Robustly Non-Convex Hypersurfaces In Contact Manifolds

Julian Chaidez Department of Mathematics
University of Southern California
Los Angeles, CA
90007
USA julian.chaidez@usc.edu

Abstract.

We construct the first examples of hypersurfaces in any contact manifold of dimension 5 and larger that cannot be $C^{2}$ -approximated by convex hypersurfaces. This contrasts sharply with the foundational result of Giroux in dimension $3$ and the work of Honda-Huang in the $C^{0}$ case. The main technical step is the construction of a Bonatti-Diaz type blender in the contact setting.

1. Introduction

A hypersurface in a contact manifold $(Y,\xi)$ is convex if there is a contact vector-field $V$ that is transverse to . Convex surface theory was first introduced by Giroux [35], and has since proven to be a deep and powerful tool for the study of contact $3$ -manifolds. Applications include classifications of contact structures [60, 40, 41, 45, 34, 24, 57, 49] and Legendrians [31, 44, 29, 19]; the construction of $3$ -manifolds with no tight contact structures [27] and tight, non-fillable contact structures [28]; and finiteness results for tight contact structures [22, 23]. There have also been many fruitful interactions with Floer homology [21, 2, 3, 30, 4, 46]

Recently, the study of higher dimensional convex surface theory was initiated by Honda-Huang [42, 43] and Breen-Honda-Huang [14], who have systematically generalized many of the foundational results of Giroux to dimensions larger than three. One such result is the following.

Theorem 1 (Giroux).

Any closed surface in a contact manifold $(Y,\xi)$ in dimension $3$ can be $C^{\infty}$ -approximated by a convex surface ^′.

In [43] (and in Eliashberg-Pancholi [26]) the following partial generalization was proven.

Theorem 2 (Honda-Huang).

Any closed hypersurface in a contact manifold $(Y,\xi)$ can be $C^{0}$ -approximated by a Weinstein convex hypersurface.

Any convex hypersurface naturally divides into two ideal Liouville manifolds meeting along their boundary, and is called Weinstein if these Liouville manifolds are Weinstein.

Theorem 2 is both stronger than Theorem 1 due to the Weinstein condition, but also weaker since it only provides $C^{0}$ -approximations. Indeed, Honda-Huang noted in [43] that the precise generalization of Giroux’s theorem was left unresolved by their work.

Question 3.

[43, Rmk 1.2.4] or [54, Problem 2.1]. Can any closed hypersurface in a contact manifold $(Y,\xi)$ be $C^{\infty}$ -approximated by a convex hypersurface?

Remark 4.

A counter-example to Question 3 was previously proposed by Mori [51, 50] but was later proven to be $C^{\infty}$ -approximable by Weinstein convex hypersurfaces by Breen [12, Cor 1.8]. A different candidate counter-example (with boundary) remains unverified [12, Rmk 1.9].

The goal of this paper is to resolve this longstanding question by proving the following theorem.

Theorem 5 (Main Theorem).

For any $n\geq 2$ , there is a closed hypersurface in standard contact ²ⁿ⁺¹ (and thus in any contact $(2n+1)$ -manifold) that cannot be $C^{2}$ -approximated by convex hypersurfaces.

Our proof uses a novel combination of constructions in contact topology and partially hyperbolic dynamics. The main idea is to use a dynamical property, namely topological mixing, as an obstruction to convexity. Specifically, we adapt a blender construction pioneered by Bonatti-Diaz [7, 10] to construct $C^{1}$ -robustly topologically mixing contactomorphisms, as perturbations of time $1$ maps of Anosov flows. We then use suspension and embedding constructions to produce hypersurfaces in ²ⁿ⁺¹ with $C^{2}$ -robustly mixing characteristic foliations. The characteristic foliation of a convex surface cannot be topologically mixing, so this will prove Theorem 5.

1.1. Characteristic Foliations

Let us briefly recall some dynamical aspects of the structure of hypersurfaces in contact manifolds, before discussing the key results in our proof.

Let be any hypersurface in a contact manifold $(Y,\xi)$ . Recall that the characteristic foliation _ξ of is the (generally singular) oriented, 1-dimensional foliation given by

{}_{\xi}=(T\Sigma\cap\xi)^{\omega}\subset T\Sigma

Here $V^{\omega}$ denotes the symplectic perpendicular of a subspace $V\subset\xi$ with respect to the symplectic structure on $\xi$ . Relatedly, a characteristic vector-field $Z$ of is a vector-field that spans _ξ everywhere and that generates the given orientation.

Any convex surface can be divided into two regions that act as a source and a sink for the flow of any characteristic vector-field for the characteristic foliation.

Definition 6 (Dividing Set).

Let be a convex surface in a contact manifold $(Y,\xi)$ with transverse contact vector-field $V$ . The dividing set with respect to $V$ is given by

\Gamma=H^{-1}(0)\cap\Sigma\quad\text{where }H=\alpha(V)\text{ for any contact form $\alpha$ for $\xi$}

The dividing set is the intersection of the negative region _- and positive region ₊ given by the inverse images $H^{-1}(-\infty,0]\cap\Sigma$ and $H^{-1}[0,\infty)\cap\Sigma$ respectively.

The dividing set is always a transversely cutout hypersurface in with a natural contact structure $T\Gamma\cap\xi$ [43]. The two regions ₊ and _- are ideal Liouville domains with Liouville forms given by the restriction of $\alpha$ , and the characteristic foliation is given by the span of the corresponding Liouville vector-field. Thus is equipped with a folded symplectic structure [37, 13]. Moreover, this structure is independent of the choices of $V$ and $\alpha$ up to isotopy, and is thus canonical up to deformation.

