On Herman's Positive Entropy Conjecture

TL;DR

The paper proves Herman's conjecture by showing that area-preserving surface diffeomorphisms with elliptic fixed points can be perturbed to exhibit positive metric entropy, indicating widespread chaotic behavior.

Contribution

It demonstrates that the identity map of the disk can be smoothly perturbed to a conservative diffeomorphism with positive entropy, confirming Herman's conjecture and advancing understanding of chaotic dynamics.

Findings

Any area-preserving surface diffeomorphism with an elliptic fixed point can be perturbed to have positive entropy.

The identity map of the disk can be approximated by diffeomorphisms with positive entropy.

Existence of a dense set of conservative diffeomorphisms either weakly stable or with chaotic islands.

Abstract

We show that any area-preserving $C^r$-diffeomorphism of a two-dimensional surface displaying an elliptic fixed point can be $C^r$-perturbed to one exhibiting a chaotic island whose metric entropy is positive, for every $1\le r\le \infty$. This proves a conjecture of Herman stating that the identity map of the disk can be $C^\infty$-perturbed to a conservative diffeomorphism with positive metric entropy. This implies also that the Chirikov standard map for large and small parameter values can be $C^\infty$-approximated by a conservative diffeomorphisms displaying a positive metric entropy (a weak version of Sinai's positive metric entropy conjecture). Finally, this sheds light onto a Herman's question on the density of $C^r$-conservative diffeomorphisms displaying a positive metric entropy: we show the existence of a dense set formed by conservative diffeomorphism which either are weakly stable (so, conjecturally, uniformly hyperbolic) or display a chaotic island of positive metric entropy.

Figures

Equations

$$\lambda (x)=\underset{n\to \infty }{\mathrm{limsup}}\frac{1}{n}\log \Vert D{f}^n(x)\Vert .$$

$$h_{\omega }(f):={\int }_{\mathbb{M}}\lambda (x)\omega (dx).$$

$$d_{\infty }(f,g)=\sum _{r=0}^{\infty }\frac{1}{r!}\min (1,d_r(f,g))$$

$$T_a(x,y)=(2x-y+a\sin 2\pi x,x).$$

$$T\mathbb{M}{|}_K=E^s\oplus E^u,\exists N\ge 1,\Vert D{f}^N{|}_{E^s}\Vert <1,\Vert D{f}^{-N}{|}_{E^u}\Vert <1.$$

$$W^s(P;f):=\{x\in \mathbb{M}:d(f^n(x),P)\underset{n\to +\infty }{\to }0\}\text{and}{W}^u(P;f):=\{x\in \mathbb{M}:d(f^n(x),P)\underset{n\to -\infty }{\to }0\}.$$

$${\hat{F}}_{Q,n}=Q^{-1}\circ f^n{|}_{Q(\mathbb{D})}\circ Q$$

$${\check{L}}_i^a\cup {\check{L}}_i^b\subset W^u({\check{P}}_i;\check{F})\cup W^s({\check{Q}}_i;\check{f}).$$

$$\bar{F}=S_{\psi }\circ F,\text{mbox}where{S}_{\psi }:=(x,y)\mapsto (x,y+\psi (x)),$$

$$F_A:(x,y)\mapsto A(x,y):=(13x+8y,8x+5y)$$

$${\Omega }_i+(x,y)\mapsto {\Omega }_i+(e^{\sigma }x,e^{-\sigma }y).$$

$$\dot{x}={\partial }_y{H}_i(x,y),\dot{y}=-{\partial }_x{H}_i(x,y)$$

$$H_i=xy.$$

$$(x,y)=(\sqrt{2\rho }\cos (\theta ),\sqrt{2\rho }\sin (\theta ))$$

$$H_i(\theta ,\rho )=\rho \sin (2\theta ).$$

$$\frac{d}{dt}\rho =2\rho \cos (2\theta ),\frac{d}{dt}\theta =-\sin (2\theta ).$$

$$\bar{\rho }=\rho p_0(\theta ),\bar{\theta }=q_0(\theta ),$$

$$\psi (\rho )=\rho -\frac{{\delta }^2}{2}\text{if ρ is close to }\frac{{\delta }^2}{2}\text{and}\psi (\rho )=\rho \text{if ρ is close to }\frac{{ϵ}^2}{2}.$$

$${\Psi }_i{|}_{V_i}:(\theta ,\rho )\mapsto (\theta ,\psi (\rho ))$$

$$\Psi ={\Psi }_i$$

$${\hat{H}}_i=(\rho -{\delta }^2/2)\sin (2\theta ).$$

$${\hat{H}}_i=(\rho -{\delta }^2/2)\sin (2\theta )\xi (\rho ),$$

$$\frac{d}{dt}\rho =2(\rho -{\delta }^2/2)\cos (2\theta ),\frac{d}{dt}\theta =-\sin (2\theta ).$$

$$\bar{\rho }=\frac{{\delta }^2}{2}+(\rho -\frac{{\delta }^2}{2})p_0(\theta ),\bar{\theta }=q_0(\theta ).$$

$$\bar{\rho }=\frac{{\delta }^2}{2}+(\rho -\frac{{\delta }^2}{2})(p_0(\theta )+p(\theta ,\rho )),\bar{\theta }=q_0(\theta )+q(\theta ,\rho ),$$

$$\bar{\rho }=\rho \cdot (p_0(\theta )+p(\theta ,\rho )),\bar{\theta }=q_0(\theta )+q(\theta ,\rho ).$$

$$\Vert {\partial }_{(x,y)}\rho \Vert \le \sqrt{2\rho },\Vert {\partial }_{(x,y)}\theta \Vert \le \frac{1}{\sqrt{2\rho }}.$$

$$\bar{x}=\sqrt{2\rho }\sqrt{p_0+p}\cos (q_0+q),\bar{y}=\sqrt{2\rho }\sqrt{p_0+p}\sin (q_0+q).$$

$$\bar{x}=\sqrt{2\rho }\sqrt{p_0}\cos (q_0),\bar{y}=\sqrt{2\rho }\sqrt{p_0}\sin (q_0)$$

$$\Vert {\partial }_{(x,y)}({\tilde{F}}_A-F_A)\Vert =\nu \Vert \frac{{\partial }_{(x,y)}\rho }{\sqrt{2\rho }}\varphi +\sqrt{2\rho }{\partial }_{(\rho ,\theta )}\varphi {\partial }_{(x,y)}(\rho ,\theta )\Vert =O(\nu ).$$

Full text

1. 1 Selected results and conjectures around the positive metric entropy conjecture
2. 2 Proof of the main Theorem
3. 3 Stochastic island
4. 4 Geometric model for a suitable stochastic island
5. 5 Restoration of broken heteroclinic links
6. 6 Rescaling Lemma

Introduction

Consider a diffeomorphism $f$ of a two-dimensional surface $\mathbb{M}$. The maximal Lyapunov exponent of $x\in\mathbb{M}$ is

[TABLE]

It quantifies the sensitivity to the initial conditions: if $\lambda(x)$ is positive, then the forward orbits of most of the points from a neighborhood of $x$ will diverge exponentially fast from the orbit of $x$.

Let $f$ preserve a smooth area form $\omega$ on $\mathbb{M}$. The metric entropy333We employ here Pesin formula for the Kolmogorov-Sinai entropy [Pes77]. of $f$ is the integral

[TABLE]

Whenever the metric entropy of a dynamical system is positive, points in $\mathbb{M}$ display a positive Lyapunov exponent with non-zero probability.

One of the most fundamental questions in conservative dynamics is

Question 0.1**.**

How typical are conservative dynamical systems with positive metric entropy?

Note that a different notion of topological entropy is one of the basic tools in describing chaotic dynamics: positive topological entropy indicates the presence of uncountably many orbits with a positive maximal Lyapunov exponent [Kat80]. However, the positivity of the metric entropy is a much stronger property, as it ensures positive maximal Lyapunov exponent for a non-negligible set of initial conditions. While numerical evidence for a large set of initial conditions corresponding to seemingly chaotic behavior in area-preserving maps is abundant, a rigorous proof for the positivity of metric entropy is available only for a small set of specially prepared examples (see Section 1). Currently no mathematical technique exists for answering Question 0.1 in full generality.

Several prominent conjectures are related to this question. In order to formulate them, let us recall the topologies involved. For $1\leq r\leq\infty$, let ${\rm{\rm Diff}}_{\omega}^{r}(\mathbb{M})$ be the space of diffeomorphisms which keep the area form $\omega$ invariant. When $r<\infty$, the space ${\rm{\rm Diff}}_{\omega}^{r}(\mathbb{M})$ is endowed with the uniform $C^{r}$-topology. The space ${\rm{\rm Diff}}_{\omega}^{\infty}(\mathbb{M})$ is endowed with the projective limit topology whose base is formed by all $C^{r}$-open subsets for all finite $r$. Let us fix a metric $d_{r}$ compatible with the $C^{r}$-topology (the space ${\rm{\rm Diff}}_{\omega}^{r}(\mathbb{M})$ with the metric $d_{r}$ is complete). The $C^{\infty}$-topology in ${\rm{\rm Diff}}_{\omega}^{\infty}(\mathbb{M})$ is defined by the following metric:

[TABLE]

(note that ${\rm{\rm Diff}}_{\omega}^{\infty}(\mathbb{M})$ with the metric $d_{\infty}$ is complete).

Consider the two-dimensional disc $\mathbb{D}:=\{(x,y)\in\mathbb{R}^{2}:x^{2}+y^{2}\leq 1\}$. Let $id$ be the identity map of $\mathbb{D}$. One of the favorite conjectures of Herman can be formulated as follows.

Conjecture 0.2** (Herman [Her98]).**

For every $\varepsilon>0$ there exists $f\in{\rm{\rm Diff}}^{\infty}_{\omega}(\mathbb{D})$ such that $d_{\infty}(f,id)<\varepsilon$ and the metric entropy of $f$ is positive: $h_{\omega}(f)>0$.

It is linked to his question:

Question 0.3** (Herman [Her98]).**

Is the set of diffeomorphisms $f$ with positive metric entropy $h_{\omega}(f)$ dense in ${\rm{\rm Diff}}^{\infty}_{\omega}(\mathbb{D})$?

In this work we prove Herman’s Conjecture 0.2. This also could be a step towards a positive answer to Question 0.3. Recall that a periodic point $P$ of $f$ is hyperbolic if the eigenvalues of $Df^{p}(P)$ (where $p$ is the period of $P$) are not equal to $1$ in the absolute value. The main result of this work is the following

Theorem A**.**

For any surface $(\mathbb{M},\omega)$, if a diffeomorphism $f\in{\rm{\rm Diff}}^{\infty}_{\omega}(\mathbb{M})$ has a periodic point which is not hyperbolic, then there is a $C^{\infty}$-small (as small as we want) perturbation of $f$ such that the perturbed map $\hat{f}\in{\rm{\rm Diff}}^{\infty}_{\omega}(\mathbb{M})$ has positive metric entropy: $h_{\omega}(\hat{f})>0$.

If $f=id$, then every point in $\mathbb{M}$ is a non-hyperbolic fixed point of $f$, so Theorem A implies Herman’s conjecture immediately. Another immediate consequence employs the notion of the weak stability [Mañ82]. The map $f\in{\rm{\rm Diff}}^{r}_{\omega}(\mathbb{M})$ is $C^{r}_{\omega}$-weakly stable if all the periodic points of any $C^{r}$-close to $f$ map $\hat{f}\in{\rm{\rm Diff}}^{r}_{\omega}(\mathbb{M})$ are hyperbolic.

Corollary B**.**

A diffeomorphism $f\in{\rm{\rm Diff}}^{\infty}_{\omega}(\mathbb{M})$ is either $C^{\infty}_{\omega}$-weakly stable or $C^{\infty}$-approximated by a diffeomorphism from ${\rm{\rm Diff}}^{\infty}_{\omega}(\mathbb{M})$ which has positive metric entropy.

This statement suggests that the answer to Question 0.3 has to be positive. The reason is that the common belief among dynamicists is that any $C^{\infty}$-weakly stable map of a closed manifold is uniformly hyperbolic (see Section 1.7). If this conjecture is true, then every $C^{\infty}_{\omega}$-weakly stable diffeomorphism has positive metric entropy (and, moreover, no $C^{\infty}_{\omega}$-weakly stable diffeomorphisms exist when $\mathbb{M}=\mathbb{D}$).

Another metric entropy conjecture regards the very popular Chirikov standard map family. This is a one-parameter family of area-preserving diffeomorphisms of $\mathbb{T}^{2}$ defined for $a\in\mathbb{R}$ by

[TABLE]

For $a\in(0,\frac{2}{\pi})$ this map has an elliptic fixed point at $(x=1/2,y=1/2)$ (a period-$p$ point $P$ of an area-preserving map $f$ is called elliptic if the eigenvalues of $Df^{p}(P)$ are equal to $e^{\pm i\alpha}$ where $\alpha\in(0,\pi)$, i.e. they are complex and lie on the unit circle). When $a$ increases, the elliptic fixed point loses stability and, at $a$ large enough, the numerically obtained phase portraits display a large set where the dynamics is apparently chaotic (the so-called “chaotic sea” [Chi79]). A conjecture due to Sinai can be formulated as follows (cf. [Sin94] P.144):

Conjecture 0.4**.**

There exists a set $\Lambda\subset\mathbb{R}$ of positive Lebesgue measure such that, for $a\in\Lambda$, the metric entropy of $T_{a}$ is positive.

This conjecture is still completely open despite intense efforts, see e.g. [GL00]. However, it is shown by Duarte [Dua94] that the map $T_{a}$ has elliptic periodic points for an open and dense set of sufficiently large values of parameter $a$. Hence, our main theorem implies the following “approximative version” of Sinai’s Conjecture 0.4:

Corollary C**.**

For every sufficiently large or sufficiently small $a\in\mathbb{R}$, there exists a $C^{\infty}$-small perturbation $\hat{T}\in{\rm{\rm Diff}}^{\infty}_{\omega}$ of the map $T_{a}$ such that $\hat{T}$ has positive metric entropy.

We note that a large set (of almost full Lebesgue measure) in a neighborhood of a generic elliptic point of an area-preserving $C^{r}$-diffeomorphism with $r\geq 4$ consists of points with zero Lyapunov exponent (the points on KAM-curves [Kol57]). Our result, nevertheless, shows that the Lebesgue measure of the points with positive maximal Lyapunov exponent in a neighborhood of any elliptic point can be positive too.

The proof of Theorem A occupies Sections 2-6 of this paper. In Section 1 we remind certain background information pertinent to this work.

This work was partially financed by the project BRNUH of USPC university, the Steklov institute, the Royal Society, and the ERC project of S. Van Strien. The second author was supported by the grant 14-41-00044 of the RSF. The first author thanks M.-C. Arnaud for informing him about this conjecture. The second author is grateful to L. M. Lerman who attracted his attention to Przytycki example. We are also grateful to M. Chaperon, S. Crovisier, F. Przytycki and E. Pujals for interesting comments.

1 Selected results and conjectures around the positive metric entropy conjecture

The study of the instability and chaos in conservative dynamics enjoys a long tradition since the seminal work by Poincaré [Poi87] on the tree-body problem.

1.1 Uniformly hyperbolic maps

An invariant compact set $K$ of the diffeomorphism $f$ of a manifold $\mathbb{M}$ is uniformly hyperbolic if the restriction $T\mathbb{M}|_{K}$ of the tangent bundle of $\mathbb{M}$ to $K$ splits into two $Df$-invariant continuous sub-bundles $E^{s}$ and $E^{u}$ such that $E^{s}$ is uniformly contracted and $E^{u}$ is uniformly expanded:

[TABLE]

Whenever $K=\mathbb{M}$, the diffeomorphism $f$ is called uniformly hyperbolic or Anosov. Note that the maximal Lyapunov exponent of a uniformly hyperbolic dynamical system is positive at every point. Hence, whenever the dynamics is conservative (i.e. $f$ keeps invariant a volume form $\omega$), the metric entropy is positive.

Such dynamics are very well understood. However, the existence of the splitting of $T\mathbb{M}$ into two non-trivial continuous sub-bundles imposes strong restrictions on the topology of $\mathbb{M}$. For instance, there exists no uniformly hyperbolic diffeomorphism of a closed two-dimensional surface different from the torus $\mathbb{T}^{2}$.

1.2 Stochastic island

The first example of a non-uniformly hyperbolic area-preserving map of the disk was given by Katok [Kat79]. His construction started with an Anosov diffeomorphism of $\mathbb{T}^{2}$ with 4 fixed points and a certain symmetry. Then the diffeomorphism is modified so that the fixed points become non-hyperbolic and sufficiently flat (while the hyperbolicity is preserved outside of the fixed points). After that, the torus is projected to a two-dimensional disc. The singularities of this projection correspond to the fixed points, and their flatness allows for making the resulting map of the disc a diffeomorphism. The positivity of the metric entropy is inherited from the original Anosov map.

This example has been pushed forward by Przytycki [Prz82], where instead of making the fixed points non-hyperbolic he made a surgery to replace each of these fixed points by an elliptic island (in this case - a neighborhood of an elliptic fixed point filled by closed invariant curves) bounded by a heteroclinic link (as defined below). Przytycki’s example was put in a more general context by Liverani [Liv04].

Recall that given a hyperbolic periodic point $P$ of a diffeomorphism $f$, the following sets are immersed smooth submanifolds: the stable and, respectively, unstable manifolds

[TABLE]

We will call a $C^{0}$-embedded circle $L$ a heteroclinic $N$-link if there exists $N$ hyperbolic periodic points $P_{1},\dots,P_{N}\in L$ satisfying $L\subset\cup_{i=1,\dots,N}W^{s}(P_{i})\cup W^{u}(P_{i})$. Note that a heteroclinic $N$-link is a piecewise $C^{\infty}$-curve with possible break points at the periodic points $P_{i}$.

Przytycki construction gives an example of the stochastic island in the followings sense.

Definition 1.1**.**

A stochastic island is a two-dimensional domain $\mathcal{I}$ bounded by finitely many heteroclinic links such that every point in $\mathcal{I}$ has positive maximal Lyapunov exponent.

In this paper (see Section 3) we build one more example of a map $f\in{\rm{\rm Diff}}^{\infty}_{\omega}(\mathbb{D})$ with a stochastic island (where $\omega$ is the standard area form $dx\wedge dy$ in the unit two-dimensional disc $\mathbb{D}$. To this aim, we adapt to the conservative setting the Aubin-Pujals blow-up construction [AP09].

One of the main difficulties in the Herman’s entropy conjecture is that no other examples of conservative maps of a disc with positive metric entropy are known. All of these constructions are very fragile (sensitive to perturbations): for example, no $C^{r}$-generic finite-parameter family of area-preserving diffeomorphisms can have a parameter value for which a heteroclinic or homoclinic link exists; no entire diffeomorphism (including e.g. the standard map and any polynomial diffeomorphism) can have a heteroclinic or homoclinic link [Ush80]. Still, we prove our main theorem by showing that stochastic islands appear near any elliptic point of an area-preserving diffeomorphism after a $C^{\infty}$-small perturbation.

1.3 Strong regularity

With the aim to extend the available examples of the non-uniformly hyperbolic behavior, Yoccoz launched a program called Strong Regularity in his first lecture at Collège de France [BY14]. The objective was to give a geometric-combinatorial definition of the non-uniformly hyperbolic dynamics which would serve both the one-dimensional (strongly dissipative) case, like e.g. in Jakobson theorem [Jak81] and the 2-dimensional case (e.g. for the positive entropy conjecture). So far there are three examples of such dynamics: one-dimensional quadratic maps (e.g. implying Jakobson theorem) [Yoc95], a non-uniformly hyperbolic horseshoe of dimension close to 6/10 [PY09], and Hénon-like endomorphisms [Ber] (implying Benedicks-Carleson Theorem [BC91]).

1.4 Isotopy to identity and renormalization

It is well-known that any symplectic diffeomorphism $F:\mathbb{D}\to\mathbb{R}^{2}$ is isotopic to identity [Cha83], which implies that it can always be represented as a composition $F=f_{n}\circ\cdots\circ f_{1}$ of $n$ symplectic diffeomorphisms $f_{i}$, each of which is uniformly $O(1/n)$-close to the identity map. Thus, one could try to prove the Herman’s conjecture by using the Ruelle-Takens construction [RT71]: take as $F$ in this formula an appropriate area-preserving map with a stochastic island, then consider $n$-disjoint $\varepsilon$-disks $\sqcup_{i}D_{i}=\psi_{i}(\mathbb{D})$ inside $\mathbb{D}$ (where $\psi_{i}$ are uniform affine contractions) and take a perturbation $f$ of the identity map such that $f(D_{i})=D_{i+1}$ and $f|_{D_{i}}$ is smoothly conjugate to $f_{i}$, i.e., $f|_{D_{i}}=\psi_{i+1}\circ f_{i}\circ\psi_{i}^{-1}$ for $i=1,\dots,n-1$, and $f|_{D_{n}}=\psi_{1}\circ f_{n}\circ\psi_{n}^{-1}$. Then, $f^{n}|_{D_{1}}$ will be smoothly conjugate to $F$ and, hence, would have positive metric entropy. However, since the conjugacies $\psi_{i}^{-1}$ expand with a rate at least $\varepsilon^{-1}$, we observe that the $C^{r}$-norm of $f$ is then $\asymp 1/(n\epsilon^{r})$. As the $n$ discs $D_{i}$ are disjoint, it follows that $n\epsilon^{2}\leq 1$, so $f$ can, a priori, be $\asymp\epsilon^{2-r}$ far, in the $C^{r}$-norm, from the identity.

