16.2Setting and main result

We fix once and for all a closed interval I ⊂ [−1, 2] of ℝ, which contains [0, 1] in its interior. Wework in the space κ of Cκ diffeomorphisms of?2 with support contained in ?2 × I2. We fix a lift I : ?2 → ℝ4 of the identity of ?2. We consider κ as a subset of by lifting each φ ∈ κ to the unique function ̃φ: ?2 → ℝ4, which coincides with I on ?2×[2,+∞[2.Wethen equip κ with the induced topology, which is clearly defined by a metric dκ. The space ( κ , dκ) is complete.

We will first define a diffusion property for diffeomorphisms in κ. In parallel, we will introduce a specific class κ(τ) ⊂ κ,whose elements do not satisfy this property and which we consider as “unperturbed systems.” Our main result then proves the existence of a large subset of suitable C k−1 perturbations of f which admit the diffusion property.

1. Let us now give precise definitions, beginning with that of diffusion orbits.

Definition 1. Fix δ > 0 and set

Given a diffeomorphism g ∈ κ, we say that a finite orbit x0, . . . , x N of g is a δ-diffusion orbit when x0 ∈ and xN ∈ 1.

2. The elements f ∈ κ(τ) are Cκ symplectic diffeomorphisms of ?2 and admit the product form

where the diffeomorphisms fi : ? satisfy some additional conditions.

∙ We begin with the conditions on the first factor f1. We denote by D(τ) the set of real numbers which are Diophantine of exponent τ > 1:

Given κ ≥ 1 and τ > 1, we introduce the set of Cκ symplectic diffeomorphisms f1 : ? , which satisfy the following conditions:

(C1) Supp f1 ⊂ ?× I.

(C2) The circles Γ0 = ? × {0} and Γ1 = ? × {1} are invariant under f1, and their rotation numbers ρ0, ρ1 are in D(τ).

To introduce our third condition, in the coordinate chart (θ1, r1) of ?, we write

(C3) The restriction of f1 to the annulus ? × [0, 1] uniformly tilts the vertical to the right, that is, there is a c > 0 such that

Note that, due to (C1) and (C2), the annulus ?×[0, 1] is invariant under f1. The condition that the rotation numbers of the circles Γi are in D(τ) will ensure their persistence under perturbation.

∙ As for the second factor, we introduce the set of Cκ symplectic diffeomorphisms f2 : ? which satisfy the following conditions:

(C4) Supp f2 ⊂ ?× I.

(C5) The diffeomorphism f2 possesses a hyperbolic fixed point O2.

(C6) The point O2 admits a transverse homoclinic point P2.

We will denote by

the maximal eigenvalue of the derivative DO2 f2. Note that O2 and P2 are contained in the support of f2.

3. We now define our set of “unperturbed” diffeomorphisms in order to guarantee additional stability properties under perturbations.

Definition 2. We define κ(τ) as the set of (symplectic) diffeomorphisms on?2 of the form (2.2), where and with

The domination condition (2.6) ensures that the invariant annulus ? × {O2} is uniformly normally hyperbolic for f , with persistence properties in the C κ topology and additional specific symplectic features (Figure 3).

4. Let us briefly show that given τ > 1, when κ is large enough, for any f ∈ κ(τ) there exists an arbitrarily small δ > 0 such that f does not possess a δ-diffusion orbit. An essential circle in? is a closed curve homotopic to ?×{0}. In the following, we denote by Ess(ϕ) the set of essential circles invariant under some diffeomorphism ϕ of ?.

Fix an f = (f1, f2) ∈ κ(τ). Then, thanks to Condition (C2) and assuming κ large enough, one deduces from Birkhoff’s normalization in the neighborhood of Γ0 (see [6, 14] and references therein for the continuous setting39)and classical KAM theorems inthe finitely differentiable setting [6, 34] the existence of an essential invariant circle Γlocated in the zone r1 > 0 (one indeed gets an infinite family of such circles). Hence there is an arbitrarily small δ > 0 such that Γ is located in the zone r1 > δ. Since f1 is assumed to have compact support, the connected components of the complement of Γ in? are invariant. This proves that f1 cannot admit an orbit with first point in the zone r1 < δ and last pointwith r1 close to 1. As a consequence, due to the product structure, the diffeomorphism f ∈ κ(τ) cannot admit a δ-diffusion orbit, which proves our claim.