Refer to caption — Figure 1. A convex surface with dividing set and characteristic foliation.

The characteristic foliation _ξ of a convex surface is always transverse to the dividing set , pointing out of ₊ and into _-, and this has some significant implications for the dynamics. For example, we recall the following definition.

Definition 7 (Topological Mixing).

A smooth flow $\Phi:\px@BbbR\times\Sigma\to\Sigma$ is topologically mixing if, for any two non-empty open subsets $U,V\subset\Sigma$ , there is a time $T$ such that

{}_{t}(U)\cap V\neq\emptyset\qquad\text{for all }t>T

We adopt the analogous definition for a diffeomorphism $\Phi:Y\to Y$ of a manifold $Y$ .

Lemma 8.

Let $\Sigma\subset(Y,\xi)$ be a closed convex surface and let $Z$ be a vector-field that is oriented tangent to _ξ. Then $Z$ is not topologically mixing.

Proof.

Choose a dividing set on and a pair of disjoint open subsets $A$ and $B$ of . Let be the flow of a characteristic vector-field $Z$ and consider the subsets

U=\Phi(\px@BbbR\times A)\quad\text{and}\quad V=\Phi(\px@BbbR\times B)

Any flowline of the characteristic foliation _ξ intersects at most once, so $U$ and $V$ are open and disjoint. Moreover, ${}_{t}(U)\cap V=U\cap V=\emptyset$ for all $t$ . Therefore is not topologically mixing. ∎

This is the only dynamical property of convex surfaces that we will need for the rest of the paper. Our goal (in view of Lemma 8) is now to construct examples of hypersurfaces that are robustly topologically mixing in the following sense.

Definition 9.

A smooth flow $\Phi:\px@BbbR\times\Sigma\to\Sigma$ is robustly mixing if there is a $C^{1}$ -open set

\mathcal{U}\subset\operatorname{Flow}(\Sigma)\quad\text{with}\quad\Phi\in\mathcal{U}

such that any $\Psi\in\mathcal{U}$ is topologically mixing. We adopt the analogous definition for a diffeomorphism $\Phi:Y\to Y$ of a manifold $Y$ .

1.2. Suspension

The first step towards this goal is to reduce the problem to a question about contactomorphisms. Fix a contact manifold $(Y,\xi)$ and a contactomorphism

\Phi:Y\to Y

We may take the suspension (or mapping torus) to get an even dimensional space

\Sigma(\Phi)=[0,1]_{s}\times Y/\sim\qquad\text{with a hyperplane field }\eta=\operatorname{span}(\partial_{s})\oplus\xi

The hyperplane field $\eta$ is an example of an even contact structure, and so $\Sigma(\Phi)$ has the structure of a contact Hamiltonian manifold (also referred to as an even contact manifold [5]). Any such space has a natural characteristic foliation, which in this case is given by

\Sigma(\Phi)_{\xi}=\operatorname{span}(\partial_{s})

In particular, the flow of the characteristic foliation is simply the suspension flow of the contactomorphism . One may also take the contactization

{}_{s}\times\Sigma(\Phi)

such that the characteristic foliation on $0\times\Sigma(\Phi)$ (as a hypersurface within the contactization) agrees with the intrinsic characteristic foliation $\Sigma(\Phi)_{\xi}$ .

As an example, we have depicted the suspension $\Sigma(\Phi)$ of a half rotation $\Phi:S^{1}\to S^{1}$ of the circle in Figure 2. This is an example of a contactomorphism that gives rise to a non-convex surface in the contactization of its suspension. It is a nice exercise for the reader to find a perturbation of this foliation that satisfies Giroux’s convexity criterion [35, 12].

In Section 2, we will discuss contact Hamiltonian manifolds and the suspension construction in detail. As an elementary application of this construction, we will prove the following result.

Theorem 10 (Proposition 2.17).

Let $\Phi:Y\to Y$ be a robustly mixing contactomorphism. Then $\Sigma(\Phi)$ has a $C^{2}$ -open neighborhood $\mathcal{U}$ , in the space of hypersurfaces in its contactization, such that every ${}^{\prime}\in\mathcal{U}$ has topologically mixing characteristic foliation.

1.3. Robustly Mixing Contactomorphisms And Blenders

The next step is the construction of robustly mixing contactomorphisms. This is the subject of our most difficult theorem.

Theorem 11 (Theorem 6.1).

Let $(Y,\xi)$ be a closed contact manifold admitting an Anosov Reeb flow with $C^{\infty}$ stable and unstable foliations. Then the $C^{1}$ -open set of robustly mixing contactomorphisms

\operatorname{Cont}_{\operatorname{RM}}(Y,\xi)\subset\operatorname{Cont}(Y,\xi)

is non-empty. More precisely, if $T$ is a (non-zero) multiple of the period of a closed Reeb orbit of , then _T is in the $C^{\infty}$ -closure of the set of robustly mixing contactomorphisms.

Before discussing the proof, we briefly note that there are examples where Theorem 11 applies.

Example 12.

Let $X$ be a closed $n$ -manifold with a hyperbolic metric $g$ . Then the geodesic flow

\Phi:\px@BbbR\times SX\to SX

on the unit cosphere bundle $SX$ is Anosov with smooth stable and unstable foliations. These foliations are precisely the quotients of the foliations by the positive and negative unit conormal bundles of the horospheres in ⁿ, respectively. Specific constructions of closed hyperbolic manifolds in any dimension $2$ or greater can be found in [11, 48, 36].