Therefore, this construction does not produce the result for $r\geq 2$, although one can create $C^{1}$-close to identity maps with positive metric entropy in this way (in fact, every area-preserving dynamics can be realized by iterations of $C^{1}$-close to identity maps exactly by this procedure ). In [NRT79], Newhouse-Ruelle-Takens pushed forward the argument to obtain the $C^{2}$-case for torus maps. Fayad [Fay] also proposed a trick to cover the $C^{2}$-case for disk maps, but his method does not work in the $C^{r}$-case if $r\geq 3$.

We bypass the problem by using symplectic polynomial approximations of [Tur03] instead of the isotopy. In this way, one $C^{r}$-approximates any symplectic diffeomorphism $F$ by the product $f_{n}\circ\cdots\circ f_{1}$ of symplectic diffeomorphisms of a very particular form (Henon-like maps). For these maps, the conjugating contractions $\psi_{i}$ can be made very non-uniform, allowing for an arbitrarily good approximation of every dynamics by iterations of $C^{r}$-close to identity maps for all $r$, see [Tur15]. The product $f_{n}\circ\cdots\circ f_{1}$ is only an approximation of $F$, and it is still not known if every area-preserving dynamics can be exactly realized by iterations of $C^{r}$-close to identity maps. The main technical novelty of this paper is to show that some maps with stochastic islands can.

1.5 Stochastic sea and elliptic islands

The main motivation for the positive metric entropy conjecture is the amazing complexity of dynamics of a typical area-preserving map. Let us stress that no conservative dynamics are understood with certainty, except for those which are semi-conjugate to a rotation [AK70] or to an Anosov map. The reason is that hyperbolic and non-hyperbolic elements are often inseparable.

Thus, it was discovered by Newhouse [New70, New79] that a uniformly-hyperbolic Cantor set can be wild, i.e., its stable and unstable manifolds can have tangencies, and these non-transverse intersections cannot all be removed by any $C^{2}$-small perturbation of the map. Moreover, Newhouse showed [New77] that a $C^{r}$-small perturbation of an area-preserving map with a wild set creates elliptic periodic orbits which accumulate to the wild hyperbolic set.

Newhouse theory was applied and further developed by Duarte. He showed in [Dua94] that for all $a$ large enough the “chaotic sea” observed in the standard map (3) contains a wild hyperbolic set $K$, and for a Baire generic subset of this interval of $a$ values the map has infinitely many generic elliptic periodic points, which accumulate on $K$. Recall that a generic elliptic point of period $k$ for a map $f$ is surrounded, in its arbitrarily small neighborhood, by uncountably many smooth circles (KAM-curves), invariant with respect to $f^{k}$. The map $f^{k}$ restricted to such curve is smoothly conjugate to an irrational rotation (so the Lyapunov exponent is zero). The set occupied by the KAM curves has positive Lebesgue measure, and their density tends to $1$ as the elliptic point is approached. An invariant curve bounds an invariant region that contains the elliptic point, such regions are called elliptic islands.

In [Dua99, Dua08], Duarte showed that small perturbations, within the class ${\rm{\rm Diff}}^{r}_{\omega}$, of any area-preserving surface diffeomorphism with a homoclinic tangency (the tangency of the stable and unstable manifolds of a saddle periodic orbit) lead to creation of a wild hyperbolic set and to infinitely many coexisting elliptic points (and elliptic islands). In turn, it was shown in [MR97, GT10] that near any elliptic point a homoclinic tangency to some saddle periodic orbit can be created by a $C^{r}$-small perturbation.

Altogether, this gives a quite complicated picture of generic conservative dynamics: within the stochastic sea there are elliptic islands, inside elliptic islands there are small stochastic seas, etc. By [GST07, GT10], it is impossible to describe such dynamics in full detail. In fact, even most general features are presently not clear: for instance, we have no idea if the observed stochastic sea represents a transitive invariant set, or if it has a positive Lebesgue measure (though, by Gorodetski [Gor12], it may contain uniformly-hyperbolic subsets of Hausdorff dimension arbitrarily close to 2).

The inherent inseparability of the hyperbolic and elliptic behavior even suggests the following provocative question, communicated to us by Fayad.

Question 1.2**.**

Does an open set of area-preserving $C^{\infty}$-diffeomorphisms exist with the following property: for each diffeomorphism belonging to this subset the complement to the union of the KAM curves has zero Lebesgue measure?

Maps with this property have zero metric entropy, so our Theorem A implies the negative answer to this question (a KAM curve is always a limit of non-hyperbolic periodic points). However, it is still possible that a generic (i.e., belonging to a countable intersection of open and dense subsets) non-hyperbolic map from ${\rm Diff}^{\infty}_{\omega}$ has zero metric entropy.

In [AB16], the complement $U$ to the set of all essential invariant curves of a symplectic twist map was considered. Any connected component of $U$ (the Birkhoff “instability zone”) is bounded by two invariant topological circles $C_{1}$ and $C_{2}$. It is shown in [AB16], that either $C_{i}$ is a heteroclinic link (a scenario as much improbable as exhibiting a stochastic island, e.g., it is impossible for entire maps), or the Lyapunov exponent of any invariant measure supported by $C_{i}$ is zero ($i=1,2$). By the results of Furman [Fur97], the latter alternative implies that the convergence444 By the works of Birkhoff, Mather, and Le Calvez [Bir22, Mat91, LC87], there exists an orbit whose $\alpha$-limit set is in $C_{1}$ and the $\omega$-limit set is in $C_{2}$. of any orbit in the instability zone to one of these curves $C_{i}$ is at most sub-exponential. Thus, it is hard to see how a transitive invariant set with strictly positive maximal Lyapunov exponents can have $C_{i}$ in its closure, if $C_{i}$ is not a heteroclinic link.

The twist property is fulfilled near a generic elliptic point, hence the results of [AB16] hold true there. Therefore, it seems probable that if the Sinai conjecture is correct, then the positive metric entropy is achieved by sets distant from KAM curves. This seems to be consistent with the numerical observations [Mei94].

1.6 Stochastic perturbation of the standard map

In higher dimension, more possibilities exist for creating examples with positive metric entropy. Thus, it was shown in [BC14] that, for large values of the parameter $a$, a skew product of the standard map over an Anosov map is non-uniformly hyperbolic and displays non-zero Lyapunov exponents for Lebesgue almost every point. Recently, Blumenthal-Xue-Young [BXY17] used a similar argument for random perturbations of the standard map with large $a$ and also showed the positivity of metric entropy.

1.7 Genericity results

A recent breakthrough by Irie and Asaoka [AI16] showed, from a cohomological argument, that for any closed surface $(\mathbb{M},\omega)$ a generic map from ${\rm{\rm Diff}}^{\infty}_{\omega}(\mathbb{M})$ has a dense set of periodic points. Hence, if such map is weakly-stable, then it has a dense set of hyperbolic periodic points. A natural conjecture is, then, the structural stability of weakly-stable maps from ${\rm{\rm Diff}}^{\infty}_{\omega}(\mathbb{M})$; this would be a counterpart of the Lambda lemma from holomorphic dynamics [MSS83, Lyu84, DL, BD17].

Another natural conjecture would be that the weakly-stable maps from ${\rm{\rm Diff}}^{r}_{\omega}(\mathbb{M})$ are uniformly hyperbolic, $1\leq r\leq\infty$. For $r=1$ this result have been proven by Newhouse [New77]. For any $r\geq 2$ this question is open, as well as its dissipative counterpart – a conjecture by Mañé [Mañ82]. Since uniformly hyperbolic maps from ${\rm{\rm Diff}}^{r}_{\omega}(\mathbb{M})$ have positive metric entropy, this conjecture and our Theorem A would imply that maps with positive metric entropy are dense in ${\rm{\rm Diff}}^{\infty}_{\omega}(\mathbb{M})$.

Because of the meagerness of the heteroclinic links, the genericity of positive metric entropy does not follow from our result. In fact, one can conjecture that a $C^{r}_{\omega}$-generic surface diffeomorphism is either uniformly hyperbolic, or of zero entropy. We do not have an opinion in this regard. In the $C^{1}$-topology, this statement was a conjecture by Mañé, now proven by Bochi [Boc02]; in higher regularity it is completely open.

A milder version of this problem can be formulated as the following question due to Herman [Her98]:

Question 1.3**.**

Given a surface $(\mathbb{M},\omega)$, is there an open subset of ${\rm Diff}^{r}_{\omega}(\mathbb{M})$ where maps with zero metric entropy are dense?

A candidate for such dense set could be a hypothetical set of maps from Question 1.2 (the maps for which the union of all KAM curves would have full Lebesgue measure). Note that by the upper semi-continuity of the maximal Lyapunov exponent, a positive answer to Question 1.3 would also imply the local genericity of maps with zero metric entropy.

1.8 Universal dynamics

In [Tur03], the richness of chaotic dynamics in area-preserving maps was characterized by the concept of a universal map. Given a $C^{r}_{\omega}$-diffeomorphism $f$ ($r=1,\dots,\infty$) of a two-dimensional surface $(\mathbb{M},\omega)$, its behavior on ever smaller spatial scales can be described by its renormalized iterations defined as follows. Let $Q$ be a $C^{r}$-diffeomorphism into $\mathbb{M}$ from some disc in $\mathbb{R}^{2}$. Assume that the domain of definition of $Q$ contains the unit disc $\mathbb{D}$ and the domain of $Q^{-1}$ in $\mathbb{M}$ contains $f^{n}(Q(\mathbb{D}))$ for some $n\geq 0$. We also assume that the Jacobian $\text{det}\;DQ$ is constant in the chart $(x,y)$ on $\mathbb{M}$ where the area-form $\omega$ is standard: $\omega=dx\wedge dy$.

Definition 1.4**.**

The map $\mathbb{D}\to\mathbb{R}^{2}$ defined as

[TABLE]

is a renormalized iteration of $f$.

Note that since the Jacobian of $Q$ is constant, all renormalized iterations of $f$ preserve the standard area-form in $\mathbb{R}^{2}$.

Definition 1.5** (Universal map).**

A diffeomorphism $f\in{\rm{\rm Diff}}^{\infty}_{\omega}(\mathbb{M})$ is universal if the set of its renormalized iterations is $C^{\infty}$-dense among all orientation-preserving, area-preserving diffeomorphisms $\mathbb{D}\to\mathbb{R}^{2}$.

By this definition, the dynamics of a single universal map approximate, with arbitrarily good precision, all symplectic maps of the unit disc.

In the general non-conservative context this notion was used in [Tur10, Tur15]. In $C^{1}$ category, the concept of universal dynamics was independently proposed by Bonatti and Diaz [BD02].

The universal dynamics might sound difficult to materialize, but it is not. It is shown in [GST07] that an arbitrarily small, in $C^{\infty}_{\omega}$, perturbation of any area-preserving map with a homoclinic tangency can create universal dynamics. Moreover, universal maps form a Baire generic subset of the Newhouse domain - the open set in ${\rm Diff}^{\infty}_{\omega}(\mathbb{M})$ comprised of maps with wild hyperbolic sets. In [GT10], it was shown that a $C^{\infty}_{\omega}$-generic diffeomorphism of $\mathbb{M}$ with an elliptic point is universal555The results in [GST07, GT10] were also proven in the space of real-analytic area-preserving maps..

Consequently, any diffeomorphism $f\in{\rm Diff}^{\infty}_{\omega}(\mathbb{M})$ with an elliptic point can be perturbed in such a way that its iterations would approximate any given area-preserving dynamics and, in particular, the dynamics with positive metric entropy. We stress that this observation is not sufficient for a proof of our Theorem A. Indeed, if $f$ is a $C^{\infty}_{\omega}$-diffeomorphism with an elliptic point and $g$ is a $C^{\infty}_{\omega}$-diffeomorphism of $\mathbb{D}$ with positive metric entropy, the only thing we can conclude from [GST07, GT10] is that arbitrarily close to $f$ in ${\rm Diff}^{\infty}_{\omega}(\mathbb{M})$ there exists a diffeomorphism whose iteration restricted to a certain disc is smoothly conjugate to a map $G$ which is as close as we want to $g$ in ${\rm Diff}^{\infty}_{\omega}(\mathbb{D})$. However, this map $G$ does not need to inherit the positive metric entropy from $g$. Overcoming this problem is the main technical point of this paper.

2 Proof of the main Theorem

We start with constructing a map with a stochastic island with certain additional properties. In section 3, we give a precise description of the construction similar to those in [Kat79, AP09, Prz82], which produces a $C^{\infty}_{\omega}$-diffeomorphism $\check{F}:\mathbb{D}\to\mathbb{D}$ with a stochastic island $\mathcal{I}$ bounded by four heteroclinic bi-links $\{\check{L}_{i}^{a}\cup\check{L}^{b}_{i}:0\leq i\leq 3\}$. Each $\check{L}_{i}^{a}\cup\check{L}_{i}^{b}$ is a $C^{\infty}$-embedded circle included in the stable and unstable manifolds of hyperbolic fixed points $\check{P}_{i},\check{Q}_{i}$:

[TABLE]

The island of $\check{F}$ is depicted in Fig. 6. For every $F$ which is $C^{1}$-close to $\check{F}$, for every $0\leq i\leq 3$, the hyperbolic continuations of $\check{P}_{i}$ and $\check{Q}_{i}$ are the uniquely defined hyperbolic $F$-periodic orbits close to $\check{P}_{i}$ and $\check{Q}_{i}$.

We also show the following

Proposition 2.1**.**

For every conservative map $F$ which is $C^{2}$-close to $\check{F}$, let $P_{i}$ and $Q_{i}$ be hyperbolic continuations of $\check{P}_{i},\check{Q}_{i}$. If $\{W^{u}(P_{i};F)\cup W^{s}(Q_{i};F):0\leq i\leq 3\}$ define four heteroclinic bi-links $\{L_{i}^{a}\cup L^{b}_{i}:0\leq i\leq 3\}$ which are $C^{2}$-close to $\{\check{L}_{i}^{a}\cup\check{L}^{b}_{i}:0\leq i\leq 3\}$, then they bound a stochastic island. In particular, the metric entropy of $F$ is positive.

The proof is given by Corollary E of Theorem D in Section 3. It follows from a new and short argument which implies also some results of [AP09].

We will call a stochastic island robust relative link preservation if it satisfies this property: for every $C^{r}$-small perturbation of the map, if the stable and unstable manifolds that form the heteroclinic link do not split, then they bound a stochastic island. Proposition 2.1 shows that the stochastic island $\check{I}$ of the map $\check{F}$ satisfies this robustness property (with any $r\geq 2$), and the same holds true for the islands bounded by the four heteroclinic bi-links $\{L_{i}^{a}\cup L^{b}_{i}:0\leq i\leq 3\}$ (if these links exist) of any map $C^{2}$-close to $\check{F}$.

In Section 4 (see Proposition 4.7), we construct a coordinate transformation $\mathring{\phi}\in{\rm Diff}^{\infty}_{\omega}(\mathbb{R}^{2})$ such that $\mathring{\mathcal{I}}:=\mathring{\phi}(\mathcal{I})$ is a “suitable” island for $\mathring{F}:=\mathring{\phi}\circ\check{F}\circ\mathring{\phi}^{-1}$. The suitability conditions are described in Definition 4.6. They include the requirement that certain segments of the stable and unstable manifolds of the hyperbolic fixed points $\mathring{P}_{i}:=\mathring{\phi}(P_{i})$ and $\mathring{Q}_{i}=\mathring{\phi}(Q_{i})$, $0\leq i\leq 3$, are strictly horizontal, i.e., they lie in the lines $y=const$ where $(x,y)$ are coordinates in $\mathbb{R}^{2}$. Moreover, the map $\mathring{F}$ near these segments has a particular form, which allows us to establish the following result in Section 5 (this is the central point of our construction):

Proposition 2.2**.**

Given any finite $r\geq 2$, for every $F\in{\rm Diff}^{r+8}_{\omega}(\mathbb{D})$ which is $C^{r+8}$-close to $\mathring{F}$, there exists a $C^{r}$-small function $\psi:\mathbb{R}\to\mathbb{R}$ such that the map $\bar{F}$ defined as

[TABLE]

has the following property: For the hyperbolic continuations $P_{i}$ and $Q_{i}$ of the fixed points $\mathring{P}_{i}$ and, respectively, $\mathring{Q}_{i}$, the union $W^{s}(P_{i};\bar{F})\cup W^{u}(Q_{i};\bar{F})$ defines a heteroclinic bi-link $L_{i}^{a}\cup L_{i}^{b}$ which is $C^{r}$-close to $\mathring{L}_{i}^{a}\cup\mathring{L}_{i}^{b}$, for each $i=0,\dots,3$.

Remark 2.3*.*

We notice that by Proposition 2.1, the map $\bar{F}=S_{\psi}\circ F$ has a stochastic island, robust relative link preservation, and its metric entropy is positive.

With this information, we can now complete the proof of the main theorem.

Proof of Theorem A. Let $f\in{\rm Diff}^{\infty}_{\omega}(\mathbb{M})$ have a non-hyperbolic periodic point. By an arbitrarily small perturbation of $f$ one can make this point elliptic. Then, by [GT10], by a $C^{\infty}_{\omega}$-small perturbation of $f$, one can create, in a neighborhood of the elliptic point, a hyperbolic periodic cycle whose stable and unstable manifolds coincide (those define two heteroclinic links), see Fig. 2.

The important thing here is that by a small perturbation of the original map $f$, we create a periodic point with a flat homoclinic tangency. After that, we apply the following result, proven in Section 6 (see Corollary F).

Proposition 2.4**.**

*Let $f\in{\rm Diff}^{\infty}_{\omega}(\mathbb{M})$ have a hyperbolic periodic point with a flat homoclinic tangency. Then, there exists a $C^{\infty}$-dense subset $\cal F$ of ${\rm Diff}^{\infty}_{\omega}(\mathbb{D},\mathbb{R}^{2})$ such that for every $F\in{\cal F}$, every $r\geq 2$, every $C^{r}$-smooth function $\psi:\mathbb{R}\to\mathbb{R}$ whose $C^{r}$-norm is bounded by $1$, and every $\varepsilon>0$, there exists a diffeomorphism $\hat{f}\in{\rm Diff}^{r}_{\omega}(\mathbb{M})$ such that

$\bullet$ $d_{r}(f,\hat{f})<\varepsilon$, where $d_{r}$ is the $C^{r}$-distance,

$\bullet$ the composition $S_{\psi}\circ F$ is equal to a renormalized iteration of $\hat{f}$, where $S_{\psi}(x,y)=(x,y+\psi(x))$.*

This statement is an enhanced version of the “rescaling lemma” (Lemma 6) of [GST07]666While in [GST07] the rescalling was done for a single round near the homoclinic tangency, here we do many rounds, similar to [Tur15].. This proposition gives us much freedom in varying the renormalized iteration of $\hat{f}$ without perturbing $F$ (by composing with $S_{\psi}$ for an arbitrarily functional parameter $\psi$).

The renormalized iterations are described by Definition 1.4. Since a renormalized iteration is $C^{r}$-conjugate to an actual iteration of the map $\hat{f}$ restricted to some small disc, it follows that by taking $F$ and $\psi$ exactly as in Proposition 2.2, (so that $S_{\psi}\circ F$ will have a stochastic island, see Remark 2.3), we will obtain that the map $\hat{f}$ has a stochastic island too, robust relative link preservation.

The stochastic island for the map $\hat{F}:=S_{\psi}\circ F$ is bounded by $4$-bi-links, each of which is equal to a $C^{r}$-embedded circle. As $S_{\psi}\circ F$ is smoothly conjugate to $\hat{f}^{n}$ for some $n>0$, it follows that the stochastic island for the map $\hat{f}$ is bounded $m=4n$ heteroclinic bilinks and is robust relative link preservation. We denote them by $(L_{i}^{a}\cup L_{i}^{b})_{i=1}^{m}$. We prove the following result in Section 5.5:

Proposition 2.5**.**

Given any map $\hat{f}\in{\rm Diff}^{r}_{\omega}(\mathbb{M})$ with a stochatics island $\mathcal{I}$ bounded by bilinks $(L^{a}_{i}\cup L_{i}^{b})_{i=1}^{m}$ such that each bilink $L^{a}_{i}\cup L_{i}^{b}$ is a $C^{r}$-embedded circle, arbitrarily close in $C^{r}$ to $\hat{f}$ there exists a map $\hat{f}_{\infty}\in{\rm Diff}^{\infty}_{\omega}(\mathbb{M})$ for which the bilinks persist (i.e., the hyperbolic continuations of the stable and unstable manifolds forming each bilink $(L_{i}^{a},L_{i}^{b})$ comprise a heteroclinic bilink for the map $\hat{f}_{\infty}$, $C^{r}$-close to $(L_{i}^{a},L_{i}^{b})$ ).

