Vert _{\infty } <\rho$, where $F_1$ denotes a continuous Hamiltonian such that $\phi ^1_{F_1} = \theta _1$,

• 7. For any neighborhood of p, there exists a compactly supported Hamiltonian diffeomorphism of B which coincides with $\theta _1$ in the complement of that neighborhood.

We see that $\alpha _1$ and $\theta _1$ are compatible with the requirements of Theorem 25.

Step 2: From a neighborhood of p to a neighborhood of q In both cases above we have set $z=\gamma (a)$ with $0\leqslant a< 1$ ($a>0$ if p is a maximum, $a=0$ otherwise), and let us now set $w=\gamma (1)$ and $\tilde{\gamma }(t)=\gamma (a+(1-a)t)$ for $t\in [0,1]$. The curve $\tilde{\gamma }$ satisfies $\tilde{\gamma }(0)=z$ and $\tilde{\gamma }(1)=w$. It also satisfies that $\frac{d}{dt} H \circ \tilde{\gamma }(t) < 0$, which implies in particular that if H is $C^2$-small enough then the map $\Gamma :(s,t)\mapsto \phi _H^t(\tilde{\gamma }(s))$ is a smooth embedding of the set $(\{0\}\times [-2,1])\cup ((0,1)\times [0,1]) \cup (\{1\}\times [-1,2])$. In the case where p is a maximum, note that since the flow of H preserves level sets, the compact set $\Gamma ([0,1]\times [0,1])$ does not intersect the open ball $B(0,\Vert x\Vert )=B(0,\Vert z\Vert )$. Moreover, if $\varepsilon$ and c are small enough, $\Gamma ([0,1]\times [0,1])$ does not intersect the $\varepsilon$-neighborhood of $\{\phi _{H}^t(x):t\in [-2,-1]\}$ either. Hence, $\Gamma ([0,1]\times [0,1])$ does not intersect the support of $\theta _1$.

We are now in the situation to apply Theorem 32: Let $U \supset \Gamma ([0,1] \times [0,1])$ be an open set chosen so small that:

• If p is not a maximum,

  $$\begin{aligned} U\cup \,\mathcal {C}(\tilde{\gamma }((0,1)))=U\cup \,\mathcal {C}(\gamma ((0,1)))\subset V_p\cup \mathcal {C}(\gamma ((0,1)))\cup V_q, \end{aligned}$$

  (10)

• If p is a maximum, U does not intersect the support of $\theta _1$ and

  $$\begin{aligned} U\cup \,\mathcal {C}(\tilde{\gamma }((0,1)))= U\cup \,\mathcal {C}(\gamma ((a,1)))\subset \mathcal {C}(\gamma ((0,1)))\cup V_q. \end{aligned}$$

  (11)

• U does not intersect the piece of orbit $\{ \phi _H^t(w) : t \in [1+\frac{1}{3},+\infty )\}$.

Then, for every $\varepsilon ,\rho >0$, we can find a curve $\tilde{\alpha }_2:[0, k+1]\rightarrow M$, with $\alpha _2(0)= x = \phi _H^{-2}(z)$ and $\alpha _2(k)=w$, and a Hamiltonian diffeomorphism $\psi$ which is $C^0$-close to $\phi _H^1$ and satisfies a certain list of properties. We set $\theta _2=\psi \circ \phi _H^{-1}$. The properties of $\alpha _2$ and $\psi =\theta _2\circ \phi _H^1$ listed in Theorem 32 have the following consequences:

• 1. $\alpha _2(t)=\alpha _1(t)=\phi _H^{t-2}(z)$ for all $t\in [0,1]$,

• 2. $\theta _2\circ \phi _H^1(\alpha _2(t))=\alpha _2(t+1)$ for all $t\in [0,k]$,