5. However, we will prove that under Cκ−1-dense small enough perturbations, any element of κ(τ) gives rise to a diffeomorphism, which admits diffusion orbits. More precisely, our main result is the following.

Theorem 1. Fix τ > 1 and fix δ > 0. Then there is a κ0 such that if κ ≥ κ0, given f ∈ κ(τ), there is an ε(f) > 0 such that for any diffeomorphism g in Bκ(f, ε(f)) there exists a Cκ−1 diffeomorphism ̃g, arbitrarily close to g in the Cκ−1 topology, which admits a δ-diffusion orbit. Moreover, there is a neighborhood of ̃g in the C0 topology in which any diffeomorphism admits a δ-diffusion orbit.

The existence of δ-diffusion orbits is clearly an open property in the C0 topology; we will therefore focus on the “density.” As mentioned in the introduction, similar results have already been obtained in related settings. One major difference is that we deal here with finitely differentiable maps. The loss of one derivative essentially comes from the fact that we have to consider the characteristic foliations of the stable and unstable manifolds of a Cκ normally hyperbolic annulus. These are Cκ hypersurfaces of ?2, so their characteristic foliations are Cκ−1 only. This problem has not been noticed before, due to the usual C∞ or analytic setting, together with the fact that the flow on the invariant annulus was assumed to be near-integrable, which increases the regularity of the various invariant manifolds at hand.

6. The proof of Theorem 1 is based on a method introduced by Moeckel in [29] to prove the existence of drifting orbits for bisystems of maps τ0, τ1 on the annulus. Given a set A, the formal definition of a bisystem (τ0, τ1) on A as a map τ : {0, 1}ℤ × A can be found in [25]. Here we will be content with the definition of its “projected orbits,” which are the result of the iteration of τ0 and τ1 on A in arbitrary order. More precisely, we say that a finite sequence (xn)0≤n≤n∗ of points of A is an orbit of the bisystem (τ0, τ1) when there exists a finite sequence (in)0≤n≤n∗−1 ∈ {0, 1}n∗ such that for 0 ≤ n ≤ n∗ − 1

We can now state the two results by Moeckel we will use in the following. Given two disjoint essential circles Γ∙ and Γ∙ in ?, we denote by A[Γ∙ , Γ∙] ⊂ ? the subannulus bounded by their union.

Theorem A [29]. Let τ0, τ1 : ? be C1 exact-symplectic diffeomorphisms with compact support in ? × I. Assume that there exist two disjoint τ0-invariant essential circles Γ∙, Γ∙ in ? × I and that τ0 is a twist map in restriction to the annulus A := A[Γ∙ , Γ∙]. Let EssA(τ0) be the set of essential τ0-invariant circles contained in A. Assume that

Then for any connected neighborhoods U∙ and U∙ of Γ∙ and Γ∙ in ?, with τ1(Γ∙) ⊂ U∙ and τ1(Γ∙) ⊂ U∙ the bisystem (τ0, τ1)admitsan orbit with first point in U∙ and last point in U∙.

This in fact is a slight generalization of the theorem of [29], since the boundaries Γ∙ and Γ∙ are not assumed to be invariant under τ1. The proof follows almost exactly the same lines. One first introduces the full orbit of U∙ under the polysystem (τ0, τ1), that is, the union of all subsets of ? of the form

where n ≥ 0 is arbitrary. One then considers the connected component c of ,which contains Γ∙ (this is the single difference with [29], where is necessarily connected), and assume that the intersection A ∩c is disjoint from U∙. Up to the classical “filling the gaps” process, one gets from A∩c an invariant subset contained in AU∙, whose upper boundary is an essential circle invariant by both maps τ0 and τ1, which is a contradiction.

The next result, based on the study of the Hausdorff dimension of the sets EssA(φ) and Ess(ψ), will provide us with the necessary tool for proving the density statement in Theorem 1.

Theorem B[29]. Fix an integer p ≥ 1. Let τ0, τ1 : ? be Cp exact-symplectic diffeomorphisms with compact support in ?×I. Assume that there exist two τ0-invariant disjoint essential circles Γ∙, Γ∙ in ?×I, and that τ0 is a twist map in restriction to A := A[Γ∙ , Γ∙]. Assume moreover that (τ0)|A has no essential invariant circle with rational rotation number. Then there exists a C∞ Hamiltonian h : ? → ℝ with support in?×I, arbitrarily small in the C∞ topology, such that

where Φh stands for the time-one map of the Hamiltonian flow generated by h.