Theorem 11 is an adaptation of a seminal theorem of Bonatti-Diaz [7, Thm A] to the contact setting, and our main task is to adapt their construction of a certain dynamical structure called a blender. Roughly speaking, a blender is a robust, horseshoe-type structure that forces certain invariant manifolds to be larger than expected. We will give a precise discussion of blenders, along with some background from partially hyperbolic dynamics, in Section 3.

Since their introduction in [7], blenders have become an essential tool in the study of robust and generic properties of smooth dynamical systems. We refer the reader to [6, 10] for a survey on this topic and [9, 8, 47, 53] for just a few examples of their applications. Although constructions of blenders in the symplectic setting have appeared previously (c.f. Nassiri-Pujals [53]), this work is (to our knowledge) the first application of this fundamental dynamical tool to an open problem in symplectic topology.

Our construction of a contact blender will occupy the majority of this paper. The original construction of Bonatti-Diaz [7] does not work without some modifications, and it is quite delicate in certain places, so we have carefully reworked it. We have also included many details that did not appear in [7]. We hope that these details will make this paper more accessible to non-experts in dynamics, e.g. readers with a background primarily in contact topology.

Remark 1.1.

The hypothesis that the Anosov Reeb flow has $C^{\infty}$ stable and unstable foliation is not present in [7, Thm A] and is quite restrictive. In our proof, we will use it in an essential way to prove one of the axioms of a blender (see Definition 3.22 and Section 5.3). However, we view it as a purely technical hypothesis that can likely be eliminated with more careful analysis.

1.4. Non-Convex Hypersurfaces

We are now ready to combine the results discussed thus far to prove the main theorem. Already, we can use Lemma 8, Theorems 10-11 and Example 12 to immediately acquire a specific case of our main theorem.

Theorem 13.

The cosphere bundle $SX$ of a closed hyperbolic manifold $X$ has a contactomorphism $\Phi:X\to X$ such that the suspension

\Sigma(\Phi)\subset\px@BbbR\times\Sigma(\Phi)

cannot be $C^{2}$ -approximated by a convex hypersurface.

In order to enhance this result to acquire the more general Theorem 5, we apply two difficult theorems. First, we have the following theorem of Sullivan.

Theorem 14.

[56] Every closed hyperbolic manifold $W$ has a finite cover $X$ that is stably parallelizable.

We also have the following existence theorem for Legendrian embeddings. This follows from the h-principle of Murphy [52], although this case follows from the earlier h-principle of Gromov.

Theorem 15.

[52] Any closed, stably parallelizable manifold $X$ has a Legendrian embedding $X\to{}^{2n+1}$ .

Finally, we need the following lemma that will be proven in Section 2. Recall that a contactomorphism is positive if it is the time $1$ map of a (possibly time-dependent) contact Hamiltonian that is positive as a smooth function.

Lemma 16 (Lemma 2.20).

Let $\Lambda\subset(Y,\xi)$ be a closed Legendrian and let $\Phi:S\Lambda\to S\Lambda$ be a positive contactomorphism of the cosphere bundle $S\Lambda$ . Then there exists a contact embedding

(-\epsilon,\epsilon)\times\Sigma(\Phi)\to(Y,\xi)\qquad\text{for small $\epsilon$}

With these preliminary results in hand, we can now proceed with the proof of the main result.

Theorem 17.

There is a closed, embedded hypersurface $\Sigma\subset{}^{2n+1}$ for any $n\geq 2$ that cannot be $C^{2}$ -approximated by convex hypersurfaces.

Proof.

Take a stably parallelizable closed hyperbolic manifold $X$ (via Theorem 14 and Example 12) with a Legendrian embedding $X\to{}^{2n+1}$ (via Theorem 15). By Theorem 6.1, there is a robustly mixing contactomorphism

\Phi:SX\to SX

that is $C^{\infty}$ -close to the time $T$ Reeb flow, where $T$ is the period of a closed Reeb orbit. Since the time $T$ Reeb flow is a positive contactomorphism, is also positive. By Lemma 16, there is a contact embedding $U\to{}^{2n+1}$ of a neighborhood $U\subset\px@BbbR\times\Sigma(\Phi)$ of the suspension $\Sigma(\Phi)$ in its contactization. By Theorem 10, $\Sigma(\Phi)$ cannot be $C^{2}$ -approximated by a convex surface. ∎

Every contact manifold contains a contact Darboux ball, so this also resolves the general case.

1.5. Open Problems

This work raises many interesting questions at the interface of contact topology and dynamics. We conclude this introduction by mentioning a few of these problems.

Definition 18 (Robust Non-Convexity).

A hypersurface in a contact manifold is robustly non-convex if there is a $C^{2}$ -neighborhood $\mathcal{U}$ in the space of embedded hypersurfaces such that

{}^{\prime}\text{ is not convex for any }{}^{\prime}\in\mathcal{U}

Similarly, a contactomorphism $\Phi:Y\to Y$ is called robustly non-convex if the suspension $\Sigma(\Phi)$ is.

Theorem 5 states that robustly non-convex hypersurfaces exist in any contact manifold of dimension five and higher. It is thus natural to ask about the diversity of such hypersurfaces.

Question 19.

Is every smoothly embedded hypersurface $\Sigma\to Y$ in a contact manifold $(Y,\xi)$ isotopic to a robustly non-convex one in dimensions $5$ and higher?