• 3. $\theta _2\circ \phi _H^1$ has the same set of fixed points as $\phi _H^1$,

• 4. There exists $t_0\in (1,2)$ such that $\alpha _2(t)=\phi _H^t(x)$ for all $t\in [0,t_0]$ and $\alpha _2([t_0,k+1])\subset U\cup \,\mathcal {C}(\tilde{\gamma }((0,1)))$. If p is not a maximum then $\{\phi _H^t(x):t\in [0,t_0]\} \subset V_p$ by definition of $V_p$; if p is a maximum, then $\{\phi _H^t(x):t\in [0,t_0]\}$ is contained in the boundary of the ball $B(0,\Vert x\Vert )$, hence in $\mathcal {C}(\gamma ((0,1)))$. Using the inclusions (10) and (11), we thus get that $\alpha _2$ takes values in $V_p\cup \mathcal {C}(\gamma ((0,1)))\cup V_q$.

• 5. $\theta _2$ is generated by a Hamiltonian $F_2$ which is supported in the open set $U\cup \,\mathcal {C}(\tilde{\gamma }((0,1)))$, which does not intersect the support of $\theta _1$,

• 6. $d_{C^0}(\theta _2,\mathrm {Id})<\varepsilon$ and $\Vert F_2\Vert _{\infty }<\rho$,

• 7. $\theta _2$ is a smooth Hamiltonian diffeomorphism.

All the points above follow immediately from Theorem 32. Note that it also follows from Theorem 32 that for any neighborhood V of $\phi _H^{\frac{1}{2}}(w)$, the curve $\alpha _2$ can be chosen so that for some $b>0$, and all $t\in [k,k+1]$,

$$\begin{aligned} \alpha _2(t) {\left\{ \begin{array}{ll} =\phi _H^{t -k}(w), &{}\quad t\in \left[ k,k+\tfrac{1}{2}-b\right] \cup \left[ k+\tfrac{1}{2}+b,k+1\right] \\ \in V, &{}\quad t\in \left[ k+\tfrac{1}{2}-b,k+\tfrac{1}{2}+b\right] . \end{array}\right. } \end{aligned}$$

(12)

Step 3: Construction in the neighborhood of q By assumption, $\phi _H^t(w)=\phi _H^t(\gamma (1))\rightarrow q$ when $t\rightarrow +\infty$. Let $\tilde{\alpha }_3:[k,+\infty )\rightarrow M$ be the smooth curve defined by

$$\begin{aligned} \tilde{\alpha }_3(t)={\left\{ \begin{array}{ll}\alpha _2(t), &{}\quad t\in [k,k+1)\\ \phi _H^{t-k}(w), &{}\quad t\in [k+1,+\infty ). \end{array}\right. } \end{aligned}$$

It has the following properties:

• $\tilde{\alpha }_3(t)\rightarrow q$ when $t\rightarrow +\infty$,

• For some small $\kappa >0$, the image of $\tilde{\alpha }_3|_{[k+1+\kappa ,+\infty )}$ does not intersect the supports of $\theta _1$ nor $\theta _2$,

• For all $t\in [k,+\infty ){\setminus }(k+\tfrac{1}{2}-b,k+\tfrac{1}{2}+b)$,

  $$\begin{aligned} \theta _2\circ \theta _1\circ \phi _H^1(\tilde{\alpha }_3(t))=\phi _H^1(\tilde{\alpha }_3(t))=\tilde{\alpha }_3(t+1),\end{aligned}$$

• According to (12), for $t\in [k+\tfrac{1}{2}-b,k+\tfrac{1}{2}+b]$, $\theta _2\circ \theta _1\circ \phi _H^1(\tilde{\alpha }_3(t))$ belongs to $\phi _H^1(V)$ which is an arbitrarily small neighborhood of $\tilde{\alpha }_3(k+1+\tfrac{1}{2})$.