The proof is exactly the same as in [29].

16.3Proof of Theorem 1

Let us first informally describe the proof. The first ingredient is the choice of ε small enough so that any g in Bκ(f, ε(f)) exhibits some of the main dynamical features off. In particular, we require that g admits a normally hyperbolic invariant annulus gclose to ? ×{O2} and a homoclinic annulus close to? ×{P2}.

We will then consider two diffeomorphisms of g.

Using Moeckel’s results, we prove that after a small perturbation of g the bisystem (φg , ψg) admits “connecting orbits,” whose initial and final points are arbitrarily close to the boundary circles of Ag.

Finally, in view of the definition of φg and the asymptotic properties of ψg, one expects that the connecting orbits of the bisystem can be uniformly approximated by genuine orbits of g. We prove that this is the case, by means of normally hyperbolic shadowing lemma, whose idea is reminiscent of [5, 13].

16.3.1Choice of ε > 0

We refer to [7, 8, 21] for the definitions and main properties of (absolutely) normally hyperbolic manifolds, and in particular [8] for a presentation in the noncompact case for Lipschitzian maps. The results we use in the following are easy consequences of [7, 8]. Recall the usual notations and basic properties. Given g ∈ κ, a g-invariant submanifold g ⊂ ?2, and a subbundle E of TAg?2 invariant under Tg, we set

Given an integer ℓ ∈ {1, . . . , κ}, the g-invariant manifold g is said to be ℓ-normally hyperbolic when there exist two Tg-invariant one-dimensional continuous subbundles E+, E− of TAg?2 satisfying

and such that

When this is the case, the manifold g is Cℓ and its stable and unstable manifolds W±(g) are well-defined and Cℓ too. These objects are persistent under Cκ perturbations of g, with Cℓ compact-open convergence to the unperturbed ones when the perturbation tends to 0. The stable and unstable manifolds W±(g) are foliated by the center-stable and unstable manifolds (W±(x))x∈Ag, which are Cℓ submanifolds of ?2. The regularity of the foliation itself is a more delicate matter; in the following we will take advantage of the symplectic features of our problem to get rid of these technicalities. Finally, normally hyperbolic manifolds enjoy natural local maximality properties.

1. Let us first examine the dynamical structure of a diffeomorphism f ∈ κ(τ), which are immediately deduced from the product form (2.2).

–The annulus = ?×{O2} is invariant under f and diffeomorphic to ?. It is moreover κ-normally hyperbolic, due to condition (2.6): E ± are the stable and unstable lines of the fixed point O2 and

The stable and unstable manifolds of inherit the product structure of f :

These are hypersurfaces of ?2 of class Cκ (since W±(O2) are Cκ). Their characteristic leaves are the one-dimensional submanifolds

–As a consequence of the normal hyperbolicity of , the manifolds W+( ) and W−( ) are foliated by the center-stable and center-unstable manifolds of the points of , respectively. In this trivial case these latter manifolds read

so they coincide with the characteristic leaves of W±( ). Let Π± : W±() → stand for the characteristic projections so that if (x, w) ∈ W±(x), then Π±(x, w) = (x, O2).

–ThelocalmanifoldsW±( ) intersect transversely in ?2, along both and the homoclinic annulus

Moreover, for each (x, O2) ∈ , the leaf W −((x, O2)) transversely intersect the manifold W+( ) at a unique point of &&&& , namely

One has a similar observation for the stable leaves. We denote by π± the restrictions of Π± to the annulus &&&& so that

–Clearly and &&&& are symplectic submanifolds of ?2 and π± are symplectic diffeomorphisms.

–There exists a pair of natural f -induced symplectic diffeomorphisms of . The first one is just the restriction φ = f|A , which here admits a natural identification with f1. The second one is the map

ψ = π+ ∘ (π−)−1,

which describes the homoclinic excursion of the orbits, we call it the homoclinic map40. Clearly ψ = Id here.

2. Our goal in this section is to prove the existence of an ε(f) small enough so that each g ∈ Bκ(f, ε(f)) admits a symplectic normally hyperbolic annulus endowed with a bisystem of symplectic maps defined in the same way as above.