Our proof of Theorem 5 relies heavily on techniques from partially hyperbolic dynamics and thus may not be applicable to address the full version of Question 19. Indeed, some spaces are known to have no partially hyperbolic diffeomorphisms. For example, we have the following result of Burago-Ivanov.

Theorem 20.

[15] The $3$ -sphere $S^{3}$ does not admit any partially hyperbolic diffeomorphisms.

A result of Bonnati-Diaz-Pujals [9, Thm 2] states that any robustly mixing diffeomorphism must be partially hyperbolic. Thus $S^{3}$ admits no robustly mixing diffeomorphisms and the obstructions developed in this paper cannot be applied. This leads naturally to the following question.

Question 21.

Does $(S^{3},\xi_{\operatorname{std}})$ admit a robustly non-convex contactomorphism?

Question 22.

More generally, is the suspension $\Sigma(\operatorname{Id}_{S})$ of the identity contactomorphism $\operatorname{Id}_{S}$ on $S^{3}$ isotopic to a robustly non-convex hypersurface in its contactization?

A negative answer to Question 21 or Question 22 would reveal an interesting connection between the global topology of hypersurfaces and their convex approximability.

Remark 23.

The result [9, Thm 2] of Bonnati-Diaz-Pujals follows from a $C^{1}$ -generic dichotomy [9, Thm 1] between maps that admit a dominated splitting and maps with infinitely many sources and sinks (i.e. exhibiting the Newhouse phenomenon), on each homoclinic class.

On the other hand, the methods of Honda-Huang [42, 43] for convexifying hypersurfaces involve the introduction of many new critical points to the characteristic foliation by many small $C^{0}$ -perturbations. It is interesting to ask if a hypersurface whose characteristic foliation satisfies a Newhouse-type property along every homoclinic class can be approximated by convex surfaces at some higher-than-expected regularity.

Outline

This concludes the introduction (Section 1) of this paper. In Section 2, we will discuss contact Hamiltonian manifolds (also known as even contact manifolds) and the suspension construction. In Section 3, we will review the necessary background from the theory of partially hyperbolic diffeomorphisms and blenders. In Sections 4 and 5 we will undertake the construction of our contact blender, following Bonatti-Diaz [7]. Finally, we prove Theorem 11 in Section 6.

Acknowledgements

Question 3 was discussed at the workshop Higher-Dimensional Contact Topology at the American Institute of Mathematics in April 2024. We thank the organizers Roger Casals, Yakov Eliashberg, Ko Honda, and Gordana Matic and the AIM staff for a fruitul week.

We also thank the members of the conformal symplectic structures breakout room (Mélanie Bertelson, Kai Celiebak, Fabio Gironella, Pacôme Van Overschelde, Kevin Sackel and Lisa Traynor) for our discussion of this problem, including a very helpful overview of even contact structures and the suspension construction. Finally, we thank Joseph Breen, Kai Celiebak and Austin Christian for input on earlier drafts.

2. Contact Hamiltonian Manifolds

In this brief section, we discuss the theory of contact Hamiltonian manifolds, which are also called even contact manifolds in the terminology of Bertelson-Meigniez [5].

This theory has satisfying parallels with the theory of (stable) Hamiltonian manifolds [20, 59, 17], which motivates our preferred nomenclature. We freely use these two terms as synonyms.

2.1. Fundamentals

We start with the basic facts, which mirror the stable Hamiltonian case.

Definition 2.1.

A contact Hamiltonian manifold $(\Sigma,\eta)$ is a $2n$ -manifold equipped with a coorientable, maximally non-integrable, plane distribution of codimension one

\eta\subset T\Sigma

Equivalently, $\eta$ is the kernel of a contact Hamiltonian form $\nu$ with $\nu\wedge d\nu^{n-1}$ is nowhere vanishing.

Example 2.2 (Product).

Let $(Y,\xi)$ be a contact manifold. Then the manifolds

\px@BbbR\times Y\qquad\text{and}\qquad\px@BbbR/\px@BbbZ\times Y

are contact Hamiltonian manifolds with distribution $\eta=\partial_{t}\oplus\xi$ , where $t$ is the -coordinate.

Every contact Hamiltonian manifold has a natural line distribution (or equivalently, foliation).

Definition 2.3.

The characteristic foliation _η of a contact Hamiltonian manifold $(\Sigma,\eta)$ is given by

{}_{\eta}=\operatorname{ker}(d\nu|_{\eta})\subset T\Sigma\qquad\text{for any contact Hamiltonian form $\nu$ for $\eta$}

A characteristic vector-field $Z$ is a section of _ξ and a framing form $\theta$ is a $1$ -form whose restriction to _ξ is nowhere vanishing. A framing form determines a characteristic vector-field by

(2.1)

\nu(Z)=0\qquad(\iota_{Z}d\nu)|_{\eta}=0\quad\text{and}\quad\theta(Z)=1

Lemma 2.4.

Any characteristic vector-field $Z$ on a contact Hamiltonian manifold $(\Sigma,\eta)$ preserves $\eta$ .

Proof.

Let be the flow of $Z$ . Note that $\iota_{Z}d\nu=f\nu$ for some smooth $f$ since $(\iota_{Z}d\nu)|_{\eta}=0$ . Thus

\mathcal{L}_{Z}\nu=d(\iota_{Z}\nu)+\iota_{Z}d\nu=f\nu\qquad\text{and therefore}\qquad{}_{t}^{*}\nu=g_{t}\cdot\nu\qed

Definition 2.5 (Hamiltonian Vector-fields).