The map $\hat{f}$ with the stochastic island lies in the $\varepsilon$-neighborhood of the original map $f$ in ${\rm Diff}^{r}_{\omega}(\mathbb{M})$. Since the stochastic island of the map $\hat{f}$ is robust relative link preservation, the map $\hat{f}_{\infty}$ also has a stochastic island and, hence, positive metric entropy.

This shows, that arbitrarily close, in $C^{r}$ for any given $r$, to the original map $f$ there exists a map $\hat{f}_{\infty}\in{\rm Diff}^{\infty}_{\omega}(\mathbb{M})$ with positive metric entropy. ∎

3 Stochastic island

In this Section we describe a particular example of a stochastic island. It is somewhat similar to the Przytycki’s development of the Katok’s construction, in the sense that the holes in the island are bounded by heteroclinic links. A difference with the Przytycki’s example is that the heteroclinic links form smooth circles in our construction. Similar examples were considered by Aubin-Pujals [AP09] in the non-conservative case.

3.1 An Anosov map of the torus

Let $\mathbb{T}^{2}$ be the torus $\mathbb{R}^{2}/\mathbb{Z}^{2}$. Let $\mathbb{S}^{1}$ be the circle $\mathbb{R}/(2\pi\mathbb{Z})$. We endow $\mathbb{R}^{2}$ and $\mathbb{T}^{2}$ with the symplectic form $\omega=dx\wedge dy$.

Consider the following linear Anosov diffeomorphism of $\mathbb{T}^{2}$:

[TABLE]

($F_{A}$ is the third iteration of the standard Anosov example $(x,y)\mapsto(2x+y,x+y)$). The map $F_{A}$ preserves the area form $\omega$ and is uniformly hyperbolic, e.g. its Lyapunov exponents are non-zero.

The map $F_{A}$ has four different fixed points $\Omega_{0}=(0,0)$, $\Omega_{1}=(\frac{1}{2},\frac{1}{2})$, $\Omega_{2}=(\frac{1}{2},-\frac{1}{2})$, $\Omega_{3}=(-\frac{1}{2},\frac{1}{2})$. Let $\sigma>0$ be the logarithm of the unstable eigenvalue of the matrix $A$: $\sigma=\ln(9+4\sqrt{5})$. We may put the origin of coordinates to one of the points $\Omega_{i}$ and make a symplectic linear transformation of the coordinates $(x,y)$ such that the map $F_{A}$ near $\Omega_{i}$ will become

[TABLE]

Observe that $F_{A}$ is the time-$\sigma$ map by the flow of the system

[TABLE]

associated with the Hamiltonian

[TABLE]

Note that the transition to polar coordinates $(\rho,\theta)$ near $\Omega_{i}$ by the rule

[TABLE]

preserves the symplectic form, i.e., we have $\omega=d\rho\wedge d\theta$. The map $F_{A}$ in these coordinates is the time-$\sigma$ map by the flow defined by the Hamiltonian function

[TABLE]

The corresponding Hamiltonian vector field is

[TABLE]

We do not need an explicit expression for the map $F_{A}$ in the symplectic polar coordinates; just note that near the point $\Omega_{i}$ this map has the form $(\rho,\theta)\mapsto(\bar{\rho},\bar{\theta})$ where

[TABLE]

and $p_{0}$ and $q_{0}$ are $C^{\infty}$-functions $\mathbb{S}^{1}\to\mathbb{R}$. Since this map preserves the symplectic form $d\rho\wedge d\theta$, its Jacobian $p_{0}(\theta)\partial q_{0}(\theta)$ equals to $1$, so $p_{0}(\theta)\neq 0$ and $\partial q_{0}(\theta)\neq 0$.

3.2 Stochastic island in $\mathbb{T}^{2}$

In order to construct a chaotic island, we shall “blow up” the four points $\Omega_{i}$. This means that we will take some small $\delta>0$, consider the closed $\delta$-discs $V_{i}$ with the center at $\Omega_{i}$ for each $i=0,1,2,3$, and build a $C^{\infty}$-diffeomorphism $\Psi$ of $\mathbb{T}^{2}\setminus\{\sqcup_{0\leq i\leq 3}V_{i}\}$ onto $\mathbb{T}^{2}\setminus\{\cup_{i}\Omega_{i}\}$. We will do it in such a way that the map $\Psi^{-1}\circ F_{A}\circ\Psi$ will be smoothly extendable to a $C^{\infty}$-diffeomorphism $\hat{F}$ of $\mathbb{T}^{2}$. Then the invariant set $\hat{I}=\mathbb{T}^{2}\setminus\{\sqcup_{i}V_{i}\}$ will be a stochastic island for $\hat{F}$. Moreover, the points $\Omega_{i}$ will be flat fixed points of $\hat{F}$ and, importantly, $\hat{F}$ will inherit the symmetry with respect to $(-id)$ from the map $F_{A}$ – this will be used at the next step in Section 3.4.

Let $\varepsilon>0$ be such that the map $F_{A}$ is the time-$\sigma$ map of the Hamiltonian flow defined by (6) in the closed $\varepsilon$-disc $V_{i}^{\prime}$ about $\Omega_{i}$, $i=0,1,2,3$; we assume that the discs $V_{i}^{\prime}$ are mutually disjoint. Let $0<\delta<\varepsilon$ and let $V_{i}\Subset V_{i}^{\prime}$ be the closed $\delta$-discs about $\Omega_{i}$, $i=0,1,2,3$.

Let $\displaystyle\psi:\left[\frac{\delta^{2}}{2},\frac{\varepsilon^{2}}{2}\right]\to\left[0,\frac{\varepsilon^{2}}{2}\right]$ be a $C^{\infty}$-diffeomorphism such that

[TABLE]

Let $\Psi_{i}\in C^{\infty}(\mathbb{T}^{2}\setminus int(V_{i}),\mathbb{T}^{2})$ be equal to the identity outside $V_{i}^{\prime}$ and let the restriction of $\Psi_{i}$ to the smaller disc $V_{i}$ be given by

[TABLE]

in the polar coordinates (5). The radius-$\delta$ circle $\partial V_{i}$ about $\Omega_{i}$ is sent by $\Psi_{i}$ to $\Omega_{i}$. Note that $\Psi_{i}$ is a diffeomorphism from $\mathbb{T}^{2}\setminus V_{i}$ onto $\mathbb{T}^{2}\setminus\Omega_{i}$ and $\Psi_{i}$ commutes with $(-id)$. Note also that in a neighborhood of $\partial V_{i}$ the map $\Psi_{i}$ preserves the form $\omega=d\rho\wedge d\theta$.

Define

[TABLE]

This map is a $C^{\infty}$-diffeomorphism from $\hat{I}=\mathbb{T}^{2}\setminus\{\sqcup_{i}V_{i}\}$ onto $\mathbb{T}^{2}\setminus\{\cup_{i}\Omega_{i}\}$ and it commutes with $(-id)$.

Denote $\hat{F}=\Psi^{-1}\circ F_{A}\circ\Psi$. By construction, this is a $C^{\infty}$-diffeomorphism of $\hat{I}$, which commutes with $(-id)$. In a small neighborhood of the circles $\partial V_{i}$ the map $\hat{F}$ preserves symplectic form $d\rho\wedge d\theta$ and, by (6),(8),(9), it coincides in this neighborhood with the time-$\sigma$ map of the flow defined by the symplectic form $d\rho\wedge d\theta$ and the Hamiltonian

[TABLE]

We can, therefore, smoothly extend $\hat{F}$ inside $V_{i}$ (i.e., to $\rho\leq\delta^{2}/2$) as the time-$\sigma$ map of the flow defined by the smoothly extended Hamiltonian $\hat{H}_{i}$:

[TABLE]

where $\xi$ is a $C^{\infty}$-function, equal to zero at all $\rho$ close to zero and equal to $1$ at all $\rho\geq\delta^{2}/2$.

We summarize some relevant properties of the map $\hat{F}$ in the following

Proposition 3.1**.**

The map $\hat{F}$ is a $C^{\infty}$-diffeomorphism of $\mathbb{T}^{2}$ such that:

$\hat{F}|_{\mathbb{T}^{2}\setminus\{\sqcup_{i}V_{i}\}}$* is conjugate to $F_{A}|_{\mathbb{T}^{2}\setminus\{\cup_{i}\Omega_{i}\}}$ via a $C^{\infty}$-diffeomorphism $\Psi$;* 2. 2.

the set $\hat{I}:=\mathbb{T}^{2}\setminus\{\sqcup_{i}V_{i}\}$ is invariant with respect to $\hat{F}$; 3. 3.

$\hat{F}$* preserves a smooth symplectic form $\hat{\omega}$;* 4. 4.

$\hat{F}$* commutes with $(-id)$, and $\hat{\omega}$ is invariant with respect to $(-id)$;* 5. 5.

$\hat{F}$* equals to the identity in a small neighborhood of the points $\Omega_{i}$;* 6. 6.

each circle $\partial V_{i}$ is a heteroclinic $4$-link.

Proof.

Claim 1 is given just by construction of $\hat{F}$. Claim 2 follows from it by continuity of $\hat{F}$: since $\Omega_{i}$ are fixed points of $F_{A}$, each disc $V_{i}$ is invariant with respect to $\hat{F}$. Claim 5 follows since $\hat{H}_{i}$ is constant near $\Omega_{i}$, so the corresponding vector field is identically zero there.

Claim 3: By claim 1, the map $\hat{F}$ preserves the symplectic form $\hat{\omega}=\Psi^{*}\omega$ in $\mathbb{T}^{2}\setminus\{\sqcup_{i}V_{i}\}$. Since $\Psi^{*}\omega=d\rho\wedge d\theta=\omega$ near $\partial V_{i}$ (as it follows from (8),(9)), we can smoothly extend $\hat{\omega}$ onto the whole torus by putting $\hat{\omega}=\omega$ in $\sqcup_{i}V_{i}$. Since $\hat{F}|_{V_{i}}$ is the time-$\sigma$ map by a Hamiltonian flow, the form $\hat{\omega}=\omega$ inside the discs $V_{i}$ is preserved by $\hat{F}$.

Claim 4 follows since both the original map $F_{A}$ and the conjugacy $\Psi$ commute with $(-id)$, the Hamiltonians $\hat{H}_{i}$ that defines $\hat{F}$ inside the discs $V_{i}$ (see (11)) are invariant with respect to $(-id):\theta\mapsto\theta+\pi$, and the symplectic form $\omega$ is invariant with respect to $(-id)$.

In order to prove claim 6, notice that by (11) the map $\hat{F}$ near $\partial V_{i}:\{\rho=\delta^{2}/2\}$ is the time-$\sigma$ map of the system

[TABLE]

This system has 4 saddle equilibria on the circle $\rho=\delta^{2}/2$: $\theta=0,\pi/2,\pi,3\pi/2$. These equilibria are hyperbolic fixed points of $\hat{F}$, and the invariant arcs of the circle $\rho=\delta^{2}/2$ between these points are formed by their stable or unstable manifolds. ∎

3.3 The island is robust relative link preservation

The following statement establishes that the set $\hat{I}$ is a stochastic island for $\hat{F}$. It also concerns the dynamics of perturbations of $\hat{F}$. Similar results were obtained by Aubin-Pujals in [AP09] for the non-conservative case. The proof is given by a new and shorter argument.

Theorem D**.**

For the map $\hat{F}$, as well as for every, not necessarily conservative, diffeomorphism $\tilde{F}$ which is $C^{2}$-close to $\hat{F}$ and keeps the circles $\partial V_{i}$ invariant ($i=0,1,2,3$), all points in $\hat{I}$ have positive maximal Lyapunov exponent. The maps $\tilde{F}$ and $\hat{F}$ are topologically conjugate on $\hat{I}$; all such maps are transitive.

Proof.

By (7),(8),(9), the map $\hat{F}$ near $\partial V_{i}$ can be written as $(\rho,\theta)\mapsto(\bar{\rho},\bar{\theta})$ where

[TABLE]

A $C^{2}$-small perturbation $\tilde{F}$ of $\hat{F}$ which keeps the circle $\partial V_{i}$ invariant must send $\rho=\frac{\delta^{2}}{2}$ to $\bar{\rho}=\frac{\delta^{2}}{2}$, so it has the form

[TABLE]

where the function $p$ is $C^{1}$-small and $q$ is $C^{2}$-small.

By reversing our surgery (10), we obtain a diffeomorphism $\tilde{F}_{A}=\Psi^{-1}\circ\tilde{F}\circ\Psi$ of $I_{A}:=\mathbb{T}^{2}\setminus\cup_{i}\{\Omega_{i}\}$, which takes the following form near $\Omega_{i}$ (see (8),(9)):

[TABLE]

The following simple and new Lemma enables us to compare $\tilde{F}_{A}$ with $F_{A}$ defined in (4).

Lemma 3.2**.**

In the Cartesian coordinates, the restrictions $\tilde{F}_{A}$ and $F_{A}$ to $I_{A}=\mathbb{T}^{2}\setminus\cup_{i}\{\Omega_{i}\}$ are uniformly $C^{1}$-close.

Proof.

The transformation $(\rho,\theta)\mapsto(x,y)=(\sqrt{2\rho}\cos\theta,\sqrt{2\rho}\sin\theta)$ to Cartesian coordinates near $\Omega_{i}$ has the following property

[TABLE]

Thus, the map $\tilde{F}_{A}$ near $\Omega_{i}$ takes the form $(x,y)\mapsto(\bar{x},\bar{y})$ where

[TABLE]

The uniformly-hyperbolic map $F_{A}$ is given by

[TABLE]

(see (7)). Recall that $p_{0}(\theta)\neq 0$ for all $\theta$.

It follows that, $\tilde{F}_{A}(x,y)=F_{A}(x,y)+\nu\sqrt{2\rho}\;\phi(\rho,\theta)$ near $\Omega_{i}$ where $\nu=\|(p,q)\|_{C^{1}}\sim\|\tilde{F}-\hat{F}\|_{C^{2}}$ is small and $\phi$ is uniformly bounded along with its first derivatives with respect to $\rho$ and $\theta$. By (13), this gives us that near the points $\Omega_{i}$

[TABLE]

∎

This lemma implies the following:

Lemma 3.3**.**

For the map $\tilde{F}$, every $z\in\hat{I}$ displays a positive Lyapunov exponent.

Proof.

We have found that the map $\tilde{F}_{A}:I_{A}\to I_{A}$ is uniformly close in $C^{1}$ to the uniformly-hyperbolic map $F_{A}$ in a neighborhood of the fixed points $\Omega_{i}$ (even though the derivative of $\tilde{F}_{A}$ may be not defined at the points $\Omega_{i}$). Since the 4 fixed points $\Omega_{i}$ are the only singularities of the surgery transformation $\Psi$, the derivative of $\tilde{F}_{A}$ is uniformly close to the derivative of $F_{A}$ everywhere on $I_{A}$, i.e., $D\tilde{F}_{A}$ is uniformly close to (x,y)\mapsto A(x,y)=\left(\begin{array}[]{cc}13&8\\ 8&5\end{array}\right)(x,y) (see (4)). In particular, $D\tilde{F}_{A}$ takes every vector with positive coordinates to a vector with positive coordinates and at least 4 times larger norm. Hence,

[TABLE]

for every point $P\in I_{A}$.

The map $\tilde{F}:\hat{I}\to\hat{I}$ is smoothly conjugate to $\tilde{F}_{A}=\Psi^{-1}\circ\tilde{F}\circ\Psi$, so

[TABLE]

for every $P\in I_{A}$. If the point $P$ is chosen such that the iterations $\tilde{F}^{n}(\Psi(P))$ do not converge to $\cup_{i}\partial V_{i}$, then there is a sequence $n_{j}\to+\infty$ such that the iterations $\tilde{F}_{A}^{n_{j}}(P)$ stay away from the points $\Omega_{i}$ - the only singularities of the conjugacy map $\Psi$. Thus, both $\|D\Psi^{-1}(\tilde{F}_{A}^{n_{j}}(P)\|$ and $\|D\Psi(P)\|$ are bounded away from zero in this case. It follows from (14) that

[TABLE]

By definition, this means that the maximal Lyapunov exponent of $\Psi(P)$ is $\geq\ln 4$, for the map $\tilde{F}$.

In the remaining case, if all iterations of the point $\Psi(P)$ by $\tilde{F}$ converge to $\cup_{i}\partial V_{i}$, they must converge to one of the saddle fixed points that lie in $\cup_{i}\partial V_{i}$ (since every $\hat{F}$-pseudo-orbit which remains close to $\partial V_{i}$, is necessarily eventually close to one of the saddle point of $V_{i}$). In this case, the maximal Lyapunov exponent of $\Psi(P)$ equals to the maximal Lyapunov exponent of the saddle point, i.e., it is positive. Thus, in any case, every point of $\hat{I}$ has positive maximal Lyapunov exponent for the map $\tilde{F}$. ∎

By the uniform hyperbolicity of $F_{A}$ (see Lemma 3.2), using a variation of the Moser’s technique [Mos69] of the proof of Anosov structural stability theorem [Ano67], we prove:

Lemma 3.4**.**

There exists a homeomorphism $h$ of $\mathbb{T}^{2}$ which conjugates $F_{A}$ and $\tilde{F}_{A}$, and leaves each $\Omega_{i}$ invariant.

This lemma implies the topological conjugacy between $\tilde{F}|_{\hat{I}}$ and $\hat{F}|_{\hat{I}}$. The transitivity of $\tilde{F}|_{\hat{I}}$ follows from the transitivity of $F_{A}|_{\mathbb{T}^{2}\setminus\{\cup_{i}\Omega_{i}\}}$ by the conjugacy. This completes the proof of Theorem D. ∎

Proof of Lemma 3.4.

The map $F_{A}$ induces an automorphism on the Banach space $\Gamma$ of bounded continuous vector fields $\gamma$ vanishing at $\Omega_{i}$, $i=0,1,2,3$:

[TABLE]

The hyperbolicity of $F_{A}$ (it uniformly expands in the unstable direction and uniformly contracts in the stable direction) implies that the linear operator $id-F_{A}^{\sharp}$ has a bounded inverse777As $F_{A}$ is a linear map, it is easy to provide an explicit formula for $id-F_{A}^{\sharp}$: if $(id-F_{A}^{\sharp})\gamma=\beta$, then $\gamma=\sum_{n=0}^{\infty}e^{-n\sigma}\beta_{s}\circ F_{A}^{-n}-\sum_{n=1}^{\infty}e^{-n\sigma}\beta_{u}\circ F_{A}^{n}$, where $\beta_{s}$ and $\beta_{u}$ are the projections of $\beta$ to the stable and, respectively, unstable directions of $F_{A}$, and $e^{\pm\sigma}$ are the eigenvalues of $F_{A}$, $\sigma>0$.. By the implicit function theorem, this implies that the fixed point $\gamma=0$ of $F_{A}^{\sharp}$ is unique, and every (nonlinear) operator on $\Gamma$ which is $C^{1}$-close to $F_{A}^{\sharp}$ has a unique fixed point close to $\gamma=0$. In particular, the operator

[TABLE]

has a unique fixed point $\gamma$. By the construction, the map $h=id+\gamma$ satisfies:

[TABLE]

Thus, $\tilde{F}_{A}$ and $F_{A}$ are semi-conjugate, and we have that

[TABLE]

for every integer $n$, positive and negative.

In order to prove the topological conjugacy between $F_{A}$ and $\tilde{F}_{A}$, it remains to show that the continuous map $h$ is injective. This is done as follows: if $h(P)=h(Q)$, then $h(F_{A}^{n}P)=h(F_{A}^{n}Q)$ for all $n\in\mathbb{Z}$, by (15). Since $h$ is uniformly close to identity, it follows that $F_{A}^{n}P$ is uniformly close to $F_{A}^{n}Q$, i.e., $A^{n}(P-Q)$ is uniformly small for all $n\in\mathbb{Z}$. By the hyperbolicity of the matrix $A$, this gives $P=Q$, as required. ∎

3.4 Stochastic island in the disc

It is easy to see that the result of the factorization $\pi$ of the 4-punctured torus $\mathbb{T}^{2}\setminus\cup_{i=0}^{3}\{\Omega_{i}\}$ over $(-id):(x,y)\mapsto(-x,-y)$ is a 4-punctured sphere. One can realize the smooth map $\pi:\mathbb{T}^{2}\setminus\cup_{i}\{\Omega_{i}\}\to\mathbb{S}^{2}$ e.g. by a Weierstrass elliptic function888or, if we realize the 4-punctured sphere as the surface $\{Z^{2}+\eta(X,Y)=1,\;|X|\leq 1,\;|Y|\leq 1\}$ in $\mathbb{R}^{3}$, where $\eta=(X^{2}+Y^{2}+\sqrt{1-X^{2}-Y^{2}+X^{4}+Y^{4}-X^{2}Y^{2}})/2$, then $\pi$ can be explicitly defined as $\pi(x,y)=(X=2|x|-1,Y=2|y|-1,Z={\rm sign}(xy)\sqrt{1-\eta(X,Y)}),\quad|x|\leq 1/2,\;|y|\leq 1/2$.; see also [Kat79].