Lemma 1. Let f = (f1, f2) ∈ κ(τ) be fixed. Then there exists such that for each

–there exists a (uniquely defined) symplectic normally hyperbolic g–invariant annulus gof the form

where ag is a Cκ function? → B2(O2, α) ⊂ ? such that ag − O2Cκ(?) → 0 when dκ(g, f) → 0 (where α > 0 is a suitable constant);

–the manifolds W±(g) are coisotropic with characteristic foliations (W±(z))z∈Ag, and the characteristic projections

–there exists a (uniquely defined) symplectic homoclinic annulus &&&&g ⊂ W+(g) ∩ W−(g), of the form

where hg is a Cκ function? → B2(P2, α) ⊂ ? such that hg − P2Cκ(?) → 0 when dκ(g, f) → 0;

–the restrictions

±±

are Cκ−1 symplectic diffeomorphisms,

–for each z ∈ g, the center-unstable manifold W −(z) intersects &&&&g at transversely in ?2, with a similar property for the center-stable manifold.

Sketch of proof. For ε small enough, the existence of the annulus g, together with the graph form(3.10), are direct consequences of the κ-normal hyperbolicity of . We have implicitly used the local uniqueness related to normally hyperbolic manifolds: the annulus g is the unique normally hyperbolic g-invariant manifold contained in a fixed small enough neighborhood of in ?2. The uniform estimates on the function ag come from the compact support condition on the diffeomorphism g. The local stable and unstable manifolds of g converge to those of in the Cκ compact-open topology when dκ(g, f) → 0. Moreover, the compact support condition on g also yields uniform convergence along the first factor ?. The existence and form of &&&&g are immediate consequences of this latter observation.

By normally hyperbolic persistence, there exist two uniquely defined Tg-invariant continuous subbundles E+, E− of TAg?2 such that TAg?2 = E+ ⊕Tg ⊕E−. Moreover, by continuous dependence, one can choose ε small enough so that

It is not difficult to deduce from these conditions the existence of a neighborhood N of g in ?2 and positive constants c < 1 < C verifiying c C < 1, such that

where stand for the projections attached to the center-stable and center-unstable foliations. As a consequence, one easily proves that g is symplectic, thatW±(g) are coisotropic, and that their characteristic foliations coincide with the center-stable and center-unstable foliations (see [26]).

From this latter property and the Cκ regularity of W±(g), one deduces that themaps are Cκ−1.Moreover, the characteristic foliations of W±(g) converge to thoseof in the Cκ−1 compact-open topology when dκ(g, f) → 0; with uniform convergence along the first factor. This yields the existence and uniqueness of an intersection point of W−(z) with &&&&g for z ∈ g, with a similar property for W+(z). Hence the maps are diffeomorphisms, which are Cκ−1 since &&&&g itself is Cκ. These maps are induced by the intersection of characteristic foliations with symplectic sections ofW±(g), hence they are symplectic.

Moreover, assuming ε small enough, by Cκ−1 convergence to the unperturbedleaves, W−(z) intersects W+(g) transversely at and W+(z) intersects W−(g) transversely at

3. We can now introduce our bisystem on g, assuming that We firstconsider the restriction

which is a Cκ symplectic diffeomorphism for the induced structure on g. As for oursecond map, we set

Therefore ψg is a Cκ−1 symplectic map. The next lemma will enable us to identify φg(and ψg) with a diffeomorphism of the standard annulus ? in a proper way.

Lemma 2. If is small enough, there exists a C κ−1 symplectic embeddingΦg of ?, equipped with the standard form, into ?2 such that

–Φg(?) = g;

–the diffeomorphism has support in ?× I and tends to f1in the Cκ−1 uniform topology when dκ(g, f) → 0.

–the diffeomorphism has support in ?× I and tends to Id in the C κ−1 uniform topology.

Proof. The map Ψg : x1 ↦ (x1, ag(x1)) is a Cκ embedding of ? into ?2 with image g, which converges to the embedding J : x1 ↦ (x1, O2) when g → f. Hence the diffeomorphism tends to f1 in the Cκ uniform topology when dκ(g, f) → 0.

Using Moser’s isotopy argument, one proves the existence of a Cκ−1 diffeomorphism Cg : ? , which sends the standard symplectic form on ? onto the pullback of the induced form on g. This loss of regularity comes from the fact that only. One also proves that Cg → Id in the uniform Cκ−1 topology when dκ(g, f) → 0.