The Reeb vector-field $R$ of a contact Hamiltonian manifold $(\Sigma,\eta)$ with contact Hamiltonian form $\nu$ and framing $\theta$ is the unique vector-field satisfying

\theta(R)=0\qquad\nu(R)=1\quad\text{and}\quad(\iota_{R}d\nu)|_{\xi}=0\quad\text{where}\quad\xi=\operatorname{ker}(\theta)

The Hamiltonian vector-field $V_{H}$ of a function $H:\Sigma\to\px@BbbR$ is the unique vector-field satisfying

\theta(V_{H})=0\qquad\nu(V_{H})=H\quad\text{and}\quad(\iota_{V_{H}}d\nu-dH(R)\cdot\nu+dH)|_{\xi}=0

Note that the last condition is equivalent to $\mathcal{L}_{V_{H}}\nu=dH(R)\cdot\nu+g\cdot\theta$ given that $H=\alpha(V_{H})$ .

2.2. Contact Hamiltonian Hypersurfaces

A natural source of contact Hamiltonian hypersurfaces are (special) hypersurfaces in contact manifolds.

Definition 2.6.

A hypersurface in a contact manifold $(Y,\xi)$ is contact Hamiltonian called if

\xi\text{ is transverse to }T\Sigma

A framing vector-field $U$ is a vector-field in a neighborhood of such that

U

is transverse to and

U

is tangent to

\xi

Lemma 2.7.

Let be a contact Hamiltonian hypersurface in $(Y,\xi)$ with framing vector-field $U$ . Then

(\Sigma,\xi\cap T\Sigma)\quad\text{is contact Hamiltonian with framing form}\quad\theta=\iota_{U}d\alpha

Proof.

Fix a contact form $\alpha$ on $Y$ . Then $d\alpha$ has a $1$ -dimensional kernel on $\operatorname{ker}(\alpha)\cap T\Sigma$ by standard symplectic linear algebra. It follows that $\nu=\alpha|_{T\Sigma}$ is a contact Hamiltonian form. Similarly, since $d\alpha$ is non-degenerate on $\xi$ , we must have

d\alpha(U,Z)\neq 0\text{ for any non-vanishing }Z\in{}_{\eta}\qed

Every contact Hamiltonian manifold arises as a hypersurface in its own contactization.

Definition 2.8.

The contactization $(C\Sigma,\alpha)$ of a closed contact Hamiltonian manifold $(\Sigma,\eta)$ with contact Hamiltonian form $\nu$ and framing form $\theta$ is given by

C\Sigma=(-\epsilon,\epsilon)_{s}\times\Sigma\qquad\text{with contact form }\alpha=s\theta+\nu

Lemma 2.9.

The contactization $(C\Sigma,\alpha)$ is a contact manifold for $\epsilon$ small, and naturally embeds as a contact Hamiltonian hypersurface

\Sigma=0\times\Sigma\subset C\Sigma\qquad\text{with}\qquad\nu=\alpha|

Proof.

Note that $\nu\wedge d\nu^{n-1}$ is nowhere vanishing and the characteristic vector-field $Z$ of $\theta$ is a nowhere vanishing vector-field that satisfies

\iota_{Z}(\nu\wedge d\nu^{n-1})=0

A $1$ -form $\mu$ on thus satisfies $\mu\wedge\nu\wedge d\nu^{n-1}$ if and only if $\mu(Z)\neq 0$ everywhere. Therefore

\mu\wedge\nu\wedge d\nu^{n-1}\qquad\text{and}\qquad\nu\wedge ds\wedge\theta\wedge d\nu^{n-1}

are volume forms on and $C\Sigma$ respectively. The second volume form above agrees with $\alpha\wedge d\alpha^{n}$ along $0\times\Sigma$ , so there is a neighborhood of $0\times\Sigma$ where $\alpha\wedge d\alpha^{n}$ is a volume form. ∎

There is a natural way to deform a contact Hamiltonian manifold as a graph in its own contactization (c.f. [18] for a stable Hamiltonian analogue).

Definition 2.10 (Deformation).

The deformation $(\Sigma,\eta_{H})$ of the contact Hamiltonian manifold $(\Sigma,\eta)$ by the Hamiltonian $H:\Sigma\to\px@BbbR$ is given by

\eta_{H}=\operatorname{ker}(\nu_{H})\qquad\text{where}\qquad\nu_{H}=H\cdot\theta+\nu

This is precisely the pullback of the induced contact Hamiltonian structure on the graph

\operatorname{Gr}H\subset C\Sigma\qquad\text{given by}\qquad\operatorname{Gr}H=\big{\{}(H(x),x)\;:\;x\in\Sigma\big{\}}

Finally, we note that the contactization provides a local model for the neighborhood of any contact Hamiltonian hypersurface. Specifically, we have the following (strict) standard neighborhood lemma.

Lemma 2.11 (Collar Neighborhood).

Let be a contact Hamiltonian hypersurface in a contact manifold $(Y,\xi)$ . Fix a contact form $\alpha$ on $Y$ and a framing vector-field $U$ of such that

\iota_{U}d\alpha\text{ is closed}

Then the flow by $U$ yields a strict contact embedding

\iota:(-\epsilon,\epsilon)\times\Sigma\to Y\qquad\text{with}\qquad\iota^{*}\alpha=s\cdot\theta+\nu\quad\text{where}\quad\text{$\nu=\alpha|$ and $\theta=\iota_{U}d\alpha|$}

Proof.

First note that we have the following calculation.