For every integer $l\in \{k+1, k+2, \ldots \}$, we let $B_l$ be a small ball centered at $\tilde{\alpha }_3(l+\tfrac{1}{2})$ and included in the open set $V_q$. We may assume V is small enough so that $V\subset U$ and $\phi _H^1(V)\subset B_{k+1}$. We also assume that the balls $B_l$ are all disjoint, and that they do not intersect the supports of $\theta _1$ nor $\theta _2$. Similarly as in Step 4 of the proof of Theorem 32, we successively make $C^0$-small perturbations of $\tilde{\alpha }_3|_{[l,l+1]}$ in $B_l$, for $l\in \{k+1, k+2, \ldots \}$, to obtain an embedded curve $\alpha _3$, such that the curves $\phi _H^1(\alpha _3|_{[l-1,l]})$ and $\alpha _3|_{[l,l+1]}$ have the same action for all l. We can choose those perturbations so that, for each l, these two curves coincide in the complement of a very small interval $J_l$ centered at $l+\frac{1}{2}$, and they both send $J_l$ into $B_l$. Using Lemma 10 (as in Step 5 in the proof of Theorem 32), this property implies that we can find Hamiltonian diffeomorphisms $\eta _l$, generated by Hamiltonians which are compactly supported in the $B_l$’s, such that $\eta _l(\phi _H^1(\alpha _3(t)))=\alpha _3(t+1)$ for all $t\in [l-1,l]$. Moreover, Lemma 10 tells us that these generating Hamiltonians can be chosen with arbitrary small $\Vert \cdot \Vert _\infty$ norm.

Let $\theta _3$ be the composition of all the $\eta _l$’s for $l\in \{k+1, k+2, \ldots \}$. By construction, $\theta _3$ is generated by a Hamiltonian $F_3$ whose support does not intersect the support of $\theta _1$ nor that of $\theta _2$.

Moreover, the $\eta _l$’s can be chosen $C^0$-close to $\mathrm {Id}$ by shrinking the balls $B_{k+1}$, $B_{k+2}$, $\dots$. If we shrink each ball $B_l$ so that $\mathrm {diam}B_l<\inf _{z\in B_l}d(z,\phi _H^1(z))$, then $\theta _3\circ \phi _H^1$ has no fixed point in $B_l$. Indeed, the triangle inequality yields for all $z\in B_l$:

$$\begin{aligned} d\left( \theta _3\circ \phi _H^1(z),z\right) \geqslant d\left( z,\phi _H^1(z)\right) - d\left( \theta _3\circ \phi _H^1(z),\phi _H^1(z)\right) >0. \end{aligned}$$

Therefore, the following list of properties is satisfied:

• 1. $\alpha _3(t)\rightarrow q$ when $t\rightarrow +\infty$,

• 2. $\theta _3\circ \phi _H^1(\alpha _3(t)) =\alpha _3(t+1)$ for all $t\in [k,+\infty )$,

• 3. $\theta _3\circ \phi _H^1$ has the same set of fixed points as $\phi _H^1$,

• 4. For all $t\in [k,+\infty )$, $\alpha _3(t)\in V_q$,

• 5. $\mathrm {supp}(F_3)\subset V_q\cup \{q\}$,

• 6. $d_{C^0}(\mathrm {Id},\theta _3)<\varepsilon$ and $\Vert F_3\Vert _{\infty }<\rho$.

• 7. For any neighborhood of q, $\theta _3$ coincides with a Hamiltonian diffeomorphism in the complement of that neighborhood.

End of the proof Let $\theta$ be the Hamiltonian homeomorphism $\theta =\theta _1\circ \theta _2\circ \theta _3$ and let $\alpha$ be the smooth curve which coincides with $\alpha _1$ on $(-\infty ,1]$, with $\alpha _2$ on $[0,k+1]$, and with $\alpha _3$ on $[k,+\infty )$. The respective Properties 1–7 established for each $\theta _1$, $\theta _2$ and $\theta _3$ in the previous steps, together with the fact that their supports are all disjoint, imply the corresponding Properties 1–7 in Theorem 25 whose proof in now achieved. $\square$