Hence Φg = Ψg ∘ Cg : ? → g is symplectic (where ? is endowed with the standard form), and Φg → J in the uniform Cκ−1 topology when dκ(g, f) → 0. Our claim easily follows.

The following corollary is an immediate application of the previous lemma and finitely differentiable KAM theory [6, 34].

Corollary 1. There is an such that for each diffeomorphism g ∈ Bκ(f, ε(f)) there exist a Cκ−1 symplectic embedding Φg of ? into g such that the map ̂φg = Φ−1 ∘ φg ∘ Φ admits two (disjoint) essential invariant circles Γ∙ and Γ∙ with rotation numbers ρ0 and ρ1, respectively (see (C2)), such that Γ∙ → Γ0 and Γ∙ → Γ1 in the C0 topology when dκ(g, f) → 0. Moreover, the map ̂φg uniformly tilts the vertical over the annulus Ag bounded by Γ∙ and Γ∙.

16.3.2Implementation of Moeckel’s method

We fix now a diffeomorphism g ∈ Bκ(f, ε(f)), where ε(f) is defined in Corollary 1, and we want to prove the existence of a perturbed diffeomorphism ̃g ∈ Bκ(f, ε(f)), arbitrarily close to g in the Cκ−1 topology, for which the associated bisystem (φ̃g , ψ̃g) satisfies condition (2.7). We proceed in two steps: we first perturb g so that φg has no rational essential circle, and we then perturb the resulting diffeomorphism again (without perturbing φg) to ensure condition (2.7). We write ε instead of ε(f) in the following.

16.3.2.1 First perturbation of g: making φg admissible

Let J be a closed interval of ℝ containing [0, 1] in its interior and contained in theinterior of I. By usual perturbation techniques (see [31, 32]), there exists a Cκ diffeomorphism ̂g ∈ Bκ(f, ε), arbitrarily close to g in the uniform Cκ topology, which satisfies

–the invariant annulus ̂g coincides with g;

–all periodic points of φ̂g = ̂g|Âg in ?2 × J2 are either hyperbolic or elliptic with nondegenerate Birkhoff invariant.

As a consequence, if κ is large enough to ensure the existence of invariant curvessurrounding each elliptic point, one easily proves that φ̂g cannot admit an essential invariant circle with rational rotation number in the compact annulus Âg defined in Corollary 1.

16.3.2.2 Second perturbation of g: making ψg admissible

In view of the last section, replacing g with ̂g, we can assume that g has no invariant circle in Ag with rational rotation number. We want now to perturb g into a new diffeomorphism ̃g such that

We first analyze the composition of g with a diffeomorphism with support localized ina small enough neighborhood of the annulus &&&&g.

Lemma 3. Let be the submanifolds (diffeomorphic to [0, 1]×?) of W±(g) boundedby g and &&&&g. Let N be a neighborhood of &&&&g such that

Assume that χ is a diffeomorphism of ?2 with support in N , which leaves the annulus&&&&g invariant, and set ̃g = χ ∘ g. Then ̃g = g,&&&&̃g = &&&&g and

Proof. By (3.19), g is still invariant and normally hyperbolic under ̃g, since g and ̃g coincide in a tubular neighborhood of g. Therefore g = ̃g and φ̃g = φg. We will write W±(̃g , ̃g) for the stable and unstable manifolds of g with respect to the diffeomorphism ̃g.

Set By (3.19) g and ̃g coincide on hence and therefore

Set now Then, again by (3.19), and

We assumed&&&&g = χ(&&&&g). Hence, since (̃g , ̃g) by (3.22).Moreover, by (3.21). As a consequence,

&&&&g ⊂ W−(̃g , ̃g) ∩ W+(̃g, ̃g)

is still a homoclinic annulus for ̃g. We set &&&& = &&&&̃g = &&&&g.

Let be the characteristic ±projections, with similar notation for ̃g. Then

Set &&&& 1 = g(&&&& ) and observe ++that by (3.19). Then

where we used (3.19) again in the second equality. Hence

As for the unstable projections, set now &&&& −1 = g−1(&&&& ) and observe that

Now

and

so that

Finally, since

which concludes the proof.

We can now use Moeckel’s Theorem B in order to produce our perturbation ̃g.