\mathcal{L}_{U}(\iota_{U}d\alpha)=d(d\alpha(U,U))+d(\iota_{U}d\alpha)=0

Now let $\iota:(-\epsilon,\epsilon)_{s}\times\Sigma\to Y$ be the tubular neighborhood coordinates of induced by $U$ . Then the previous calculation and the fact that $\iota_{U}\theta=0$ shows that the $1$ -form

\theta=\iota^{*}(\iota_{U}d\alpha)=\iota_{\partial_{s}}d(\iota^{*}\alpha)\qquad\text{satisfies}\qquad\mathcal{L}_{\partial_{s}}\theta=0\text{ and }\theta(\partial_{s})=0

Thus $\theta$ is the pullback of a differential form on to $(-\epsilon,\epsilon)\times\Sigma$ . Moreover, we see that

\mathcal{L}_{U}\alpha=d(\iota_{U}\alpha)+\iota_{U}d\alpha=\iota_{U}d\alpha\qquad\text{and thus}\qquad\mathcal{L}_{\partial_{s}}\iota^{*}\alpha=\theta

It follows that $\iota^{*}\alpha$ and $s\theta+\nu$ satisfy the same ODE and have the same restriction to $0\times\Sigma$ . Therefore they are equal on the given tubular neighborhood. ∎

2.3. Suspensions

The key examples of contact Hamiltonian manifolds for the purposes of this paper are suspensions of contactomorphisms (or synonymously, mapping tori). This is analogous to the mapping torus construction of stable Hamiltonian manifolds [20, §2.1].

Fix a contact manifold $(Y,\xi)$ with a contactomorphism of $Y$ . Recall that the suspension $\Sigma(\Phi)$ of is the quotient of $\px@BbbR\times Y$ by the map

\bar{\Phi}:\px@BbbR\times Y\to\px@BbbR\times Y\qquad\text{given by}\qquad\bar{\Phi}(t,y)=(t-1,\Phi(y))

Since $\xi$ is preserved by , the contact Hamiltonian structure $\operatorname{span}(\partial_{t})\oplus\xi$ on the product $\px@BbbR\times Y$ (see Example 2.2) is $\bar{\Phi}$ -invariant. It descends to a contact Hamiltonian structure $\eta$ on the suspension.

Definition 2.12 (Contact Suspension).

The contact suspension of a contactomorphism $\Phi:(Y,\xi)\to(Y,\xi)$ is the contact Hamiltonian manifold given by

(\Sigma(\Phi),\eta)\qquad\text{with the framing form}\qquad\theta=dt

The characteristic foliation and a natural framing form are given by the coordinate vector-field and covector-field in the $t$ -direction.

\Sigma(\Phi)_{\eta}=\operatorname{span}(\partial_{t})\qquad\text{and}\qquad\theta=dt

Remark 2.13.

In this case, the contact structure on the contactization extends to all of $\px@BbbR\times\Sigma(\Phi)$ , and we will refer to this latter space as the contactization.

The most important result of this section is the following lemma, which relates graph-like perturbations of the suspension hypersurface to perturbations of the underlying contactomorphism.

Lemma 2.14 (Hamiltonian Perturbation).

Let $H:\Sigma(\Phi)\to\px@BbbR$ be a smooth function on the suspension of $\Phi:(Y,\xi)\to(Y,\xi)$ such that $\theta=dt$ frames $\eta_{H}$ . Then there exists a contactomorphism

{}^{H}:(Y,\xi)\to(Y,\xi)\qquad\text{with an isomorphism}\qquad{}^{H}:(\Sigma({}^{H}),\eta)\to(\Sigma(\Phi),\eta_{H})

such that $\operatorname{dist}_{C^{1}}({}^{H},\Phi)\leq C\cdot\|H\|_{C^{2}}$ for any Riemannian metric $g$ and a constant $C=C(g)$ .

Proof.

Let $Z_{H}$ denote the characteristic vector-field of $\eta_{H}$ with respect to the framing form $dt$ . The characteristic flow ^H of $Z_{H}$ satisfies ${}^{H}_{t}(0\times Y)=t\times Y$ since $Z_{H}(t)=1$ and ^H preserves $\eta_{H}$ by Lemma 2.4. Finally, note that $\nu_{H}$ restricts to $\alpha$ on $0\times Y$ and thus $\eta_{H}\cap(0\times Y)$ is $\xi$ . By restriction to $\px@BbbR\times Y$ where $Y$ is identified with $0\times Y$ in $\Sigma(\Phi)$ , we get a map

{}^{H}:\px@BbbR\times Y\to\Sigma(\Phi)\qquad\text{with}\qquad({}^{H})^{*}dt=dt\text{ and }({}^{H})^{*}\eta_{H}=\operatorname{span}(\partial_{s})\oplus\xi

We now define ^H to be the time 1 map ${}^{H}_{1}$ of the flow. Then the map ^H satisfies

{}^{H}(s,x)={}^{H}(s-1,{}^{H}(x))

In particular, ^H descends to a map ${}^{H}:\Sigma({}^{H})\to\Sigma(\Phi)$ with $({}^{H})^{*}\eta_{H}=\eta$ . Finally, note that by Definition 2.3, $Z_{H}$ is defined by the formulas

\theta(Z_{H})=1\qquad\iota_{Z_{H}}(H\cdot\theta+\nu)=0\qquad\iota_{Z_{H}}(dH\wedge\theta+d\nu)=0

It follows that there is a smooth linear bundle map

T:{}^{0}(\Sigma(\Phi))\oplus{}^{1}(\Sigma(\Phi))\oplus{}^{2}(\Sigma(\Phi))\to T\Sigma(\Phi)\qquad\text{such that}\qquad Z_{H}=T(H,\nu,d\nu)

In particular, for any choice of metric on $\Sigma(\Phi)$ , there is a constant $C>0$ and an estimate

\|Z_{H}-Z\|_{C^{1}}\leq C\cdot\|H\|_{C^{1}}

The same estimate holds for the flow and the time-1 maps. ∎

Example 2.15 (Mapping Torus Of Identity).