Each fiber of $\pi$ is a pair of points $(x,y)$ and $-(x,y)$. Since $\hat{F}$ commutes with $(-id)$ in our construction, and $\hat{F}$ is the identity map in a neighborhood of each of the points $\Omega_{i}$ ($i=0,1,2,3$) where $\pi$ is singular, the push-forward $\check{F}=\pi\circ\hat{F}\circ\pi^{-1}$ of $\hat{F}$ is a well-defined $C^{\infty}$-diffeomorphism of the sphere $\mathbb{S}^{2}$. As the symplectic form $\hat{\omega}$, which is preserved by $\hat{F}$, is invariant with respect to $(-id)$, it follows that the push-forward of $\hat{\omega}$ by $\pi$ is a smooth symplectic form $\check{\omega}$ on $\mathbb{S}^{2}\setminus\cup_{i}\{\Omega_{i}\}$, and $\check{\omega}$ is invariant by $\check{F}$. Note that the form $\check{\omega}$ can get singular at the points $\pi\Omega_{i}$, but we can smoothen $\check{\omega}$ in an arbitrary way near these points; since $\check{F}$ is the identity map there, it preserves any smooth area form near $\pi\Omega_{i}$. Hence $\check{F}$ leaves invariant a smooth sympletic form $\check{\omega}$.

By construction, the set $\check{I}=\pi\hat{I}$ is the stochastic island for the map $\check{F}$ (the island $\hat{I}$ is at a bounded distance from the singularities $\Omega_{i}$, so $\pi^{-1}$ realizes a smooth conjugacy between $\check{F}|_{\check{I}}$ and $\hat{F}|_{\hat{I}}$, which takes heteroclinic links to heteroclinic links and keeps the maximal Lyapunov exponent positive). Note that the 4-links $\partial V_{i}$ are invariant with respect to $(-id)$. The map $\pi$ glues the opposite points of $\partial V_{i}$ together, hence the circles $\pi(\partial V_{i})$ that bound the island $\check{I}$ are heteroclinic bi-links.

Now we will transform the stochastic island $\check{I}$ on $\mathbb{S}^{2}$ to a stochastic island for a map of the plane. Let us identify $\mathbb{S}^{2}$ with the one-point compactification of $\mathbb{R}^{2}$, where $\pi\Omega_{0}$ is identified with $\infty$; this can be done e.g. by the stereographic projection $\pi_{0}:(\mathbb{S}^{2}\setminus\pi\Omega_{0})\to\mathbb{R}^{2}$. As $\check{F}$ is equal to the identity at a neighborhood of $\Omega_{0}$, after the projection to $\mathbb{R}^{2}$, the map $\check{F}$ will be a $C^{\infty}$-diffeomorphism and will be equal to the identity at a neighborhood of infinity. The form $\check{\omega}$ will become a symplectic form on $\mathbb{R}^{2}$ and it will be preserved by $\check{F}$. Let $\check{\omega}=\beta(x,y)dx\wedge dy$ for some smooth function $\beta\neq 0$. The diffeomorphism $\pi_{1}:(x,y)\mapsto(X=x,Y=\int_{0}^{y}\beta(x,s)ds)$ of $\mathbb{R}^{2}$ onto a domain $D\subset\mathbb{R}^{2}$ transforms $\check{\omega}$ to the standard symplectic form $dX\wedge dY$. The map $\check{F}$ takes $D$ to $D$ in the coordinates $(X,Y)$ and is equal to identity near the boundary $\partial D$, so it can be extended onto the whole of $\mathbb{R}^{2}$ as the identity map outside of $D$; it will still preserve the standard form $dX\wedge dY$. By performing an additional scaling $\pi_{2}:(X,Y)\mapsto(\kappa X,\kappa Y)$ we can achieve that $\check{F}=id$ everywhere outside the unit disc $\mathbb{D}$. Thus the image by $\pi_{2}\circ\pi_{1}\circ\pi_{0}$ of the stochastic island $\check{I}$ on $\mathbb{S}^{2}$ will lie inside $\mathbb{D}$; it is a stochastic island $\mathcal{I}$ for the map $\check{F}$ (because $\check{I}$ is separated from $\pi\Omega_{0}$, so the map $\pi_{2}\circ\pi_{1}\circ\pi_{0}$ is a smooth conjugacy).

We have shown the existence of a diffeomorphism $\check{F}\in{\rm Diff}^{\infty}_{dX\wedge dY}(\mathbb{D})$ with a stochastic island $\mathcal{I}$. Note that Proposition 2.1 from Section 2 is satisfied for this island, as easily follows from Theorem D:

Corollary E**.**

The stochastic island $\mathcal{I}$ for the map $\check{F}\in{\rm Diff}^{\infty}(\mathbb{D})$ is robust relative link preservation.

Proof.

The island $\mathcal{I}$ is bounded by four heteroclinic bi-links ${\mathcal{L}}_{i}=\pi_{2}\circ\pi_{1}\circ\pi_{0}\circ\pi(\partial V_{i})$, $i=0,1,2,3$. For every $F$ which is $C^{2}$-close to $\check{F}$, if $F$ does not split the links, then it is $C^{2}$-conjugate to a $C^{2}$-diffeomorphism of $\mathbb{D}$ which is $C^{2}$-close to $\check{F}$ and keeps the links ${\mathcal{L}}_{i}$ invariant. Lifting this diffeomorphism to the torus $\mathbb{T}_{2}$ by $\pi^{-1}\circ\pi_{0}^{-1}\circ\pi_{1}^{-1}\circ\pi_{2}^{-1}$, we obtain a diffeomorphism $\tilde{F}$ of $\mathbb{T}_{2}\setminus\{\cup_{i=0,1,2,3}\Omega_{i}\}$ which preserves the links $\partial V_{i}$ and is $C^{2}$-close to $\hat{F}$ on $\hat{I}$. By Proposition D, the map $\tilde{F}$ has positive maximal Lyapunov exponent at every point of $\hat{I}$. Since the smooth conjugacy does not change the Lyapunov exponent, the map $F$ has positive maximal Lyapunov exponent a every point of the island bounded by the continuations of the links $\mathcal{L}_{i}$. ∎

4 Geometric model for a suitable stochastic island

Now, we choose a particular coordinate system in $\mathbb{R}^{2}$ such that the map $\check{F}$ and its stochastic island $\mathcal{I}$ we just constructed will acquire certain suitability properties (as given by Definition 4.6; see Fig. 4).

Let $F\in{\rm Diff}^{\infty}_{\omega}(\mathbb{R}^{2})$ have a heteroclinic link $L$.

Definition 4.1**.**

A fundamental interval of $L$ for the map $F$ is a closed segment $D_{1}\subset L$ such that $F(D_{1})\cap D_{1}$ is exactly one point – an endpoint both to $D_{1}$ and $F(D_{1})$. Given $m\geq 1$, an $m$-fundamental interval $D_{m}$ of $L$ is the union $D_{m}=\cup_{i=0}^{m-1}F^{i}(D_{1})$ of the $m$ first iterates of a certain fundamental interval $D_{1}$.

Let $(x,y)$ be symplectic coordinates in $\mathbb{R}^{2}$, so $\omega=dx\wedge dy$. Below we always fix the orientation in $\mathbb{R}^{2}$ such that the $x$-axis looks to the right and the $y$-axis looks up.

Definition 4.2**.**

An $m$-fundamental interval $D_{m}$ of a heteroclinic link $L$ will be called straight if $D_{m}$ is included in a straight line $y=const$.

It is a well-known fact (see [GL01]) that any $m$-fundamental interval $D_{m}$ can be straightened and the so-called time-energy coordinates can be introduced in its neighborhood, i.e., the map near $D_{m}$ becomes a translation to a constant vector. We formulate this result as

Lemma 4.3**.**

If $L$ is a link between two hyperbolic fixed points $P$ and $Q$ for $F\in{\rm Diff}^{\infty}_{\omega}(\mathbb{R}^{2})$ and $D_{m}\in L$ is an $m$-fundamental interval, then there exists a symplectic $C^{\infty}$-diffeomorphism $\phi$ from a neighborhood of $D_{m}$ into $\mathbb{R}^{2}$ such that $\phi(D_{m})$ is a straight $m$-fundamental interval for $\phi\circ F\circ\phi^{-1}$ and, in a neighborhood of $\phi(D_{m})$,

[TABLE]

for some constant $\tau\neq 0$.

Proof.

Put $P$ to the origin of coordinates and bring the map to the Birkhoff normal form by a symplectic $C^{\infty}$ coordinate transformation [Cha83, Thm. 1]. This means that we introduce symplectic coordinates $(x,y)$ near $P$ such that the map $F$ near $P$ will be given by

[TABLE]

for some function $q\in C^{\infty}(\mathbb{R},\mathbb{R})$ with $q(0)>0$. Note that the unstable manifold of $P$ is given by $y=0$ in these coordinates. By iterating $F$ forward, we can extend the domain of the Birkhoff coordinates to a small neighborhood of any compact subset of $W^{u}(P)$. In particular, we may assume that the map $F$ is given by (16) near the $m$-fundamental interval $D_{m}$. Observe that $D_{m}:=\{(x,y):\;\tilde{x}\in[x_{0},e^{mq(0)}x_{0}],\;y=0\}$ in these coordinates, for some $x_{0}\neq 0$ (by making, if necessary, the coordinate change $(x,y)\to-(x,y)$, we can always make $x_{0}>0$).

Let $h=\int q$. Obviously, $F$ is the time-1 map of the flow defined by the Hamiltonian $H(x,y)=h(xy)$. Put $\displaystyle X(x,y):=\frac{\ln x}{q(xy)}$ and $Y(x,y)=h(xy)$. The map $(x,y)\mapsto(X,Y)$ is a $C^{\infty}_{\omega}$-coordinate change near $D_{m}$ which conjugates $F$ with $(X,Y)\mapsto(X+1,Y)$. ∎

Note that the map $(x,y)\mapsto(x+\tau,y)$ is the time-$\tau$ map by the vector filed $\dot{x}=1,\;\dot{y}=0$ defined by the Hamiltonian $H(x,y)=y$. Therefore, $x$ plays the role of time and the conserved quantity $y$ can be viewed as energy, which justifies the “time-energy” terminology.

4.1 Making a bi-link suitable

A vertical strip $V$ is the region $\{(x,y):\;x\in[c_{1},c_{2}],y\in\mathbb{R}\}$ in $\mathbb{R}^{2}$ for some $c_{1}<c_{2}$.

Definition 4.4** (Suitable intersection of a heteroclinic bi-link with two strips).**

Two vertical strips $V^{a}$ and $V^{b}$ intersect a heteroclinic bi-link $(\mathring{L}^{a},\mathring{L}^{b})$ of a dynamics $\mathring{F}$ in a suitable way if:

• •

the intersection of $V^{a}$ with $\mathring{L}^{a}$ is a straight $2$-fundamental interval $D_{2}^{a}$;

• •

the intersection of $V^{b}$ with $\mathring{L}^{b}$ is the disjoint union of a straight $2$-fundamental interval $\mathring{D}_{2}^{b}$ and a straight $4$-fundamental interval $\mathring{D}_{4}^{b}$;

• •

the strip $V^{b}$ does not intersect $\mathring{L}^{a}$;

• •

there exists $\tau>0$ such that the map $\mathring{F}$ in restriction to a neighborhood of $\mathring{D}_{2}^{a}$ is given by $(x,y)\mapsto(x-\tau,y)$ and, in restriction to a neighborhood of $\mathring{D}_{2}^{b}$, it is given by $(x,y)\mapsto(x+\tau,y)$;

• •

there exists $n\geq 1$ such that $\mathring{F}^{n}$ sends $\mathring{D}_{2}^{b}\cup\mathring{F}^{2}(\mathring{D}_{2}^{b})$ to $\mathring{D}_{4}^{b}$, and the restriction of $\mathring{F}^{n}$ to a neighborhood of $\mathring{D}_{2}^{b}$ is

[TABLE]

for some $\theta\in\mathbb{R}^{2}$.

See Fig. 3 for an illustration.**

In Lemma 4.5 below, we are going to show that any conservative diffeomorphism $F$ with a bi-link $(L^{a},L^{b})$ is smoothly conjugate to a conservative diffeomorphism $\mathring{F}$ with a suitable bi-link $(\mathring{L}^{a},\mathring{L}^{b})$. This means that we have a lot of geometric freedom in the choice of the bi-link $(\mathring{L}^{a},\mathring{L}^{b})$, that we shall explain.

Take any two parallel straight lines in $\mathbb{R}^{2}$: $\{y=y_{1}\}$ and $\{y=y_{2}\}$ with $y_{2}<y_{1}$. Let $\tau>0$ and $x_{a}<x_{b}$ so that $x_{a}+\tau<x_{b}-\tau$. Consider the two vertical strips:

[TABLE]

Consider any $C^{\infty}$-smooth circle $\mathring{L}$ in $\mathbb{R}^{2}$ equal to the union of two curves $\mathring{L}^{a}$, $\mathring{L}^{b}$ satisfying:

• •

$\mathring{L}=\mathring{L}^{a}\cup\mathring{L}^{b}$ and $\mathring{L}^{a}\cap\mathring{L}^{b}=\partial\mathring{L}^{a}=\partial\mathring{L}^{b}$  .

• •

$\mathring{L}^{a}\cap V^{a}=[x_{a}-\tau,x_{a}+\tau]\times\{y_{1}\}$ and $\mathring{L}^{a}\cap V^{b}=\varnothing$.

• •

$\mathring{L}^{b}\cap V^{b}=[x_{b}-\tau,x_{b}+\tau]\times\{y_{1},y_{2}\}$.

Now we can state:

Lemma 4.5**.**

Let $F\in{\rm Diff}^{\infty}_{\omega}(\mathbb{R}^{2})$ have a heteroclinic bi-link $(L^{a},L^{b})$. Then there exists a symplectic $C^{\infty}$-diffeomorphism $\mathring{\phi}$ of a small neighborhood of $L^{a}\cup L^{b}$ into $\mathbb{R}^{2}$ such that:

• •

$\mathring{\phi}(L^{a}\cup L^{b})=\mathring{L}$, $\mathring{\phi}(L^{a})=\mathring{L}^{a}$, $\mathring{\phi}(L^{b})=\mathring{L}^{b}$ ;

• •

the vertical strips $V^{a}$ and $V^{b}$ intersect the bi-link $(\mathring{L}^{a},\mathring{L}^{b})$ of $\mathring{\phi}\circ F\circ\mathring{\phi}^{-1}$ in a suitable way (with $n=4$ in Definition 4.4).

Proof.

Let us take fundamental intervals $D^{a}$ of $L^{a}$ and $D^{b}$ of $L^{b}$. Observe that:

• •

$D_{2}^{a}:=D^{a}\cup F(D^{a})$ and $D_{2}^{b}:=D^{b}\cup F(D^{b})$ are 2-fundamental intervals of $L^{a}$ and $L^{b}$, respectively;

• •

$D_{4}^{b}:=\cup_{i=4}^{7}F^{i}(D^{b})$ is a 4-fundamental interval of $L^{b}$.

We remark that $D_{2}^{b}$ and $D_{4}^{b}$ are included in the $8$-fundamental interval $D^{b}_{8}:=\cup_{i=0}^{7}F^{i}(D^{b})$ of $L^{b}$.

By Lemma 4.3, there exist symplectic diffeomorphisms $\phi^{a}$ and $\phi^{b}$ acting from a neighborhood of $\hat{D}^{a}_{2}:=D^{a}_{2}\cup F(D^{a}_{2})$ and, respectively, a neighborhood of $\hat{D}^{b}_{8}=D_{8}^{b}\cup F(D^{b}_{8})$ into $\mathbb{R}^{2}$ such that:

• •

$\phi^{a}(D^{a}_{2})=[-2,0]\times\{0\}$ and $\phi^{a}\circ F\circ(\phi^{a})^{-1}$ is the translation to $(-1,0)$ in a neighborhood of $\phi^{a}(D^{a}_{2})$;

• •

$\phi^{b}(D^{b}_{8})=[0,8]\times\{0\}$ and $\phi^{b}\circ f\circ(\phi^{b})^{-1}$ is the translation to $(1,0)$ in a neighborhood of $\phi^{b}(D^{b}_{8})$.

Observe that $\phi^{b}(D^{b}_{2})=[0,2]\times\{0\}$ and $\phi^{b}(D^{b}_{4})=[4,8]\times\{0\}$.

Let $\delta>0$ be small and denote $J:=[-\delta\tau,\delta\tau]$ and $J^{\prime}:=[-\frac{\delta\tau}{2},\frac{\delta\tau}{2}]$. Let

[TABLE]

For $\delta>0$ small enough, the sets $B_{2}^{b}$, $B_{4}^{b}$ and $B_{2}^{a}$ are disjoint. Take two linear area-preserving maps:

[TABLE]

Notice that the maps

[TABLE]

send, respectively, $B_{2}^{a}$, $B_{2}^{b}$ and $B_{4}^{b}$ onto translations of $R:=[-\tau,\tau]\times[-\delta,\delta]$.

Let $\mathring{\phi}_{0}$ be a symplectic embedding of a neighborhood of the disjoint union $B_{0}:=B_{2}^{a}\cup B_{2}^{b}\cup B_{4}^{b}$ into $\mathbb{R}^{2}$ such that:

• •

$B_{2}^{a}$, $B_{2}^{b}$ and $B_{4}^{b}$ are sent by $\mathring{\phi}_{0}$ onto, respectively, (see Fig. 3):

[TABLE]

• •

the restriction of $\mathring{\phi}_{0}$ to neighborhoods of $B_{2}^{a}$, $B_{2}^{b}$ and $B_{4}^{b}$ is the composition of respectively $\mathring{\phi}^{a}_{2}$, $\mathring{\phi}^{b}_{2}$ and $\mathring{\phi}^{b}_{4}$ with some translations.

Note that $\mathring{\phi}_{0}$ sends the fundamental intervals $D_{2}^{a}$, $D_{2}^{b}$ and $D_{4}^{b}$ onto, respectively,:

[TABLE]

so these images lie in the curve $\mathring{L}$, in the intersection with the vertical strips $V^{a}$ and $V_{b}$.

Note that the map $\mathring{\phi}_{0}\circ F^{4}\circ\mathring{\phi}_{0}^{-1}$ sends $\mathring{D}_{2}^{b}$ into $\mathring{D}_{4}^{b}$ and its restriction to a neighborhood of $\mathring{D}_{2}^{b}$ is the composition of a translation with the linear map $(x,y)\mapsto(-x/2,-2y)$, as required by Definition 4.4 with $n=4$. Therefore, to prove the lemma, it suffices to construct a symplectic $C^{\infty}$-diffeomorphism $\mathring{\phi}$ of a small neighborhood of $B_{0}\cup L^{a}\cup L^{b}$ into $\mathbb{R}^{2}$ which would send $L^{a},L^{b}$ to $\mathring{L}^{a},\mathring{L}^{b}$, such that its restriction to a neighborhood of $B_{0}$ would be equal to $\mathring{\phi}_{0}$.

Without the symplecticity requirement, the map $\mathring{\phi}$ would be given by Whitney extension theorem [Whi34]. Making the diffeomorphism $\mathring{\phi}$ symplectic requires an extra effort, as it is done below.

Consider an annulus $\mathbb{A}:=(\mathbb{R}/\mathbb{Z})\times[-\eta,\eta]$ for a sufficiently small $\eta>0$. Let $t\in\mathbb{R}/\mathbb{Z}$ and $h\in[-\eta,\eta]$ be coordinates in $\mathbb{A}$. By the Weinstein’s Lagrangian neighborhood theorem [Wei71], if $\eta$ is sufficiently small, then there exists an area-preserving diffeomorphism $N$ from the annulus $\mathbb{A}$ to a neighborhood of the bi-link $L^{a}\cup L^{b}$, which sends the central circle $S:=\{h=0\}$ to $L^{a}\cup L^{b}$. Similarly, there exists an area-preserving diffeomorphism $\mathring{N}$ from $\mathbb{A}$ to a small neighborhood of the curve $\mathring{L}=\mathring{L}^{a}\cup\mathring{L}^{b}$, which sends $S$ to $\mathring{L}$.

Let $\delta>0$ be small enough, so that the sets $B_{0}$ and $\mathring{\phi}_{0}(B_{0})$ will be contained in $N(\mathbb{A})$ and, respectively, $\mathring{N}(\mathbb{A})$. By Whitney extension theorem, there exists $G\in{\rm Diff}^{\infty}(\mathbb{A},\mathbb{A})$ such that $G(S)=S$ and $\mathring{N}\circ G\circ N^{-1}$ restricted to a neighborhood $U$ of $B_{0}$ is $\mathring{\phi}_{0}$. In particular, $\text{det}DG|_{N^{-1}(U)}=1$. The map $G$ is orientation-preserving but it is not, a priori, area-preserving outside of $U$.