Lemma 4. There exists a diffeomorphism ̃g ∈ κ−1, arbitrarily close to g in the Cκ−1 topology such that ̃g = g, φ̃g = φg and the maps τ0 = φg and τ1 = ψ̃g satisfy Condition (2.7) of Theorem A.

Proof. Let ϕ := Φg : ? → g be the symplectic embedding exhibited in Lemma 2,and set

By Theorem B, there exists a C∞ Hamiltonian h : ? → ℝarbitrarily close to 0 suchthat ̂φg and the modified diffeomorphism

satisfy (2.7).

In view of Lemma 3, equation (3.20), let us introduce the perturbed diffeomorphism

where χ : &&&&g is a diffeomorphism we want to determine (and which we will have to continue to a diffeomorphism χ defined in a neighborhood of &&&&g in the following). Set

We want to choose χ in order to solve the equation

Straightforward computation yields

where ℓ = h ∘ ϕ−1.

Therefore χ is a Cκ−1 Hamiltonian diffeomorphism of the annulus &&&&g, with compact support, which tends to Id in the Cκ−1 topology when dκ(g, f) → 0. As a consequence, there is a Cκ function ξ : ℝ×&&&&g →ℝ, with support in ]0, 1[ ×&&&&g such that

where Φξ is the time-one map starting at 0 generated by ξ. Using the Moser isotopy argument, one proves the existence of a Cκ−1 symplectic diffeomorphism

where N is a neighborhood of &&&&g in ?2, α is a positive constant, and the first factor is endowed with the usual symplectic structure. Fix a C∞ bump function η : B2(0, α) → ℝ equal to 1 in a neighborhood of 0 and define a function L : ℝ×?×B2(0, α) → ℝby

Then clearly the time-one map χ = T ∘ ΦL leaves &&&&g invariant, with χ|Hg = χ and the support of χ is contained in N . Moreover, χ tends to the identity in the Cκ−1 topology when ℓ tends to 0 in the Cκ topology. Setting ̃g = χ ∘ g provides us with the perturbed diffeomorphism we were looking for.

16.3.3Normally hyperbolic shadowing

The aim of this section is to prove the following result, from which our main theorem easily follows. Let d stand for the product metric on ?2.

Theorem 2. Fix f ∈ κ(τ) with κ so that the statements of the last section hold. Fix g in Bκ(f, ε(f)) and fix an orbit x0, . . . , xn of the polysystem (φg , ψg) on g. Then for any δ > 0 there is an orbit z0, . . . , zN of g in ?2 such that d(z0, x0) < δ and d(zN , xn) < δ. One can moreover choose z0 so that for each i ∈ {0, . . . , n}, there is an m(i) with

One main ingredient in the proof is the Poincaré recurrence theorem, following an idea introduced in [17]. Since φg has compact support and preserves the symplectic area on g, by the Poincaré recurrence theorem almost every point of g is positively and negatively recurrent for φg. In the following we use recurrent as a shorthand for positively and negatively recurrent.

The other main tool of the proof is the following λ-lemma.

Normally hyperbolic inclination lemma. Fix f ∈ κ(τ) with κ so that the statements of the last section hold, and fix g in B(f, ε(f)). Let (jx)x∈Ag be a continuous family of C1 parametrizations of the local center-unstable manifolds attached to g, that is, a C0 map j : g × [−1, 1] → W−(g) such that, setting jx = j(x, ⋅)

and jx is C1. Then for any C1 submanifold Δ of ?2 which intersects W+(g) transversely in ?2 at some point ξ ∈ W+(x), there exist a sequence (Δn)n∈ℕ such that

and for n ∈ ℕ, a C1 diffeomorphism ℓn : [−1, 1] → gn(Δn) such that

We refer to [33] for a proof with detailed estimates in the compact setting, which directly applies here, thanks to our compactness assumption on the support of g.

Proof of Theorem 2. We will write φ, ψ instead of φg, ψg. Fix an orbit x0, . . . , xn of the polysystem (φ, ψ) on g and fix δ > 0. We fix a tubular neighborhood N of g in ?2 such that N ∩ W−(g) is invariant by g−1 and for each z ∈ N ∩ W−(g) with z ∈ W−(y):

Setting τ0 = φ and τ1 = ψ, by definition, there exists a sequence ω0, . . . , ωn−1 in {0, 1} such that for 0 ≤ j ≤ n − 1

Choose r > 0 small enough so that if D0 = g ∩ B(x0, r) and if

then Dj ⊂ g ∩ B(xj , δ/2) for 0 ≤ j ≤ n (which is possible by continuity of both maps τj).