Let $(Y,\xi)$ be a contact manifold with contact form $\alpha$ and consider the suspension of the identity

\Sigma(\operatorname{Id}_{Y})=(\px@BbbR/\px@BbbZ)_{t}\times Y\qquad\text{with contact Hamiltonian form }\nu=\beta

Fix a Hamiltonian $H:\px@BbbR/\px@BbbZ\times Y\to\px@BbbR$ and let $V_{H}:\px@BbbR/\px@BbbZ\times Y\to TY$ be the contact vector-field of $H$ . Consider the deformation

(\px@BbbR/\px@BbbZ\times Y,\nu_{-H})\qquad\text{with}\qquad\nu_{-H}=-Hdt+\alpha

It is simple to check that in this case the characteristic vector-field for framing form $dt$ is given by

Z_{H}=\partial_{t}+V_{H}

It follows that the contactomorphism ^H constructed in Lemma 2.14 is precisely the time 1 map of the contactomorphism generated by $-H$ and map defines an isomorphism

(\Sigma({}^{H}),\eta)\simeq(\px@BbbR/\px@BbbZ\times Y,\eta_{-H})

We easily derive the following analogue of Lemma 2.14 for perturbations of hypersurfaces.

Lemma 2.16 (Surface Perturbation).

Let be a contactomorphism. Then there is a $C^{2}$ -neighborhood

\mathcal{U}\subset\operatorname{Emb}(C\Sigma(\Phi))\qquad\text{of the sub-manifold}\qquad\iota:\Sigma(\Phi)\to C\Sigma(\Phi)

in the space of embedded smooth sub-manifolds such that any element $\Sigma\in\mathcal{U}$ has an isomorphism

:(\Sigma,\eta\cap\Sigma)\to(\Sigma(),\eta)\qquad\text{where}\qquad\operatorname{dist}_{C^{1}}(,\Phi)\leq C\cdot\operatorname{dist}_{C^{2}}(\Sigma,\Sigma(\Phi))

Proof.

Any surface in the contactization $(-\epsilon,\epsilon)\times\Sigma(\Phi)$ that is $C^{2}$ -close to $0\times\Sigma(\Phi)$ is the graph of a function $H$ on $\Sigma(\Phi)$ with $C^{2}$ -norm controlled by the $C^{2}$ -distance of to $0\times\Sigma$ . Thus this lemma is immediate from Lemma 2.14 and Definition 2.10. ∎

We are now ready to prove Theorem 10 from the introduction, which is very simple.

Proposition 2.17 (Theorem 10).

Let $\Phi:Y\to Y$ be a $C^{1}$ -robustly mixing contactomorphism. Then the suspension $\Sigma(\Phi)$ has a $C^{2}$ -neighborhood

\mathcal{U}\subset\operatorname{Emb}(C\Sigma(\Phi))

consisting of hypersurfaces with topologically mixing characteristic foliation.

Proof.

By Lemma 2.16, there is a neighborhood $\mathcal{U}$ of $\Sigma(\Phi)$ such that every $\Sigma\in\mathcal{U}$ is the suspension of a contactomorphism that is $C^{1}$ -close to . Since is robustly mixing (Definition 9), we can assume after shrinking $\mathcal{U}$ that is topologically mixing for any in $\mathcal{U}$ . ∎

2.4. Constructions Of Contact-Hamiltonian Hypersurfaces

We conclude this section with constructions of contact Hamiltonian hypersurfaces. The following elementary embedding lemma will be our main tool.

Lemma 2.18 (Disk Neighborhood).

Let be a closed contact manifold and let $\Phi:\Gamma\to\Gamma$ be a positive contactomorphism. Fix a contact form $\beta$ and let

H:\px@BbbR/\px@BbbZ\times\Gamma\to(0,\infty)

be the corresponding generating Hamiltonian such that $\Phi={}^{H}_{1}$ . Then there is a contact embedding

(2.2)

(-\epsilon,\epsilon)_{s}\times\Sigma(\Phi)\to(D^{2}\times\Gamma,-a\cdot r^{2}d\theta+\beta)\qquad\text{ for any }a>\frac{1}{2\pi}\cdot\operatorname{max}H

Proof.

We use the following smooth map in radial coordinates.

\iota:(-2\pi a,0)_{s}\times\px@BbbR/{}_{t}\times\Gamma\to(D^{2}-0)\times\Gamma\quad\text{given by}\quad\iota(s,t,x)=((-s/2\pi a)^{1/2},2\pi t,x)

This map satisfies $\iota^{*}(-ar^{2}d\theta+\beta)=sdt+\beta$ . Fix $a$ satisfying $2\pi a>\operatorname{max}H$ . By Example 2.15, the graph of $-H$ defines an embedding

(\Sigma(\Phi),\eta)\simeq(\px@BbbR/\px@BbbZ\times\Gamma,-Hdt+\beta)\to((-2\pi a,0)_{s}\times\px@BbbR/{}_{t}\times\Gamma,sdt+\beta)

This embedding extends to a contactomorphism (2.2) by the flow of $\partial_{s}$ (see Lemma. 2.11). ∎

The next lemma shows that small regions of the jet bundle contains arbitrarily large tubular neighborhoods of the cosphere bundle.