Our goal is to correct $G$ in order to make it area-preserving. More precisely, we are going to construct a $C^{\infty}$-diffeomorphism $\Psi$ of $\mathbb{A}$ such that $\text{det}D\Psi=\text{det}DG$, $\Psi(S)=S$, and the restriction of $\Psi$ to $N^{-1}(B_{0})$ is the identity. Then the Lemma will be proved by taking $\mathring{\phi}:=\mathring{N}\circ G\circ\Psi^{-1}\circ N^{-1}$.

Let us keep fixed the neighborhood $U$ of $B_{0}$ where $G$ is area-preserving, and let us take $\delta>0$ small. Then $B_{0}$ can be made as close as we want to $D_{2}^{b}\cup D_{4}^{b}\cup D_{2}^{a}$. Hence, for $\delta>0$ small enough, if the image by $N$ of a vertical segment $\{t=const,\;|h|\leq\eta\}$ intersects $B_{0}$, then it lies entirely in $U$, i.e., $\text{det}DG=1$ everywhere on this segment. Therefore, if we define the map

[TABLE]

then $\Psi=id$ in the restriction to $N^{-1}(B_{0})$. It is also obvious, that $\Psi=id$ in restriction to the central circle $S=\{h=0\}$, and $\text{det}D\Psi=\text{det}DG$.∎

4.2 Making the stochastic island suitable

Consider the map $\check{F}\in{\rm Diff}^{\infty}_{\omega}$ with the stochastic island $\mathcal{I}$. Recall that $\mathcal{I}$ is bounded by 4 heteroclinic bi-links $(L_{i}^{a},L_{i}^{b})$, $i=0,1,2,3$, each of which is a $C^{\infty}$-smooth circle. We take a convention that $L_{0}^{a}\cup L_{0}^{b}$ is the outer circle, i.e., the bi-links $(L_{i}^{a},L_{i}^{b})$ with $i=1,2,3$ lie inside the region bounded by $L_{0}^{a}\cup L_{0}^{b}$.

Below, we will construct symplectic coordinates $\mathring{\phi}$ in $\mathbb{R}^{2}$ such that the island $\mathring{\phi}({\mathcal{I}})$ will satisfy the following suitability conditions.

Definition 4.6** (Suitable intersection of 4 heteroclinic bi-link-s with 4 pairs of vertical strips).**

We say that 4 pairs of vertical strips $(V_{i}^{a},V_{i}^{b})$, $i=0,1,2,3$, intersect bi-links $(\mathring{L}_{j}^{a},\mathring{L}_{j}^{b})$, $j=0,1,2,3$ in a suitable way if the following conditions hold true.

1. $(H_{1})$

For every $0\leq i\leq 3$, the intersection of $V^{a}_{i}\sqcup V^{b}_{i}$ with $(\mathring{L}_{i}^{a},\mathring{L}_{i}^{b})$ is suitable in the sense of Definition 4.4, with $n=4$. 2. $(H_{2})$

For every $j\geq 1$ and every $i\not=j$, the strips $V^{a}_{i}$ and $V^{b}_{i}$ do not intersect the circle $\mathring{L}_{j}^{a}\cup\mathring{L}_{j}^{b}$

See Fig. 4 for an illustration.**

Proposition 4.7**.**

There exist $\mathring{\phi}\in{\rm Diff}^{\infty}_{\omega}(\mathbb{R}^{2})$ and 4 pairs of vertical strips $V_{i}^{a}$, $V^{b}_{i}$, $i=0,1,2,3$, which intersect the heteroclinic bi-links $(\mathring{L}_{j}^{a}=\mathring{\phi}(L_{j}^{a}),\mathring{L}_{j}^{b}=\mathring{\phi}(L_{j}^{b}))$, $j=0,1,2,3$, of the map $\mathring{F}=\mathring{\phi}\circ\check{F}\circ\mathring{\phi}^{-1}$ in a suitable way.

Proof.

By Lemma 4.5, for every $i=0,1,2,3$ there exists a pair of vertical strips $V_{i}^{a}$, $V^{b}_{i}$ and a symplectic $C^{\infty}$ diffeomorphism $\mathring{\phi}_{i}$ of a small neighborhood of the bi-link $(L_{i}^{a},L_{i}^{b})$ such that the strips $V_{i}^{a}$, $V_{i}^{b}$ intersect $(\mathring{L}^{a}_{i},\mathring{L}^{b}_{i}):=(\mathring{\phi}_{i}(L^{a}_{i}),\mathring{\phi}_{i}(L^{b}_{i}))$ of the map $\mathring{F}_{i}:=\mathring{\phi}_{i}\circ\check{F}\circ\mathring{\phi}_{i}^{-1}$ in a suitable way, ensuring the fulfillment of Condition $(H_{1})$ of Definition 4.6.

Note that in Lemma 4.5 there is a freedom in the choice of the curve $\mathring{L}_{i}=\mathring{\phi}_{i}(L^{a}_{i}\cup L^{b}_{i})$. So, we take $\mathring{L}_{i}$ such that it will bound a disc of the same volume as $L_{i}=L_{i}^{a}\cup L_{i}^{b}$ does. Also, by choosing the constants $y_{1},y_{2},x_{0},x_{1}$ in Lemma 4.5 in an appropriate way for each $i$, we can assure that $\mathring{\phi}_{i}(L_{i})$ do not intersect for different $i$ and $\mathring{\phi}_{1}(L_{1})\cup\mathring{\phi}(L_{2})\cup\mathring{\phi}(L_{3})$ lies inside the disc bounded by $\mathring{\phi}_{0}(L_{0})$, and the strips $V_{i}^{a}$ and $V_{i}^{b}$ are positioned where we wish, thus ensuring Condition $(H_{2})$ of Definition 4.6.

Let $\mathbb{A}_{i}$ be a sufficiently small closed annulus around $L_{i}$, $i=0,1,2,3$. Let us prove the proposition by showing the existence of a symplectic extension $\mathring{\phi}$ of the symplectic maps $\mathring{\phi}_{i}$ from a neighborhood of the annuli $\mathbb{A}_{i}$ to the whole of $\mathbb{R}^{2}$, i.e., a diffeomorphism $\mathring{\phi}\in{\rm Diff}^{\infty}_{\omega}$ such that $\mathring{\phi}|_{\mathbb{A}_{i}}=\mathring{\phi}_{i}|_{\mathbb{A}_{i}}$ for all $i=0,1,2,3$. Since the curve $\mathring{L}_{i}$ bound the disc of the same volume as $L_{i}=L_{i}^{a}\cup L_{i}^{b}$ does, for each $i$, the annuli ${\mathbb{A}}_{i}$ is necessarily such that the volume of each of the connected components of $\mathbb{R}^{2}\setminus\sqcup_{i}\mathbb{A}_{i}$ equals to the volume of a corresponding component of $\mathbb{R}^{2}\setminus\sqcup_{i}\mathring{\phi}_{i}(\mathbb{A}_{i})$. Then the existence of the sought sympletic extension $\mathring{\phi}$ is a standard consequence of Dacorogna-Moser theorem [DM90], as given by

Corollary 4.8** (Cor. 4 [Avi10]).**

Let $K\subset\mathbb{R}^{2}$ be a compact set, $U$ be a neighborhood of $K$ and let $\psi\in C^{\infty}_{\omega}(U,\mathbb{R}^{2})$ be close to the identity. Assume that for every bounded connected component $W$ of $\mathbb{R}^{2}\setminus U$ and its corresponding one in $\mathbb{R}^{2}\setminus\psi(U)$ have the same volume. Then there exists $\phi\in Diff^{\infty}_{\omega}(\mathbb{R}^{2})$ which is $C^{\infty}$-close to the identity and so that $\phi|K=\psi|K$.

∎

5 Restoration of broken heteroclinic links

Let $\mathring{F}\in Diff^{\infty}_{\omega}(\mathbb{R}^{2})$ be the map constructed in the previous Section. It has a stochastic island $\mathring{\mathcal{I}}$ bounded by 4 smooth circles - heteroclinic bi-links $(\mathring{L}_{i}^{a},\mathring{L}_{i}^{b})$, $i=0,1,2,3$. The bi-link $\mathring{L}_{0}^{a}\cup\mathring{L}_{0}^{b}$ forms the outer boundary of $\mathring{\mathcal{I}}$. By construction, there exist 4 pairs of vertical strips $(V_{i}^{a},V_{i}^{b})$ which intersect $(\mathring{L}_{i}^{a},\mathring{L}_{i}^{b})$ in a suitable way in the sense of Definition 4.6. We denote $V_{i}^{a}=I_{i}^{a}\times\mathbb{R}$ and $V_{i}^{b}=I_{i}^{b}\times\mathbb{R}$, where $I_{i}^{a}$, $I_{i}^{b}$ are closed disjoint intervals in the $x$-axis.

In this Section we consider perturbations of the map $\mathring{F}$ and prove Proposition 2.2. Namely, we show that for every $r\geq 1$, for every $\eta>0$, for every $F\in{\rm Diff}^{r+8}_{\omega}$ which is sufficiently close to $\mathring{F}$ in $C^{r+8}$, there exists $\psi\in C^{r}(\mathbb{R},\mathbb{R})$, supported in $\cup_{i}(I_{i}^{a}\cup I_{i}^{b})$ and with $C^{r}$-norm smaller than $\eta$, such that the map $\bar{F}=S_{\psi}\circ F$ has 4 heteroclinic bi-links $(L_{i}^{a},L_{i}^{b})$ close to $(\mathring{L}_{i}^{a},\mathring{L}_{i}^{b})$. We recall that given a function $\psi$, we denote

[TABLE]

It is pretty much obvious that to prove this statement, it suffices to show

Proposition 5.1**.**

Let $r\geq 1$. Let $\mathring{F}\in{\rm Diff}^{r+4}_{\omega}$ have a heteroclinic bi-link $(\mathring{L}^{a},\mathring{L}^{b})$ which intersects two vertical strips $V^{a}=I^{a}\times\mathbb{R}$ and $V^{b}=I^{b}\times\mathbb{R}$ in a suitable way. For every $F\in{\rm Diff}^{r+4}_{\omega}$ which is $C^{r+4}$-close to $\mathring{F}$, there exists $\psi\in C^{r}(\mathbb{R},\mathbb{R})$ which is $C^{r}$-small and supported in $I^{a}\sqcup I^{b}$, such that the map $S_{\psi}\circ F$ has a bi-link $(L^{a},L^{b})$ close to $(\mathring{L}^{a},\mathring{L}^{b})$.

Proposition 2.2 is inferred from this statement as follows.

Proof of Proposition 2.2.

Let $F$ be a $C^{r+8}$-small perturbation of $\mathring{F}$. By Proposition 5.1, for each $i=1,2,3$ there exists a $C^{r+4}$-small function $\psi_{i}$ supported in $I^{a}_{i}\sqcup I^{b}_{i}$ such that the map $S_{\psi_{i}}\circ\mathring{F}$ has a bi-link $(\bar{L}_{i}^{a},\bar{L}_{i}^{b})$ close to $(\mathring{L}_{i}^{a},\mathring{L}_{i}^{b})$. By property $(H_{2})$ of the suitable intersection (see Definition 4.6), the vertical strips $V^{a}_{i}$ and $V^{b}_{i}$ do not intersect the bi-links $(\mathring{L}_{j}^{a},\mathring{L}_{j}^{b})$ for $j\neq i$, $j>0$. Thus the map $S_{\psi_{i}}$ is identity near the bi-links $(\mathring{L}_{j}^{a},\mathring{L}_{j}^{b})$ with $j\neq i$, $j>0$. Therefore, the map $S_{\psi_{1}+\psi_{2}+\psi_{3}}\circ F$ has 3 bi-links $(L^{a}_{i},L^{b}_{i})$ close to $(\mathring{L}^{a}_{i},\mathring{L}^{b}_{i})$, respectively.

The map $S_{\psi_{1}+\psi_{2}+\psi_{3}}\circ F$ is an $\omega$-preserving diffeomorphism and is $C^{r+4}$-close to $\mathring{F}$. Therefore, by applying Proposition 5.1 to this map and the link $(\mathring{L}^{a}_{0},\mathring{L}^{b}_{0})$, we obtain that there exists a $C^{r}$-small function $\psi_{0}$ localized in $I_{0}^{a}\cup I_{0}^{b}$ such that the map $S_{\psi_{0}}\circ S_{\psi_{1}+\psi_{2}+\psi_{3}}\circ F$ has a bi-link $(L^{a}_{0},L^{b}_{0})$ close to $(\mathring{L}^{a}_{0},\mathring{L}^{b}_{0})$. Since $V^{a}_{0}$ and $V^{b}_{0}$ do not intersect the bi-links $(L_{i}^{a},L_{i}^{b})$ for $i>0$ (by property $(H_{2})$ of the suitable intersection), the map $S_{\psi_{0}}$ is identity near these bi-links, hence it does not destroy them. Thus, the map $\bar{F}=S_{\psi}\circ F$ with $\psi=\psi_{0}+\psi_{1}+\psi_{2}+\psi_{3}$ has all 4 bi-links $(L_{i}^{a},L_{i}^{b})$ as required. ∎

Proof of Proposition 5.1.

This Proposition follows from the two lemmas below which we prove in Sections 5.3 and 5.4 respectively.

Lemma 5.2**.**

Under the hypotheses of Proposition 5.1, for every $F\in{\rm Diff}^{k}_{\omega}$ which is $C^{k}$-close to $\mathring{F}$, $k\geq 3$, there exists a $C^{k-2}$-small function $\psi_{a}$ supported in $I^{a}$ and such that the map $S_{\psi_{a}}\circ F$ has a link $L^{a}$ close to $\mathring{L}^{a}$.

Lemma 5.3**.**

Under the hypotheses of Proposition 5.1, for every $F\in{\rm Diff}^{k}_{\omega}$ which is $C^{k}$-close ($k\geq 3$) to $\mathring{F}$ and has a link $L^{a}$ close to $\mathring{L}^{a}$, there exists a $C^{k-2}$-small function $\psi^{b}$ supported in $I^{b}$ and such that the map $S_{\psi_{b}}\circ F$ has a link $L^{b}$ close to $\mathring{L}^{b}$.

Indeed, if an $\omega$-preserving diffeomorphism $F$ is a $C^{r+4}$-small perturbation of $F$, then, by Lemma 5.2, there exists a $C^{r+2}$-small $\psi_{a}$ such that the map $S_{\psi_{a}}\circ F$ has the link $L_{a}$. This map is $\omega$-preserving and is $C^{r+2}$-close to $\mathring{F}$. Therefore, applying Lemma 5.3 to this map, we find that there exists a $C^{r}$-small $\psi_{b}$ supported in $I^{b}$ and such that the map $S_{\psi_{b}}\circ S_{\psi_{a}}\circ F=S_{\psi_{a}+\psi_{b}}\circ F$ has the link $L_{b}$. As the strip $I^{b}\times\mathbb{R}$ does not intersect $L^{a}$, the link $L^{a}$ also persists for the map $S_{\psi_{a}+\psi_{b}}\circ\hat{F}$, which gives Proposition 5.1 with $\psi=\psi_{a}+\psi_{b}$. ∎

5.1 Evaluation of the link splitting

The map $\mathring{F}$ has two saddle fixed points $P$ and $Q$ on the circle $\mathring{L}^{a}\cup\mathring{L}^{b}$. The point $P$ is repelling on the circle, while $Q$ is attracting on the circle.

Let $W^{a}(P;\mathring{F})$ and $W^{b}(P;\mathring{F})$ denote the halves of the unstable manifolds of $P$ equal to, respectively, $\mathring{L}^{a}\setminus\{Q\}$ and $\mathring{L}^{b}\setminus\{Q\}$. Let $W^{a}(Q;\mathring{F})$ and $W^{b}(Q;\mathring{F})$ be the halves of the stable manifolds of $Q$ equal to respectively $\mathring{L}^{a}\setminus\{P\}$ and $\mathring{L}^{b}\setminus\{P\}$.

The points $P$ and $Q$ persist for every $C^{1}$-close map $F$, and depend continuously on $F$. The corresponding manifolds $W^{a}(P;F)$, $W^{b}(P;F)$, $W^{a}(Q;F)$ and $W^{b}(Q;F)$ also persist, and depend continuously on $F$ as embedded curves of the same smoothness as $F$. To avoid ambiguities, we will fix a sufficiently small neighborhood of the point $Q$ and then $W^{a}(P;F)$ and $W^{b}(P;F)$ will denote the two arcs of $W^{u}(P,F)$ which connect $P$ with the boundary of this neighborhood and are close, respectively, to $W^{a}(P;\mathring{F})$ and $W^{b}(P;\mathring{F})$. Similarly, $W^{a}(Q;F)$ and $W^{b}(Q;F)$ are the arcs in $W^{s}(Q,F)$ which connect $Q$ with the boundary of a small neighborhood of $P$ and are close, respectively, to $W^{a}(Q;\mathring{F})$ and $W^{b}(Q;\mathring{F})$.

In general, the links are broken when the map $\mathring{F}$ is perturbed, so $W^{a}(P;F)$ and $W^{b}(P;F)$ do not need to coincide with, respectively, $W^{a}(Q;F)$ and $W^{b}(Q;F)$.

In the next two Sections we will show, for a given $F\in{\rm Diff}^{\omega}_{k}(\mathbb{D})$ which is $C^{k}$-close to $\mathring{F}$, how to find a $C^{k-2}$-function $\psi$ such that each of the unions $W^{a}(P;S_{\psi}\circ F)\cup W^{a}(Q;S_{\psi}\circ F)$ and $W^{b}(P;S_{\psi}\circ F)\cup W^{b}(Q;S_{\psi}\circ F)$ forms a heteroclinic link between $P$ and $Q$.

In this Section, we obtain formulas for the defect of coincidence between $W^{a}(P;S_{\psi}\circ F)$ and $W^{a}(Q;S_{\psi}\circ F)$ or $W^{b}(P;S_{\psi}\circ F)$ and $W^{b}(Q;S_{\psi}\circ F)$. In order to do that, we shall use the so-called time-energy coordinates near the fundamental interval $\mathring{D}_{2}^{a}=V_{a}\cap L_{a}$ of the link $L_{a}$ and the fundamental domain $\mathring{D}_{2}^{b}\subset V_{b}\cap L_{b}$ of the link $L_{b}$. Recall that by the suitability conditions (see Definition 4.4) there exist $\tau>0$ and $(x_{a},y_{a})\in\mathbb{D}$, $(x_{b},y_{b})\in\mathbb{D}$ such that $\mathring{D}_{2}^{a}=\{x\in[x_{a}-2\tau,x_{a}]\}\times\{y=y_{a}\}$, $\mathring{D}_{2}^{b}=\{x\in[x_{b},x_{b}+2\tau]\}\times\{y=y_{b}\}$, and the map $\mathring{F}$ restricted to a small neighborhood $N^{a}$ of $\mathring{D}_{2}^{a}$ or a small neighborhood $N^{b}$ of $\mathring{D}_{2}^{b}$ is given by

[TABLE]

Definition 5.4**.**

For an $\omega$-preserving map $F$, which is $C^{k}$-close to $\mathring{F}$, an $N^{a}$-time-energy chart $\phi^{a}$ is an $\omega$-preserving diffeomorphism from $N^{a}\cup F(N^{a})$ to $\mathbb{D}$ which is $C^{k-1}$-close to identity and satisfies

[TABLE]

An $N^{b}$-time-energy chart $\phi^{b}$ is an $\omega$-preserving diffeomorphism from $N^{b}\cup F(N^{b})$ to $\mathbb{D}$ which is $C^{k-1}$-close to identity and satisfies

[TABLE]

We notice that the identity map is a time-energy chart for $\mathring{F}$. The time-energy charts are not uniquely defined, so we will fix their choice below. In our construction the time-energy charts will be identity near $\{x=x_{a}\}$ and $\{x=x_{b}\}$.

Once certain time-energy coordinates are introduced in $N^{a}\cup F(N^{a})$, the curves $W^{a}(P;F)\cap\{N^{a}\cup F(N^{a})\}$ and $W^{a}(Q;F)\cap\{N^{a}\cup F(N^{a})\}$ become graphs of $\tau$-periodic functions: the manifolds $W^{a}(P;F)$ and $W^{a}(Q;F)$ are invariant with respect to $F$ which means that in the time-energy coordinates they are invariant with respect to the translation to $(-\tau,0)$, see (18),(17). We denote as $w^{u}_{a}(F,\phi^{a})$ and $w^{s}_{a}(F,\phi^{a})$ the $\tau$-periodic functions whose graphs are the curves $\phi^{a}(W^{a}(P;F))$ and $\phi^{a}(W^{a}(Q;F))$, respectively.