We will prove the existence of an orbit (yj)1≤j≤n of (τ0, τ1) associated with the same sequence (ωj) such that the point yj belongs to Dj and is recurrent for τ0 = φ, and the existence of a sequence of balls (Bj)0≤j≤n of ?2, which satisfy the following two properties:

∙ (Cj): for 0 ≤ j ≤ n, Bj is centered at some point zj ∈ W−(yj) ∩ N and Bj ⊂ B(yj , δ/2),

∙ (Tj): for 0 ≤ j ≤ n − 1, ∃mj > 0 such that gmj (Bj) ⊂ Bj+1.

We will construct these objects backward, by finite induction. It is enough to prove that given some recurrent point yj+1 ∈ Dj+1 together with a ball Bj+1 satisfying (Cj+1), one can find a recurrent point yj ∈ Dj, a ball Bj satisfying (Cj), and a positive mj which satisfies (Tj).

1. Assume first that xj+1 = φ(xj), so Dj+1 = φ(Dj). By assumption, the point yj+1 ∈ Dj+1 is recurrent for φ, hence the point yj = φ−1(yj+1) is in Dj and is recurrent for φ too. By (Cj+1), the ball Bj+1 is centered at some zj+1 ∈ W−(yj+1). By our assumption on W−(g) ∩ N and since g coincides with φ on g, setting zj = g−1(zj+1),

Therefore, by continuity of g, there exists a ball Bj centered at zj and contained in B(yj , δ/2) such that g(Bj) ⊂ Bj+1.

2. Assume now that xj+1 = ψ(xj) so that Dj+1 = ψ(Dj). Let Rj and Rj+1 be the fullmeasure subsets of Dj and Dj+1 formed by the recurrent points for φ. Since ψ is measure-preserving, Rj+1 ∩ ψ(Rj ) is a full measure subset of Dj+1. Therefore, there exists a recurrent point yj ∈ Rj such that yj+1 = ψ(yj) is recurrent, and so close to yj+1 that, by continuity of the center-unstable foliation, the leaf W−(yj+1) intersects the ball Bj+1. By definition of ψ and by the last item in Lemma 1, the submanifold Δ = W−(yj) intersects W+(g) transversely in ?2 at some point ξ ∈ W+(yj+1). Apply the inclination lemma to Δ in the neighborhood of ξ, together with the positive recurrence property of yj+1: there exists an arbitrarily large integer m′′ such that gm′ (Δ) intersects Bj+1. Fix

then

Now, by definition of and since is negatively recurrent, there is an (arbitrarilylarge) integer m″ such that

Set so that yj is recurrent and the point zj = g−(m″+m′)(z) ∈ W−(yj)satisfies

Hence by continuity there exists a ball Bj centered at zj such that conditions (Cj) and (Tj) are satisfied.

3.As a consequence, there exists a sequence of integers (mi )1≤≤n such that for 1 ≤ i ≤ n

By construction, any z0 ∈ B0 satisfies our statement.

16.3.4Conclusion of the proof of Theorem 1

Fix f ∈ κ(τ) and δ > 0.We assume that κ is large enough so that all the conclusions of the last sections hold. Set

By Lemma 2 and Corollary 1, one can choose ε ∈ ]0, ε(f)[ small enough so that for any diffeomorphism g ∈ B(f, ε), with the notation of Lemma 2, the invariant circles Γ∙ and Γ∙ of ̂φg satisfy

and are such that moreover

We then proved the existence of a C κ−1 diffeomorphism ̃g ∈ B(f, ε(f)) arbitrarily close to g in the Cκ−1 topology, which admits the same annulus g and such that the bisystem (φ̃g , ψ̃g) associated with ̃g satisfies (2.7). In particular, (φ̃g , ψ̃g) admits an orbit with first point in ∙ and last point in ∙. Theorem 2 applied with δ/2 instead of δ to the bisystem (φ̃g , ψ̃g) and the previous orbit produces an orbit of ̃g with first point in 0 and last point in 1 (see Definition 1). This concludes the proof.