Definition 5.5**.**

The link-splitting function $M^{a}(F,\phi^{a})$ associated to $(N^{a},F,\phi^{a})$ is the $\tau$-periodic function equal to $w^{u}_{a}(F,\phi^{a})-w^{s}_{a}(F,\phi^{a})$ at $x\in[x_{a}-\tau,x_{a}]$.

Similarly, let $w^{s}_{b}(F,\phi^{b})$ and $w^{u}_{b}(F,\phi^{b})$ be the $\tau$-periodic functions whose graphs are the curves $\phi^{b}(W^{b}(Q;F)$ and $\phi^{b}(W^{b}(P;F))$.

Definition 5.6**.**

The link-splitting function $M^{b}(F,\phi^{b})$ associated to $(N^{b},F,\phi^{b})$ is the $\tau$-periodic function equal to $w^{u}_{b}(F,\phi^{b})-w^{s}_{b}(F,\phi^{b})$ at $x\in[x_{b},x_{b}+\tau]$.

By the definition, the link $L^{a}$ or $L^{b}$ is restored when the function $M^{a}$ or, respectively, $M^{b}$ is identically zero.

We start with constructing a $C^{k-1}$-smooth time-energy chart for the map $F$.

Lemma 5.7**.**

There exists a small neighborhood $N^{a}$ of $\mathring{D}_{2}^{a}$ and a small neighborhood $N^{b}$ of $\mathring{D}_{2}^{b}$ such that for every $\omega$-preserving diffeomorphism $F$ which is $C^{k}$-close to $\mathring{F}$, $k\geq 3$, there exists $C^{k-1}$-smooth time-energy chart $\phi^{a}$ and $\phi^{b}$, which depend continuously on $F$ and equal to identity if $F=\mathring{F}$.

Proof.

We will show the proof only for the existence of $\phi^{a}$. The proof for $\phi^{b}$ is identical up to the exchange of index $a$ to $b$ and $(-\tau)$ to $\tau$.

Let $\rho\in C^{\infty}(\mathbb{R},[0,1])$ be zero everywhere near $x=x_{a}$ and $1$ everywhere near $x=x_{a}-\tau$. Let $\phi_{0}(x,y):=(x,y)(1-\rho(x))+\rho(x)\mathring{F}\circ F^{-1}(x,y)$. The map $\phi_{0}$ is a $C^{k}$-diffeomorphism from a small neighborhood of $\{x\in[x_{a}-\tau,x_{a}],\;y=y_{a}\}$ into $\mathbb{D}$, it is $C^{k}$-close to identity and equals to the identity near $(x_{a},y_{a})$ and to $\mathring{F}\circ F^{-1}$ near $(x_{a}-\tau,y_{a})$.

Thus, $\phi_{0}$ satisfies

[TABLE]

in a neighborhood of $(x_{a},y_{a})$ (see (17)). Take a small neighborhood of $D^{a}$ and define there $\phi^{a}(x,y)=\phi_{0}(x,\sigma(x,y))$ where the $C^{k-1}$-function $\sigma$ satisfies $\sigma(x,y_{a})=y_{a}$ and $\partial_{y}\sigma=\text{det}D\phi_{0}^{-1}(x,\sigma)$. By construction, $\text{det}D\phi^{a}\equiv 1$, i.e., it is an $\omega$-preserving $C^{k-1}$-diffeomorphism and, since $\phi_{0}$ is $C^{k}$-close to the identity, $\phi^{a}$ is $C^{k-1}$-close to the identity. Since $\text{det}D\phi_{0}=1$ everywhere near $(x_{a},y_{a})$ and $(x_{a}-\tau,y_{a})$, we have that $\sigma\equiv y$ near these points, so $\phi^{a}\equiv\phi_{0}$ there. In particular,

[TABLE]

near $(x_{a},y_{a})$.

It follows that we obtain the required time-energy chart if we extend $\phi^{a}$ to a small neighborhood of $\mathring{D}^{a}_{2}\cup\mathring{F}\mathring{D}^{a}_{2}$ by the rule

[TABLE]

∎

Now, take some sufficiently small $\delta>0$. Consider any map $F$ close enough to $\mathring{F}$ and let $\phi^{a,b}$ be the $C^{k-1}$ time-energy charts for $F$, defined in Lemma 5.7. Given any close to zero smooth function $\psi(x)$ supported inside $[x_{a}-2\tau+\delta,x_{a}-\delta]$, we consider the map

[TABLE]

and define for it the time-energy chart $\phi_{\psi}^{a}$ in the open set $N^{a}\cup F(N^{a})$ such that

[TABLE]

Recall that $\psi$ vanishes for $x$ close to $x_{a}$ and $x$ larger than $x_{a}$ and for $x$ close to $x_{a}-2\tau$ and smaller than that. Furthermore, if $x$ is close to $x_{a}-\tau$, then the $x$-coordinate of $\bar{F}^{-1}(x,y)$ is close to $x_{a}-2\tau$. Thus, at this point $F\circ\bar{F}^{-1}=id$ and so

[TABLE]

Hence $\phi_{\psi}^{a}$ has no discontinuities at $x=x_{a}-\tau$ and the following required conjugacy holds true:

[TABLE]

Similarly, for any close to zero smooth function $\psi(x)$ which is supported inside $[x_{b}+\delta,x_{b}+2\tau-\delta]$, for the map $\bar{F}$ given by (20), we define the time-energy chart $\phi_{\psi}^{b}$ in $N^{b}\cup F(N^{b})$ by the rule

[TABLE]

such that the identity

[TABLE]

holds.

With this choice of the time-energy charts, for small $\psi$ we have the link-splitting function $M^{a}(S_{\psi}\circ F,\phi_{\psi}^{a})$ (if $\psi$ is supported inside $[x_{a}-2\tau,x_{a}]$) or $M^{b}(S_{\psi}\circ F,\phi_{\psi}^{b})$ (if $\psi$ is supported inside $[x_{b},x_{b}+2\tau]$). This defines the operators $\mathcal{M}^{a}:\psi\to M^{a}(S_{\psi}\circ F,\phi_{\psi}^{a})$ and $\mathcal{M}^{b}:\psi\to M^{b}(S_{\psi}\circ F,\phi_{\psi}^{b})$ acting from the space of smooth functions supported inside $[x_{a}-2\tau,x_{a}]$ or, respectively, inside $[x_{b},x_{b}+2\tau]$, to the space of $\tau$-periodic functions of the same smoothness.

5.2 Regularity of the graph transform operator

The regularity of the operators $\mathcal{M}^{a}$ and $\mathcal{M}^{b}$ depends on the smoothness class of $\psi$. We choose it to be $C^{k-2}$. For $\delta>0$ small, for every $x\in\mathbb{R}$ we denote by $C^{k-2}_{0}([x-2\tau+\delta,x-\delta],\mathbb{R})$ the Banach space of real $C^{k-2}$-functions supported inside $[x-2\tau+\delta,x-\delta]$, endowed with the $C^{k-2}$-norm. We have the crucial result:

Proposition 5.8**.**

The operators

[TABLE]

[TABLE]

are of class $C^{1}$ in a small neighborhood of zero, and depend continuously on the map $F$.

Proposition 5.8 follows immediately from the two following lemmas:

Lemma 5.9**.**

The functions $w^{u}_{a}(S_{\psi}\circ F,\phi_{\psi}^{a})$ and $w^{u}_{b}(S_{\psi}\circ F,\phi_{\psi}^{b})$ are independent of $\psi$, and depend continuously on $F$:

[TABLE]

Lemma 5.10**.**

The following operators are of class $C^{1}$ and depends continuously on $F$:

[TABLE]

[TABLE]

Lemma 5.10 was rather unexpected: the proof works because $S_{\psi}$ induces a graph transform which is a translation (which is a smooth operator).

Proof of Lemma 5.9.

Let $\psi$ be supported inside $[x_{a}-2\tau+\delta,x_{a}-\delta]$. Denote $\bar{F}:=S_{\psi}\circ F$. The graph of the function $w^{u}_{a}$ is the curve $\phi_{\psi}^{a}(W^{a}(P;\bar{F}))\cap[x_{a}-\tau,x_{a}]\times\mathbb{R}$. It follows from our choice of the chart $\phi_{\psi}^{a}$ (see the first line of (21)) that this curve is the image by $\phi^{a}\circ F$ of an arc of the curve $\ell^{u}_{a}=\bar{F}^{-1}(W^{a}(P;\bar{F}))\cap[x_{a}-\tau,x_{a}]\times\mathbb{R}$, which is an arc of $W^{a}(P;\bar{F})$ lying at $x\geq x_{a}$. The set $W^{a}(P;\bar{F})\cap\{x\geq x_{a}\}$ is a part of the unstable manifold of $P$ which depends only on the dynamics at $\{x\geq x_{a}\}$. Since $\psi$ is zero at $x\geq x_{a}$, the map $S_{\psi}$ is identity there, hence $\bar{F}|_{\{x\geq x_{a}\}}$ equals $F|_{\{x\geq x_{a}\}}$, and $W^{a}(P;\bar{F})\cap\{x\geq x_{a}\}$ equals $W^{a}(P;F)\cap\{x\geq x_{a}\}$, in particular the curve $\ell^{u}_{a}$ does not depend on $\psi$, i.e. it is the same for $F$ and $\bar{F}$.

Thus,

[TABLE]

so $w^{u}_{a}(\bar{F},\phi_{\psi}^{a})=w^{u}_{a}(F,\phi^{a})$ is the same for all small $\psi$.

Exactly in the same way, just by changing the index $a$ to $b$ and the interval $[x_{a}-\tau,x_{a}]$ to $[x_{b},x_{b}+\tau]$, we obtain that when $\psi$ is supported inside $[x_{b}+\delta,x_{b}+2\tau-\delta]$ the function $w^{u}_{b}(\bar{F},\phi_{\psi}^{b})$ is independent of $\psi$. ∎

Proof of Lemma 5.10.

For $c\in\{a,b\}$, let us show that $w^{s}_{c}(\bar{F},\phi_{\psi}^{c})$ is a $C^{1}$ function of $\psi\in C^{k-2}(\mathbb{R})$, with $\bar{F}:=S_{\psi}\circ F$. Again, we start with the case $c=a$ and $\psi$ supported inside $[x_{a}-2\tau+\delta,x_{a}-\delta]$ and derive the expression for $w^{s}_{a}(\bar{F},\phi_{\psi}^{a})$ in this case.

The graph of $w^{s}_{a}(\bar{F},\phi_{\psi}^{a})$ is the curve $\phi_{\psi}^{a}\bar{\ell}_{s}^{a}$ where $\bar{\ell}_{s}^{a}$ is the piece of $W^{a}(Q;\bar{F})$ in the intersection with $(\phi_{\psi}^{a})^{-1}([x_{a}-\tau,x_{a}]\times\mathbb{R})$. The curve $W^{s}(Q;\bar{F})$ (a half of the stable manifold of $Q$) is obtained by iterations of its small, adjoining to $Q$ part by the map $\bar{F}^{-1}$. The maps $\bar{F}^{-1}$ and $F^{-1}$ coincide at $x\leq x_{a}-2\tau+\delta$, and both are close to the map $\mathring{F}^{-1}$ which takes the line $\{x=x_{a}-2\tau+\delta\}\cap N^{a}$ into $x=x_{a}-\tau+\delta$, so the piece of $W^{s}(Q;\bar{F})$ between $Q$ and $x=x_{a}-\tau+\delta/2$ does not move as $\psi$ varies, i.e., it is the same as for the map $F$.

This means that for all small $\psi$, the image by $\bar{F}$ of the curve $\bar{\ell}^{s}_{a}$ lies inside the piece of $W^{s}(Q;F)$ to the left of $x=x_{a}-\tau+\delta/2$, i.e., $\bar{F}\bar{\ell}^{s}_{a}\subset F\ell^{s}_{a}$, where the $\psi$-independent curve $\ell^{s}_{a}$ is a piece of $W^{s}(Q;F)$ in the intersection with $\{x\in[x_{a}-\tau-\delta/2,x_{a}+\delta/2]\}$. So

[TABLE]

as given by (21),(20).

Formula (25) states that the graph of $w^{s}_{a}(\bar{F},\phi_{\psi}^{a})$ is the image of the $\psi$-independent curve $\ell^{s}_{a}$ by the map $\phi^{a}\circ S_{-\psi}\circ F^{-1}\circ S_{-\psi}\circ F$. Thus, the regularity of the map $\psi\mapsto w^{s}_{a}(\bar{F},\phi_{\psi}^{a})$ is determined by the regularity of the corresponding graph transform operators.

The following defines the graph-transform operator $\mathcal{F}^{\#}$ associated with a smooth map $\mathcal{F}$.

Fact 5.11** (See Thm 2.2.5. P. 145 [Ham82]).**

Consider a curve ${\mathcal{L}}=\{(x,y):y=w(x)\}$ in an $(x,y)$-plane, where $w$ is a $C^{s}$-smooth function defined on some closed interval, and a $C^{n}$-smooth map ${\mathcal{F}}:(x,y)\mapsto(p(x,y),q(x,y))$ ($n\geq s$) defined in a neighborhood $U$ of ${\mathcal{L}}$. Then, under the condition $\partial_{x}p(x,y)+\partial_{y}p(x,y)Dw(x)\neq 0$ everywhere in $U$, the image $\mathcal{FL}$ is a curve of the form $y=\tilde{w}(x)$ where $\tilde{w}\in C^{s}$. Moreover the operator $\mathcal{F}^{\#}$ which takes the $C^{s}$-function $w$ to the $C^{s}$-function $\tilde{w}$ is of regularity class $C^{n-s}$.

Since $\phi^{a}(\ell^{s}_{a})$ is the graph of $w^{s}_{a}(F,\phi^{a})$, we have from (25) that

[TABLE]

Since $\phi^{a}$ and $F$ are at least of class $C^{k-1}$, the graph-transform operators $(\phi^{a})^{\#}$, $(F^{-1})^{\#}$, and $(F\circ(\phi^{a})^{-1})^{\#}$ have regularity at least $C^{1}$ when act from $C^{k-2}$-smooth functions to $C^{k-2}$-smooth functions. We cannot use the same fact for the graph transform operator induced by $S_{-\psi}$ since the latter map is only of class $C^{k-2}$. However, the map $S_{-\psi}$ is given by $(x,y)\mapsto(x,y-\psi(x))$, so the associated graph transform operator $(S_{-\psi})^{\#}$ sends a function $w$ to $w-\psi$. Thus, it is linear in both $w$ in $\psi$, i.e., it is of class $C^{\infty}$ with respect to both $w$ and $\psi$ (irrespective of their class of smoothness).

Altogether, this implies that the map $\psi\mapsto w^{s}_{a}(\bar{F},\phi_{\psi}^{a})$ given by (26) is of class $C^{1}$.

Now, let us handle the case $c=b$.

We need to derive the expression for $w^{s}_{b}(\bar{F},\phi_{\psi}^{b})$. The graph of this function is the curve $\phi_{\psi}^{b}\bar{\ell}^{s}_{b}$ where $\bar{\ell}^{s}_{b}$ is the piece of $W^{b}(Q;\bar{F})$ in the intersection with $(\phi_{\psi}^{b})^{-1}(\{x\in[x_{b},x_{b}+\tau]\})$. The curve $W^{b}(Q;\bar{F})$ is close to $W^{b}(Q;\mathring{F})$. Recall that $W^{b}(Q,\mathring{F})$ coincides with $W^{b}(P,\mathring{F})$ and forms a heteroclinic link $\mathring{L}_{b}$. It intersects the vertical strip $V_{b}=\{x\in[x_{b},x_{b}+2\tau]\}$ twice, along two straight line segments $\mathring{D}_{2}^{b}=[x_{b},x_{b}+2\tau]\times\{y=y_{b}\}$ and $\mathring{D}_{4}^{b}=\mathring{F}^{4}(\mathring{D}_{2}^{b})\cup\mathring{F}^{6}(\mathring{D}_{2}^{b})=[x_{b},x_{b}+2\tau]\times\{y=y_{b}^{\prime}\}$ for some $y_{b}^{\prime}<y_{b}$. The piece of $W^{b}(Q,\mathring{F})$ between $Q$ and the left end of $\mathring{D}_{4}^{b}$ lies entirely in the region $x\leq x_{b}$, i.e., to the left of the vertical strip $V_{b}$. Also, the map $\mathring{F}$ in a small neighborhood of $\mathring{D}_{4}^{b}$ is given by

[TABLE]

(as implied by the link suitability Definition 4.4). It follows that the curve $W^{b}(Q,\bar{F})$ intersects the straight line $x=x_{b}$ at a point $Z_{b}$ with the $y$-coordinate close to $y_{b}$ such that the piece of $W^{b}(Q,\bar{F})$ between $Q$ and $\bar{F}^{7}Z_{b}$ lies entirely in the region $x<x_{b}+\tau/2+\delta/2$. Since $\psi$ is supported inside $[x_{b}+\delta,x_{b}+2\tau-\delta]$, it follows that $F=\bar{F}$ near this piece of $W^{b}(Q,\bar{F})$. Therefore, the piece of $W^{b}(Q,\bar{F})$ between $Q$ and $x=x_{b}+\tau/2+\delta/2$ is unmoved as $\psi$ varies, so it coincides with the corresponding piece of $W^{b}(Q,F)$.

This piece contains the point $\bar{F}^{7}Z_{b}$, hence it contains the curve $\bar{F}^{7}\bar{\ell}_{s}^{b}$ for all small $\psi$. It follows that there exists a $\psi$-independent curve $\ell^{s}_{b}$ so that $\bar{F}^{7}\bar{\ell}^{s}_{b}\subset F^{7}\ell^{s}_{b}$ and $\ell^{s}_{b}$ is a piece of of $W^{s}(Q;F)$ in the intersection with $\{x\in[x_{b}-\delta/2,x_{b}+\tau+\delta/2]\}$. So

[TABLE]

see (23),(20).

Since $\phi^{b}(\ell^{s}_{b})$ is the graph of $w^{s}_{b}(F,\phi^{b})$, we have from (28) that

[TABLE]

Like we did it for the function $w^{s}_{a}$ given by (26), we obtain that the map $\psi\mapsto w^{s}_{b}(\bar{F},\phi_{\psi}^{b})$ defined by (29) is of class $C^{1}$. ∎

The continuity of the operators $\mathcal{M}^{a}$ and $\mathcal{M}^{b}$ with respect to $F$ allows us to obtain enough information about them by computing them for $F=\mathring{F}$, which we do in the two following lemmas.

Lemma 5.12**.**

When $F=\mathring{F}$ :

[TABLE]

Proof.

For $F=\mathring{F}$, the link $L^{a}$ exists and the fundamental interval $D^{a}$ is straight, so the curves $W^{a}(P;\mathring{F})$ and $W^{a}(Q;\mathring{F})$ coincide for $x\in[x_{a}-\tau,x_{a}]$ and lie in the straight line $y=y_{a}$. The map $\phi^{a}$ for $\mathring{F}$ is identity, so we have $w^{u}_{a}(\mathring{F},\phi^{a})=w^{s}_{a}(\mathring{F},\phi^{a})=y_{a}$. The map $\mathring{F}$ is the translation to $(-\tau,0)$ (see (17)). Plugging this information into (26) gives (30) immediately. ∎

Lemma 5.13**.**

When $F=\mathring{F}$,

[TABLE]

for $x\in[x_{b},x_{b}+\tau]$.

Proof.

When $F=\mathring{F}$, we have $\phi^{b}=id$, so

[TABLE]

(see (29)). Recall that $W^{b}(Q,\mathring{F})$ coincides with $W^{b}(P,\mathring{F})$ and intersects the vertical strip $V_{b}$ on two straight line segments, $\mathring{D}_{2}^{b}=[x_{b},x_{b}+2\tau]\times\{y=y_{b}\}$ and $\mathring{D}_{4}^{b}=[x_{b},x_{b}+2\tau]\times\{y=y_{b}^{\prime}\}$, so

[TABLE]

Also, the map $\mathring{F}^{7}$ from a small neighborhood of $[x_{b},x_{b}+\tau]\times\{y=y_{b}\}$ to a small neighborhood of $[x_{b},x_{b}+\tau]\times\{y=y_{b}^{\prime}\}$ acts as

[TABLE]

(see Definition 4.4). Therefore,

[TABLE]

and

[TABLE]

Next, by (27), we obtain

[TABLE]

Repeating the same procedure two more times, we obtain

[TABLE]

The map $\mathring{F}^{-1}$ takes the rectangle $K:=[x_{b}+3\tau/2,x_{b}+2\tau]\times[y_{b}^{\prime}-\varepsilon,y_{b}^{\prime}+\varepsilon]$ (for some small $\varepsilon$) to the right of the strip $V_{b}$, and its next image $\mathring{F}^{-2}K$ also lies to the right of $V_{b}$. We recall that the the function $\psi$ vanishes there and $S_{-\psi}=id$. Therefore, $F^{-1}\circ(S_{-\psi}\circ F^{-1})^{2}=F^{-3}$ on $K$. As the curve $\{y=w_{4}(x)\}$ lies in $K$ for small $\psi$, this implies that

[TABLE]

By the link suitability conditions (Definition 4.4), the map $F^{-3}$ takes $K$ into a small neighborhood of $\mathring{D}_{2}^{b}$ and is given by

[TABLE]

This gives

[TABLE]

Since $\mathring{F}^{-1}$ is the translation to $(-\tau,0)$ near $\mathring{F}(D_{2}^{b})$, we finally obtain that

[TABLE]

which implies (31) by (32),(33). ∎

5.3 Proof of Lemma 5.2: Restoring the link $L^{a}$

In order to prove Lemma 5.2, we show that for every $\omega$-preserving $F$ which is $C^{k}$-close to $\mathring{F}$ there exists a $C^{k-2}$-smooth function $\psi$, supported in $x\in(x_{a}-2\tau,x_{a})$, such that

[TABLE]

where the $C^{k-2}$-smooth time-energy chart $\phi_{\psi}^{a}$ for the map $S_{\psi}\circ F$ is defined by (21). We will solve this equation for $\psi$ by reducing it to a fixed point problem for a certain contracting operator.

First, we restrict the class of perturbation functions $\psi$; namely, we will take them in the form

[TABLE]

where $\tilde{\psi}$ is a $\tau$-periodic function, and $\rho\in C^{\infty}(\mathbb{R},[0,1])$ has support in $[x_{a}-2\tau,x_{a}]$ and satisfies $\rho(x)+\rho(x-\tau)=1$ for every $x\in[x_{a}-\tau,x_{a}]$. Then the operator

[TABLE]

which provides a correspondence between $\tilde{\psi}$ and the link-splitting function for the map $S_{\rho\tilde{\psi}}\circ F$, takes a small ball around zero in the space $C^{k-2}(\mathbb{R}/(\tau\mathbb{Z}),\mathbb{R})$ of $\tau$-periodic functions into the same space.

By Lemma 5.8, the operator $\mathcal{M}^{a}_{\rho}$ is of class $C^{1}$ on $C^{k-2}(\mathbb{R}/(\tau\mathbb{Z}),\mathbb{R})$. For $F=\mathring{F}$, we have $\mathcal{M}^{a}_{\rho}=id$. Indeed, by (30),

[TABLE]

Due to a continuous dependence on $F$, the operator

[TABLE]

is a contraction in a neighborhood of zero in $C^{k-2}(\mathbb{R}/(\tau\mathbb{Z}),\mathbb{R})$ for all $F\in{\rm Diff}^{k}_{\omega}(\mathbb{D})$ which are $C^{k}$-close to $\mathring{F}$. Therefore, it has a unique fixed point $\tilde{\psi}$ near zero in $C^{k-2}(\mathbb{R}/(\tau\mathbb{Z}),\mathbb{R})$, for each such $F$.

The operator $\mathcal{M}^{a}_{\rho}$ vanishes at this fixed point. By construction, the corresponding function $\psi=\rho\tilde{\psi}$ solves equation (34), i.e., the link-splitting function is identically zero for the map $\bar{F}=S_{\psi}\circ F$, meaning that the link $L^{a}$ persists for this map. Lemma 5.2 is proven.∎

5.4 Proof of Lemma 5.3: Restoring the link $L^{b}$

In order to prove Lemma 5.3, we show that for every $\omega$-preserving $F$ which is $C^{k}$-close to $\mathring{F}$, if the link $L_{a}$ persists for the map $F$, then there exists a $C^{k-2}$-smooth function $\psi$, supported in $x\in(x_{b},x_{b}+2\tau)$, such that

[TABLE]

where the $C^{k-2}$ time-energy chart $\phi_{\psi}^{b}$ for the map $\bar{F}=S_{\psi}\circ F$ is defined by (23).

In order to resolve equation (35), we will use the following property of the link-splitting function $M^{b}$.

Lemma 5.14**.**

If the link $L^{a}$ is persistent for the map $F$, then the link-splitting function $M^{b}(S_{\psi}\circ F,\phi_{\psi}^{b})$ has zero mean for every $\psi$ supported by $(x_{b},x_{b}+2\tau)$:

[TABLE]

Proof.

We may always assume that the map $\phi_{\psi}^{b}$ is the restriction to $N_{a}$ of an area-preserving diffeomorphism or $\mathbb{R}^{2}$ (the possibility of the symplectic extension of an area-preserving diffeomorphism from a disc to the whole $\mathbb{R}^{2}$ is a standard fact; see e.g. Corollary 4 in [Avi10]).

Let the points $Z^{u}\in W^{b}(P,\bar{F})\cap N^{b}$ and $Z^{s}\in W^{b}(Q,\bar{F})\cap N^{b}$ have the same $x$-coordinate $x_{b}$, i.e., $Z^{u}=(x_{b},w^{u}(\bar{F},\phi_{\psi}^{b})(x_{b}))$, $Z^{s}=(x_{b},w^{s}(\bar{F},\phi_{\psi}^{b})(x_{b}))$. As the map $\bar{F}$ is the translation to $(\tau,0)$ in the time-energy coordinates in $N^{b}$, the points $\bar{F}Z^{s}$ and $\bar{F}Z^{u}$ have the same $x$-coordinate $x_{b}+\tau$, and the image of the vertical segment connecting $Z^{s}$ and $Z^{u}$ is the vertical segment connecting $\bar{F}Z^{s}$ and $\bar{F}Z^{u}$.

Since the support of $\psi$ lies in $(x_{b},x_{b}+2\tau)$, the maps $F$ and $\bar{F}$ coincide in a neighborhood of $L_{a}$, so the link $L_{a}$ is not split for all $\psi$ under consideration. Thus, we may consider a region $\mathcal{D}$ bounded by the link $L^{a}$, the piece of $W^{b}(P,\bar{F})$ between $Z^{u}$ and $P$, the piece of $W^{b}(Q,\bar{F})$ between $Q$ and $Z^{s}$, and the vertical segment that connects $Z^{s}$ and $Z^{u}$. The region $\bar{F}\mathcal{D}$ has the same area as $\mathcal{D}$. It is bounded by the link $L^{a}$, the piece of $W^{b}(P,\bar{F})$ between $\bar{F}Z^{u}$ and $P$, the piece of $W^{b}(Q,\bar{F})$ between $Q$ and $\bar{F}Z^{s}$, and the vertical segment that connects $\bar{F}Z^{s}$ and $\bar{F}Z^{u}$. The equality of the areas means that the area of the region between the curves $W^{b}(P,\bar{F}):\{y=w^{u}(\bar{F},\phi_{\psi}^{b})(x)\}$ and $W^{b}(Q,\bar{F}):\{y=w^{s}(\bar{F},\phi_{\psi}^{b})(x)\}$ at $x\in[x_{b},x_{b}+\tau]$ is zero (see fig. 5).

This means that $\int_{x_{b}}^{x_{b}+\tau}(w^{u}(\bar{F},\phi_{\psi}^{b})-w^{s}(\bar{F},\phi_{\psi}^{b}))dx=0$, which proves the lemma (see Definition 5.6 of the link-splitting function). ∎

Now, like in the previous Section, we restrict the class of perturbation functions $\psi$:

[TABLE]

where $\tilde{\psi}$ is a $\tau$-periodic function with the zero mean, and $\rho\in C^{\infty}(\mathbb{R},[0,1])$ has support in $[x_{b}+\delta,x_{b}+2\tau-\delta]$ for $\delta>0$ small, and satisfies $\rho(x)+\rho(x+\tau)=1$ for every $x\in[x_{b},x_{b}+\tau]$. Then, if the link $L_{a}$ persists for the map $F$, the operator

[TABLE]

takes a small ball around zero in the space $C^{k-2}_{0}(\mathbb{R}/(\tau\mathbb{Z}),\mathbb{R})$ of $\tau$-periodic functions with the zero mean into the same space (by Lemma 5.14).

By Lemma 5.8, the operator $\mathcal{M}^{b}_{\rho}$ is of class $C^{1}$ on $C^{k-2}_{0}(\mathbb{R}/(\tau\mathbb{Z}),\mathbb{R})$. It follows easily from (31) that if $F=\mathring{F}$, then

[TABLE]

Note that the space $C^{k-2}_{0}(\mathbb{R}/(\tau\mathbb{Z}),\mathbb{R})$ is a Banach space when endowed with the following norm:

[TABLE]

This norm includes evaluation of the derivatives only (still it is a well-defined norm - if two functions with the zero mean have the same derivative, they coincide).

By (31) and by the continuous dependence of $\mathcal{M}^{b}$ on $F$ (see Lemma 5.8), the operator $id-\mathcal{M}^{b}_{\rho}$ is $C^{1}$-close to the linear operator $\tilde{\psi}\mapsto\bar{\psi}$ where

[TABLE]

We have

[TABLE]

so this operator is, obviously, a contraction in the norm (37). Therefore, $id-\mathcal{M}^{b}_{\rho}$ is a contraction on $C^{k-2}_{0}(\mathbb{R}/(\tau\mathbb{Z}),\mathbb{R})$ for all $F$ which are $C^{k}$-close to $\mathring{F}$, provided $F$ is area-preserving and the link $L_{a}$ persists for $F$.

Thus, $id-\mathcal{M}^{b}_{\rho}$ has a fixed point $\tilde{\psi}$. The corresponding function $\psi=\rho\tilde{\psi}$ solves equation (35); Lemma 5.3 is proven.∎

5.5 Proof of Proposition 2.5

The previous results allow for construction of $C^{r}$-maps with stochastic islands for any finite $r$. Below we prove Proposition 2.5 which deals with the $C^{\infty}$ case. Let $\hat{f}\in{\rm Diff}^{r}_{\omega}(\mathbb{M})$ have a stochatics island $\mathcal{I}$ bounded by bilinks $(L^{a}_{i}\cup L_{i}^{b})_{i=1}^{m}$ so that each bilink $C_{i}:=L^{a}_{i}\cup L_{i}^{b}$ is a $C^{r}$-smooth circle (without break points). Let us show that arbitrarily close in $C^{r}$ to $\hat{f}$ there exists a map $\hat{f}_{\infty}\in{\rm Diff}^{\infty}_{\omega}(\mathbb{M})$ for which the bilinks persist.

Choose a map $f\in{\rm Diff}^{\infty}_{\omega}(\mathbb{R}^{2})$ which is sufficiently close in $C^{r}$ to $\hat{f}$; such exists by Zehnder smoothing theorem [Zeh77, Thm. 1]. The bilinks do not need to persist for $f$. To restore them, the idea is to smoothen the circles $C_{i}$ to $C^{r}$-close circles $\tilde{C}_{i}$ which are of class $C^{\infty}$. Then we will perform a local surgery to construct $\hat{f}_{\infty}$ of class $C^{\infty}_{\omega}$ which is $C^{r}$-close to $f$ and such that $\hat{f}_{\infty}(\sqcup_{i}\tilde{C}_{i})=\sqcup_{i}\tilde{C}_{i}$. By local maximality of the hyperbolic continuation $(\tilde{P}_{i},\tilde{Q}_{i})_{i}$ of saddle points $(P_{i},Q_{i})_{i}$ defining the bilinks $(L^{a}_{i}\cup L_{i}^{b})_{i=1}^{m}$, it follows that each $\tilde{P}_{i},\tilde{Q}_{i}$ belongs to $\tilde{C}_{i}$. Moreover, as the unique invariant curves which contain $\tilde{P}_{i}$ or $\tilde{Q}_{i}$ are their local stable and unstable manifolds, we obtain that the circles $\tilde{C}_{I}$ are heteroclinic bi-links, i.e., the bilinks are persistent for $\hat{f}_{\infty}$.

Consequently, we need only to prove the following

Lemma 5.15**.**

There exists a collection of $C^{\infty}$-circles $\tilde{C}_{i}$ which are $C^{r}$-close to $C_{i}$, and a map $\hat{f}_{\infty}$ of class $C^{\infty}_{\omega}$ which is $C^{r}$-close to $f$, such that $\hat{f}_{\infty}(\sqcup_{i}\tilde{C}_{i})=\sqcup_{i}\tilde{C}_{i}$.

To prove this lemma, we show below

Lemma 5.16**.**

There exists $\epsilon>0$, so that for all $i$, there exist:

• •

a $C^{\infty}$-circle $\tilde{C}_{i}$ which is $C^{r}$-close to $C_{i}$, such that $C_{i}$ and $\tilde{C}_{i}$ bound disks of the same area,

• •

a $C^{\infty}_{\omega}$ tubular neighborhood $N:=\sqcup_{i}\tilde{C}_{i}\times[-\epsilon,\epsilon]\to\mathbb{R}^{2}$ such that $N^{-1}(\hat{f}(\sqcup_{i}\tilde{C}_{i}))$ is a graph of a $C^{r}$-small section.

Proof of Lemma 5.15.

By Lemma 5.16, the collection of circles $N^{-1}(f(\sqcup_{i}\tilde{C}_{i}))$ is given by the equation $h=\sigma(x,i)$ where $x$ and $h$ are coordinates in $\tilde{C}_{i}$ and, respectively, $[-\epsilon,\epsilon]$, and $\sigma$ is a $C^{r}$-small function of class $C^{\infty}$. Denote $\phi:(x,h)\mapsto(x,h-\sigma(x))$; this map is of class $C^{\infty}$ and $C^{r}$-close to the identity.

Observe that $V_{i}:=N(\tilde{C}_{i}\times[-\epsilon/2,\epsilon/2])$ is a neighborhood of $\tilde{C}_{i}$ which is bounded by two smooth circles close to $N(\tilde{C}_{i}\times\{-\epsilon/2,\epsilon/2\})$. Both $V_{i}$ and $N(\tilde{C}_{i}\times[-\epsilon/2,\epsilon/2])$ bound a disk of the same volume. Furthermore, $\psi:=N\circ\phi\circ N^{-1}|_{V_{i}}$ is of class $C^{\infty}$ and $C^{r}$-close to the canonical inclusion of $\sqcup_{i}V_{i}\hookrightarrow\mathbb{R}^{2}$. Consequently, by Corollary 4.8, we can extend $\psi$ to a $C^{\infty}_{\omega}$-diffeomorphism of $\mathbb{R}^{2}$. We observe that $\psi$ sends $\sqcup_{i}f(\tilde{C}_{i})$ to $\sqcup_{i}\tilde{C}_{i}$. Thus $\hat{f}_{\infty}:=\psi\circ f$ satisfies the requirements of the lemma.∎

Proof of Lemma 5.16.

The existence of a sympletic tubular neighborhood is given by [Wei71], however, the statement of this theorem does not imply that $N^{-1}(\hat{f}(\tilde{C}_{i}))$ is $C^{r}$-small when $\tilde{C}_{i}$ is close to $C_{i}$. Hence we need to develop this theorem for our particular case.

Let $\rho_{i}$ be a $C^{r}$-submersion from a neighborhood of $C_{i}$ into $\mathbb{R}$, so that $C_{i}=\rho_{i}^{-1}(\{0\})$.

Let $\mathring{\rho}_{i}$ be of class $C^{\infty}$ and $C^{r}$-close to $\rho_{i}$. Let $\mathring{C}_{i}$ be the circle $\mathring{\rho}_{i}^{-1}(\{0\})$. By adding, if necessary, a small constant to $\mathring{\rho}_{i}$, we can achieve that $\mathring{C}_{i}$ and $C_{i}$ bound disks of the same area. Let $L_{h}$ be the length of the circle $\mathring{\rho}_{i}^{-1}(\{h\})$. Let $H$ be such that $\nabla_{z}H:=L_{\tilde{\rho}_{i}(h)}\nabla\mathring{\rho}_{i}/\|\nabla\mathring{\rho}_{i}\|$. We observe that the Hamitonian flow $\phi^{t}$ of $H$ leaves $\mathring{\rho}_{i}$ invariant. Also, all the orbits close to $\mathring{C}_{i}$ are periodic with period 1. Hence, the obits of $H$ define a trivial $C^{\infty}$-fibration by circles, and $\phi^{t}$ defines a $C^{\infty}_{\omega}$-conservative diffeomorphism $\mathring{N}_{i}$ from $\mathring{C}_{i}\times[-\epsilon,\epsilon]$ onto a neighborhood $V_{i}$ of $C_{i}$. In this chart, $\mathring{N}_{i}^{-1}\mathring{C}_{i}$ is the graph of a section $\sigma_{i}$ of class $C^{r}$, with zero integral, but a priori only $C^{r-1}$-small.

Take a $C^{\infty}$-circle $\tilde{C}_{i}$, which bounds a disk of the same volume as $C_{i}$ and is $C^{r}$-close to $C_{i}$. The circle $\tilde{C}_{i}$ is the image by $N_{i}$ of the graph of a section $\tilde{\sigma}_{i}$ which is $C^{r}$-close to $\sigma_{i}$ and has zero integral. We endow $\tilde{C}_{i}$ with the tubular neighborhood:

[TABLE]

As $f(\sqcup_{i}\tilde{C}_{i})$ is $C^{r}$-close to $\sqcup\tilde{C}_{i}$, it is the graph of a $C^{r}$-small section in this tubular neighborhood. Now note that the tubular neighborhood $N$ of $\sqcup_{i}\tilde{C}_{i}$ whose restriction to $\tilde{C}_{i}\times[-\epsilon,\epsilon]$ is $N_{i}$ satisfies the required properties. ∎

6 Rescaling Lemma

Consider a symplectic $C^{r+1}$-diffeomorphism $f$ of a two-dimensional unit disc $\mathbb{D}$ into $\mathbb{R}^{2}$, $r\geq 1$. Let $f$ have a saddle periodic point $O$. Assume that there exists a homoclinic band, i.e. the intersection of the stable and unstable manifolds of $O$ contains a non-empty open interval $J$.

Every orbit starting at $cl(J)$ is homoclinic, i.e., it tends to the orbit of $O$ both at forward and backward iterations of $f$, so the closure $C$ of the set of these orbits is this set itself and the orbit of $O$. Let $\mathcal{U}$ be a small neighborhood of $C$.

The following statement is a different999and more developed, since it includes many rounds near a homoclinic tangency, see also Remark 6.1. version of the rescaling lemma from [GST07].

Rescaling lemma. Take any odd natural $N$ and any $\varepsilon>0$. There exist $N$ area-preserving diffeomorphisms $\Phi_{i}:\mathbb{R}^{2}\to\mathbb{R}^{2}$, $\varepsilon$-close to identity in the $C^{r}$-norm, and a $C^{r}$-smooth diffeomorphism $Q:\mathbb{R}^{2}\to\mathbb{R}^{2}$ with a constant Jacobian, such that $Q(\mathbb{D})\subset U$, for which the following holds. For every sequence of $C^{r}$-smooth functions $\psi_{i}:\mathbb{R}^{1}\to\mathbb{R}^{1}$, $i=1,\dots,N$, and any, arbitrarily small $\delta>0$, $\nu>0$, there exist an area-preserving diffeomorphism $g:\mathbb{R}^{2}\to\mathbb{R}^{2}$, such that $(g-id)$ is bounded by $\delta$ in the $C^{r}$-norm and supported in a radius-$\nu$ disc in $\mathcal{U}$, and a positive integer $n$ such that the renormalization of the $n$-th iteration of the perturbed diffeomorphism $\hat{f}=g\circ f$ restricted to the disc $Q(\mathbb{D})$ is the product of Henon-like maps as given by the following formula

[TABLE]

where the symplectic diffeomorphisms $H_{\psi_{i}}:(x,y)\mapsto(\bar{x},\bar{y})$ are given by

[TABLE]

Remark 6.1*.*

It is important for us that the maps $\Phi_{i}$ in formula (38) are independent of the choice of the functions $\psi_{1},\dots,\psi_{N}$.

In particular, the Rescaling Lemma implies Proposition 2.4, which we formulate in a more detailed fashion as follows:

Corollary F**.**

Take any symplectic $C^{r}$-diffeomorphism $F:{\mathbb{D}}\to\mathbb{R}^{2}$. For any $\varepsilon>0$ there exists a $C^{r}$-smooth diffeomorphism $Q:\mathbb{R}^{2}\to\mathbb{R}^{2}$ with a constant Jacobian, such that $Q(\mathbb{D})\subset U$, and a symplectic diffeomorphism $\hat{F}$, which is $\varepsilon$-close to $F$ in the $C^{r}$-norm, for which the following holds. For every $C^{r}$-smooth function $\Psi:\mathbb{R}^{1}\to\mathbb{R}^{1}$ and any, arbitrarily small $\delta>0$, $\nu>0$, there exist an area-preserving diffeomorphism $g:\mathbb{R}^{2}\to\mathbb{R}^{2}$, such that $g-id$ is bounded by $\delta$ in the $C^{r}$-norm and supported in a radius-$\nu$ disc in $\mathcal{U}$, and a positive integer $n$ such that the renormalization of the $n$-th iteration of the perturbed diffeomorphism $\hat{f}=g\circ f$ restricted to the disc $Q(\mathbb{D})$ is given by

[TABLE]

where the symplectic diffeomorphism $S_{\psi}:(x,y)\mapsto(\bar{x},\bar{y})$ is given by

[TABLE]

Proof.

By Theorem 2 of [Tur03], every symplectic $C^{r}$-diffeomorphism of any two-dimensional disc to $\mathbb{R}^{2}$ can be arbitrarily well approximated by a composition of an even number of maps of the form (39). In particular, for the given $C^{r}$-diffeomorphism $F$, there exists an even number $N^{\prime}$ of the functions $\psi_{1},\dots,\psi_{N^{\prime}}$ such that

[TABLE]

where ${\cal R}$ is the linear rotation $(x,y)\mapsto(-y,x)$. Since ${\cal R}^{-1}=H_{\psi=0}$ is the map of the form (39), it follows that

[TABLE]

where $\psi_{N^{\prime}+1}\equiv 0$. Since the map $S_{\psi}$ from (41) satisfies $S_{\psi}=H_{\psi}\circ{\cal R}^{-1}$, by applying Rescaling Lemma with the sequence of functions $\psi_{1},\dots,\psi_{N^{\prime}},\psi_{N^{\prime}+1}\equiv 0,\psi_{N^{\prime}+2}\equiv 0,\psi_{N^{\prime}+3}=\psi$, we find that the perturbed map $\hat{f}$ can be constructed such that (40) is fulfilled with

[TABLE]

[TABLE]

(we can use the Rescaling Lemma because the number $N=N^{\prime}+3$ of the functions $\psi_{i}$ is odd here).

Since the maps $\Phi_{i}$ can be made as close to identity as we want, we can make

[TABLE]

which implies, by (42), that $\|\hat{F}-F\|_{C^{r}({\mathbb{D}},\mathbb{R}^{2})}<\varepsilon$, as required. Since the maps $\Phi_{i}$ are independent of the choice of $\psi$, it follows that $\hat{F}$ is independent of the function $\psi$ either. ∎

In the following Sections we complete the proof of the main theorem by proving the Rescaling Lemma.

6.1 Local behavior near a nonlinear saddle.

Let $U_{0}$ be a sufficiently small neighborhood of $O$. Let $s$ be the period of the saddle point $O$; denote:

[TABLE]

It is straightforward to show (see Section 2.1 in [GST07]) that one can introduce symplectic $C^{r+1}$-coordinates $(x,y)$ in $U_{0}$ such that the local stable and unstable manifolds $W^{s}_{loc}(O)$ and $W^{u}_{loc}(O)$ in $U_{0}$ get straightened (i.e., they acquire equations $y=0$ and $x=0$, respectively) and the restrictions of $T_{0}$ onto $W^{s}_{loc}(O)$ and $W^{u}_{loc}(O)$ become linear. This means that the map $T_{0}:(x,y)\mapsto(\bar{x},\bar{y})$ takes the following form

[TABLE]

where $0<|\lambda|<1$ and

[TABLE]

It also follows that

[TABLE]

Proof of equalities (45).

Since $T_{0}$ is symplectic, $\text{det}(DT_{0})\equiv 1$, so

[TABLE]

and it is a trivial exercise to check that this identity and (44) imply (45).∎

Identities (44), (45) are important, because they imply nice uniform estimates for arbtrarily long iterations of $T_{0}$. Namely, the following result holds true:

Lemma 6.2** (Lem. 7 in [GST08]).**

there exist $\alpha>0$ and a sequence of functions $\xi_{k}$, $\eta_{k}$ from $[-\alpha,\alpha]^{2}\subset U_{0}$ into $\mathbb{R}^{1}$, such that for all $k$ large enough, we have:

[TABLE]

and the functions $\xi_{k}$ and $\eta_{k}$ are $C^{k}$-small:

[TABLE]

This Lemma means that arbitrarily long iterations of $T_{0}$ are well approximated by the iteratons of its linearization. Note that our perturbation $g$ will be supported outside of $f(U_{0})$, so formulas (46) will stay valid for the perturbed map $\hat{f}=g\circ f$ with the same $\xi_{k}$, $\eta_{k}$, and $\lambda$.

6.2 Formulas for the iterations near the homoclinic band

As all the points of the homoclinic band $J$ are homoclinic, by taking it possibly smaller, we may assume it included in a fundamental domain and so that $J\cap f^{m}(J)\not=\varnothing$, for every $m\not=0$. Furthermore, some forward iteration $J^{+}$ of $J$ lies in $W^{s}_{loc}(0):=[-\alpha,\alpha]\times\{0\}$ and a backward iteration $J^{-}$ lies in $W^{u}_{loc}(0):=\{0\}\times[-\alpha,\alpha]$:

[TABLE]

Let $m>0$ be such that $f^{m}(J^{-})=J^{+}$. Choose $N$ different points $M_{i}^{-}:=:(0,y_{i}^{-})\in J^{-}$ and put $M_{i}^{+}:=(x_{i}^{+},0):=f^{m}M_{i}^{-}$.

Let $U_{1}$ be a small neighborhood of $\{M_{i}^{-}:1\leq i\leq N\}$, so that $U_{1}\subset(-\alpha,\alpha)^{2}\subset U_{0}$ and denote as $T_{1}$ the restriction of $f^{m}$ to $U_{1}$:

[TABLE]

The perturbation $g$ will be supported in small neighborhoods of the points $M_{i}^{+}$, so it can be supported in a disc of any small radius $\nu$, as claimed, provided the points $M_{i}^{-}$ are chosen sufficiently close to each other.

As $J^{-}\subset\{0\}\times(-\alpha,\alpha)$ is sent to $J^{+}\subset(-\alpha,\alpha)\times\{0\}$, the map $T_{1}:(x,y)\mapsto(\bar{x},\bar{y})$ near the point $M_{i}^{-}=(0,y_{i}^{-})$ can be written in the form

[TABLE]

where the $C^{r+1}$-functions $\varphi_{1\,i}$, $\varphi_{2\,i}$ satisfy

[TABLE]

[TABLE]

Note also that the area-preserving property of $T_{1}$ implies that the coefficients $b_{i}$, $c_{i}$ in (48) satisfy

[TABLE]

6.3 Scaling transformation

We will further assume that the indices $i$ are defined modulo $N$, i.e. hereafter $i+1=1$ if $i=N$ and $i-1=N$ if $i=1$. We will need a sequence of positive real numbers $R_{i}$, $i=1,\dots N$, that satisfy

[TABLE]

Such sequence indeed exists when $N$ is odd: we define $R_{i}$ by (52) inductively $R_{1},$ $R_{3},$ $\dots,$ $R_{N},$ $R_{2}$ $\dots,$ $R_{N-1}$ until we arrive to $R_{N+1}=(-1)^{N}\prod_{i=1}^{N}c_{i}\prod_{i=1}^{N}b_{i}$ and notice that the constraint $R_{N+1}=R_{1}$ is satisfied by virtue of (51).

Let us define affine rescaling coordinates:

[TABLE]

In order to define $\bar{Q}_{i}$ and $Q_{i}$ we chose a scaling constant $\mu<1$ such that:

[TABLE]

Furthermore, we assume the the integer $k$ large. The affine rescaling coordinates $\bar{Q}_{i}$ and $Q_{i}$ are defined by the rules:

[TABLE]

[TABLE]

where the constant terms $\gamma_{i\,k}$ and $\beta_{i\,k}$ are given by

[TABLE]

Fact 6.3**.**

When $k$ is large, the map $\bar{Q}_{i}$ sends $\mathbb{D}$ into a small neighborhood of $M_{i}^{+}$, whereas the map $Q_{i}$ sends $\mathbb{D}$ into a small neighborhood of $M_{i}^{-}$.

Proof.

First note that $\gamma_{i\,k}\to 0$ and $\beta_{i\,k}\to 0$ as $k\to+\infty$, by virtue of (47). Hence $\mathbb{D}$ is sent into a small neighborhood of $(x_{i}^{+},0)=M_{i}^{+}$ by $\bar{Q}_{i}$ and into a small neighborhood of $(0,y_{i}^{-})=M_{i}^{-}$ by $Q_{i}$. ∎

The map $\bar{Q}_{1}$ is the map $Q$ in the statement of the lemma. Thus, $Q({\mathbb{D}})$ lies close to $M_{1}^{+}$ if $k$ is large enough, as required.

Below, we derive formulas (see (57),(72)) for the maps $\bar{Q}_{i+1}^{-1}T_{1}Q_{i+1}$ and $(Q_{i+1})^{-1}T_{0}^{k}\bar{Q}_{i}$ for the original map $f$ and the perturbed map $\hat{f}$. As the composition $\prod_{i=1}^{N}(\bar{Q}_{i+1}^{-1}T_{1}Q_{i+1})(Q_{i+1})^{-1}T_{0}^{k}\bar{Q}_{i}$ equals to $\bar{Q}^{-1}\circ\hat{f}^{n}\circ\bar{Q}$ for $n=N(ks+m)$, we will obtain formulas (38),(39) in this way and, thus, prove the lemma.

6.4 Renormalized iterations of the unperturbed map

We start with the map $(Q_{i+1}^{\prime})^{-1}T_{0}^{k}Q_{i}$, i.e. the map $T_{0}^{k}$ in the rescaled coordinates.

Lemma 6.4**.**

There are two real functions $\hat{\xi}_{ik}$, $\hat{\eta}_{ik}$ which are $C^{r}$-small when $k$ is large such that every $(\bar{X}_{i},\bar{Y}_{i})\in\mathbb{D}$ sent to $(X_{i+1},Y_{i+1})$ by $(Q_{i+1})^{-1}T_{0}^{k}\bar{Q}_{i}$ satisfies:

[TABLE]

Moreover, the functions $\tilde{\xi}_{ik}^{\prime}$, $\tilde{\eta}_{ik}^{\prime}$ vanish at $(\bar{X}_{i},\bar{Y}_{i})=0$ and uniformly tend to zero in the $C^{r}$-norm:

[TABLE]

Proof.

By definition, it holds:

[TABLE]

Put $(x_{i+1},y_{i+1})=Q_{i+1}(X_{i+1},Y_{i+1})$ and $(\bar{x}_{i},\bar{y}_{i}):=\bar{Q}_{i}(\bar{X}_{i},\bar{Y}_{i})$. By (46), (54) and (55), it holds:

[TABLE]

Consequently:

[TABLE]

By definition of $\beta_{i+1\,k}$ and $\gamma_{i\,k}$, given in (56), formula (61) shows that the zero value of $(\bar{X}_{i},\bar{Y}_{i})$ corresponds to the zero value of $(X_{i+1},Y_{i+1})$. So we can rewrite (61) as

[TABLE]

where, as follows from (47), the functions $\tilde{\xi}_{ik},\tilde{\eta}_{ik}$ uniformly tend to zero in the $C^{r}$-norm; moreover, they vanish when $(\bar{X}_{i},Y_{i+1})=0$. Hence the $C^{r}$-norms of $\mu^{-k}\tilde{\xi}_{ik}(\mu^{k}\cdot)$ and $\mu^{-k}\tilde{\eta}_{ik}(\mu^{k}\cdot)$ are uniformly $C^{r}$-small and vanish at [math]. By the implicit function theorem there are $C^{r}$-functions $\tilde{\eta}_{ik}^{\prime}$ and $\tilde{\xi}^{\prime}_{ik}$, so that:

[TABLE]

where $\tilde{\xi}_{ik}^{\prime}$, $\tilde{\eta}_{ik}^{\prime}$ uniformly tend to zero in the $C^{r}$-norm and vanish at $(\bar{X}_{i},\bar{Y}_{i})=0$. We notice that (57) and (58) hold true with $\hat{\xi}_{ik}(\bar{X}_{i},\bar{Y}_{i}):=\mu^{-k}\tilde{\xi}_{ik}^{\prime}(\mu^{k}\bar{X}_{i},\mu^{k}\bar{Y}_{i})$ and $\hat{\eta}_{ik}(\bar{X}_{i},\bar{Y}_{i}):=\mu^{-k}\tilde{\eta}_{ik}^{\prime}(\mu^{k}\bar{X}_{i},\mu^{k}\bar{Y}_{i})$. ∎

As the next step, we consider the map $\bar{Q}_{i+1}^{-1}T_{1}Q_{i+1}$ (the map $T_{1}$ in the rescaled coordinates) for the unperturbed map $f$. We denote $d_{i}=\partial_{xy}\varphi_{2\,i}(0,0)$.

Lemma 6.5**.**

There are two real functions $\hat{\phi}_{1ik}$, $\hat{\phi}_{2ik}$ which are $C^{k}$-small when $k$ is large such that every $(X_{i+1},Y_{i+1})\in\mathbb{D}$ sent to $(\bar{X}_{i+1},\bar{Y}_{i+1})$ by $\bar{Q}_{i+1}^{-1}T_{1}Q_{i+1}$ satisfies:

[TABLE]

where the constant term $C_{ik}=[c_{i+1}(x_{i}^{+}+\beta_{i+1\,k})-y_{i+2}^{-}-\gamma_{i+1\,k}]/R_{i+1}$ and $A_{i}=d_{i+1}x_{i}^{+}R_{i}/R_{i+1}$ are uniformly bounded, and for the uniform $C^{r}$-norm:

[TABLE]

Proof.

By definition, it holds:

[TABLE]

Put $(\bar{x}_{i+1},\bar{y}_{i+1}):=\bar{Q}_{i+1}(\bar{X}_{i+1},\bar{Y}_{i+1})$ and $(x_{i+1},y_{i+1})=Q_{i+1}(X_{i+1},Y_{i+1})$. By (48), (54) and (55) it holds:

[TABLE]

We inject the third line into the second and the third lines, and then eliminate the term $x_{i+1}^{+}$ which appears on both side of the new first line.

[TABLE]

Let us now isolate the terms $\bar{X}_{i+1}$ and $\bar{Y}_{i+1}$, and replace $y_{i+1}-y_{i+1}^{-}$ by $R_{i}\mu^{k}Y_{i+1}$:

[TABLE]

By injecting the last line in the latter first line, it comes:

[TABLE]

with:

[TABLE]

and with $C_{ik}$ defined in the statement of the Lemma, the number $\frac{c_{i+1}b_{i}R_{i-1}}{R_{i+1}}=-1$ by (52), and:

[TABLE]

By (49), $\varphi_{1\,i}(0)=\partial_{y}\varphi_{1\,i}(0)=0$. As $\phi_{1\,i}$ is at least of class $C^{r}$, with $r\geq 3$, $\tilde{\phi}_{1ik}(0)$ is bounded by $\mu^{-k}(\lambda^{k}+(\mu^{k})^{2})$ which is small. Furthermore, $\partial_{y}\tilde{\phi}_{1ik}(0)$ is bounded bounded $\mu^{k}$ which is small. Finally, by the definition of $\tilde{\phi}_{1ik}$ all other derivatives (up to the order $r$) are uniformly small on $\mathbb{D}$. Thus, the $C^{r}$-norm of $\tilde{\phi}_{1ik}$ is small.

Observe that all the derivatives of the form $\partial^{n}_{X}\partial^{m}_{Y}\tilde{\phi}_{2ik}$ are uniformly small whenever $n+m\geq 2$ with $n\geq 1$. To carry the case $m\geq 2$ and $n=0$, we use (50) which implies that $\partial^{m}_{Y}\tilde{\phi}_{2ik}$ is uniformly small.

The Lagrange formula integrated from $(0,0)$ to $\lambda^{k}(x_{i}^{+}+\beta_{i+1\,k})$ shows that both $\partial_{X}\tilde{\phi}_{2ik}(0)$ and $\tilde{\phi}_{2ik}(0)$ are close to $\partial_{x}\varphi_{2\,i+1}(0)$ and $\varphi_{2\,i+1}(0)$, which are small by (49).

Consequently, all the derivatives of $\tilde{\phi}_{2ik}$ are small except for, possibly, $\partial_{Y}\tilde{\phi}_{2ik}$. In view of the above estimates, it it is close to a constant. Hence it is suffices to evaluate it at [math]. Since $\partial_{y}\varphi_{2\,i+1}(0)=0$, we have:

[TABLE]

which converges to $\frac{\lambda^{-k}R_{i}\lambda^{k}(x_{i}^{+}+\beta_{i+1\,k})}{R_{i+1}}\partial_{x}\partial_{y}\varphi_{2\,i+1}(0)\sim\frac{R_{i}x_{i}^{+}}{R_{i+1}}\partial_{x}\partial_{y}\varphi_{2\,i+1}(0)=A_{i}.$

∎

6.5 Construction of the perturbation $g$

The perturbation we will now add to the map $f$ does not change the map $T_{0}$ (because the difference $\hat{f}-f$ is supported outside of $U_{0}$), so Lemma 6.4 and its formula (57) for $(Q_{i+1})^{-1}T_{0}^{k}\bar{Q}_{i}$ remains the same when $f$ is replaced by the perturbed map $\hat{f}=g\circ f$. The map $T_{1}$ will be affected by the perturbation, hence Lemma 6.5 and its formula (64) for the map $\bar{Q}_{i+1}^{-1}T_{1}Q_{i+1}$ will be modified. In order to construct the perturbation map $g$, we take the functions $\psi_{1},\dots,\psi_{N}$ from the statement of the lemma and define

[TABLE]

where the constants $C_{ik}$, $A_{i}$ are the same as in (64). As (53) states that $\mu^{r}>\lambda$, it holds:

[TABLE]

We use the notation $(\bar{x},\bar{y})$ for the non-rescaled coordinates near $\{M_{i}^{+};1\leq i\leq N\}=\{(x_{i}^{+},0);1\leq i\leq N\}\subset W^{u}_{loc}(O)$. Let $\delta>0$ be small and put:

[TABLE]

Hence $V:=\sqcup_{i}V_{i}$ and $V^{\prime}:=\sqcup_{i}V^{\prime}_{i}$ are close to $\{M_{i}^{+};1\leq i\leq N\}$. We recall that $\bar{Q}_{i}(\mathbb{D})$ is close to $M_{i}^{+}$ when $k$ is large. For $\delta>0$ small enough, independent of $k$ large enough, so that for every $i$, $f^{j}(\bar{Q}_{i}(\mathbb{D}))$ is disjoint from $V$ for every $1\leq j<sk+m$. Let $\psi\in C^{r}(\sqcup_{i}[-\delta+x_{i}^{+},\delta+x_{i}^{+}],\mathbb{R})$ be so that its restriction to $[-\delta+x_{i}^{+},\delta+x_{i}^{+}]$ satisfies $D\Psi=\hat{\psi}_{i}$. Let $\rho\in C^{\infty}(\mathbb{R},[0,1])$ be the bump function equal to $1$ over $V^{\prime}$ with support in $V$.

Let the symplectic perturbation map $g$ be equal to the time-1 map by the Hamiltonian flow with the Hamiltonian $H(\bar{x},\bar{y})=-\Psi(\bar{x})\rho(\bar{x},\bar{y})$. Since $\|\hat{\psi}_{i}\|_{C^{r}}\to 0$ as $k\to+\infty$, it follows that $g-id$ can be made arbitrarily small in $C^{r}$.

By construction, the restriction $g|_{\bar{Q}_{i}(\mathbb{D})}$ is given by

[TABLE]

Plugging this into (48), we find that for the perturbed map $\hat{f}=g\circ f$ the transition map $T_{1}=\hat{f}^{m}=g\circ f^{m}$ from a neighborhood of $M_{i+1}^{-}$ to a neighborhood of $M_{i+1}^{+}$ is given by

[TABLE]

with the same functions $(\varphi_{1\,i+1},\varphi_{2\,i+1})$ and coefficients $x^{+}_{i+1}$, $y^{-}_{i+1}$, $b_{i+1}$, $c_{i+1}$ as for the unperturbed map $f$, and the only additional term $\hat{\psi}_{i+1}(\bar{x})$ in the second equation. Therefore, the only correction to the map $\bar{Q}_{i+1}^{-1}T_{1}Q_{i+1}$ due to this perturbation, will be the additional term

[TABLE]

in the right-hand side of the respective equations of (64) (we use here the rescaling formula (54) with $i$ replaced by $i+1$). By (70), this term equals to

[TABLE]

so the equation (64) for the map $Q_{i+1}^{-1}T_{1}Q_{i+1}^{\prime}$ changes to

[TABLE]

By this formula and Lemma 6.4 (57), the map $T_{1}T_{0}^{k}=\hat{f}^{m+sk}=g\circ f^{m+sk}$ from a small neighborhood of $M_{i}^{+}$ to a small neighborhood of $M_{i+1}^{+}$ is written, in the rescaled coordinates, as

[TABLE]

where

[TABLE]

By (58),(65), the functions $\phi_{1i},\phi_{2i}$ tend to zero uniformly in $C^{r}$ on any compact as $k\to+\infty$. Importantly, the functions $(\phi_{1\,1},\phi_{2\,i})_{i}$ do not depend on the choice of the perturbation functions $(\psi_{i})_{i}$.

Formula (73), in fact, completes the proof of the lemma. Indeed, it can be rewritten as

[TABLE]

where

[TABLE]

Thus, the map $(T_{1}T_{0}^{k})^{N}$ from a small neighborhood of $M_{1}^{+}$ takes indeed the required form (38) in the rescaled coordinates $(X,Y)$.