Lemma 2.19.

Let $X$ be a closed smooth manifold with cosphere bundle $SX$ and jet bundle $JX$ . Fix a neighborhood $U$ of $X\subset JX$ and a contact form $\beta$ on $SX$ . Then for any $a>0$ , there is a contact embedding

(D^{2}\times SX,\xi_{a,\beta})\to U\qquad\text{where}\qquad\xi_{a,\beta}=\operatorname{ker}(-a\cdot r^{2}d\theta+\beta)

Proof.

Recall that the jet bundle $JX$ is given by ${}_{t}\times T^{*}X$ with the standard contact form $\alpha_{\operatorname{std}}=dt+\lambda_{\operatorname{std}}$ . We break the proof into two steps.

Step 1. First assume that $U=JX$ (so that we may ignore the neighborhood). There is a standard embedding of the symplectization of $\px@BbbR\times SX$ into $T^{*}X$ via the Liouville flow of $T^{*}X$ . By using $t$ -translation, we can extend this to an embedding

\kappa:{}_{\rho}\times{}_{t}\times SX\to JX\qquad\text{with}\qquad\kappa^{*}\alpha_{\operatorname{std}}=dt+e^{\rho}\beta

By applying a further change coordinates by taking $s=-e^{-\rho}$ , we get a map

\jmath:(-\infty,0]_{s}\times{}_{t}\times SX\to JX\qquad\text{with}\qquad\jmath^{*}(dt+e^{\rho}\beta)=dt-s^{-1}\beta=-s^{-1}\cdot(-s\cdot dt+\beta)

Now we construct an embedding to $(-\infty,0]_{s}\times{}_{t}\times SX$ . Take a disk $D\subset(-\infty,0]_{s}\times{}_{t}$ of radius $(2a)^{1/2}$ centered at a point $(s_{0},t_{0})$ . We consider the Liouville form

\lambda=\frac{1}{2}((t-t_{0})ds-(s-s_{0})dt))\qquad\text{satisfying}\qquad d\lambda=-ds\wedge dt

Next, let $\tau:{}^{2}\to\px@BbbR$ be a primitive such that $\lambda=-sdt+d\tau$ and consider the diffeomorphism

\Psi:{}_{s}\times{}_{t}\times SX\to{}_{s}\times{}_{t}\times SX\quad\text{given by}\quad\Psi(s,t,x)=(s,t,{}^{R}(\tau(s,t),x)

Here ${}^{R}:{}_{\tau}\times SX\to SX$ denote the Reeb flow of $\beta$ . We compute that

{}^{*}(-sdt+\beta)=-sdt+\beta(R)\cdot d\tau+({}^{R})^{*}\beta=-sdt+d\tau+\beta=\lambda+\beta

Finally, we compose with the map $\Phi=\phi\times\operatorname{Id}_{SX}$ where

\phi:D^{2}\to{}_{s}\times{}_{t}\qquad\text{given by}\qquad(r,\theta)\mapsto(ar\cos(\theta)+s_{0},ar\sin(\theta)+t_{0})

The composition $\Psi\circ\Phi$ now restricts to a contact embedding

D^{2}\times SX\to(-\infty,0]_{s}\times\times SX\qquad\text{with}\qquad(\Psi\circ\Phi)^{*}(-s\cdot dt+\beta)=-\frac{a}{2}\cdot r^{2}d\theta+\beta

The composition $\jmath\circ\Psi\circ\Phi:D^{2}\times SX\to JX$ is the desired embedding in the lemma.

Step 2. Now consider the general case where $U\subset JX$ is a proper open set. There is a natural flow of contactomorphisms ${}^{\prime}:{}_{r}\times JX\to JX$ given by ${}_{r}^{Z}(t,z)=(e^{r}t,{}^{Z}_{r}(z))$ where ^Z is the Liouville flow on $T^{*}X$ . This flow is generated by a vector-field $U=t\partial_{t}+Z$ satisfying

\mathcal{L}_{U}\alpha_{\operatorname{std}}=\alpha_{\operatorname{std}}

In particular, any compact set in $JX$ can be pushed into $U$ by ${}^{\prime}_{r}$ for $r$ sufficiently negative. We may then compose the embedding from Step 1 with ${}^{\prime}_{r}$ to acquire the desired embedding. ∎

By using the Weinstein neighborhood theorem for Legendrians [33] to convert a Legendrian into an embedding of the cosphere bundle of the Legendrian, we acquire the following corollary.

Lemma 2.20 (Lemma 16).

Let $\Lambda\subset(Y,\xi)$ be a closed Legendrian sub-manifold and let $\Phi:S\Lambda\to S\Lambda$ be a positive contactomorphism of the cosphere bundle $S\Lambda$ . Then there is a contact embedding

(-\epsilon,\epsilon)\times\Sigma(\Phi)\to(Y,\xi)

Proof.

By the Weinstein neighborhood theorem, we have a contact embedding

(U,\xi_{\operatorname{std}})\to(Y,\xi)\qquad\text{for a neighborhood }U\subset JX\text{ of }X

By Lemma 2.18 and Lemma 2.19, for any contact form $\beta$ on $S\Lambda$ , there are constants $\epsilon,a>0$ and a contact embedding of the form

((-\epsilon,\epsilon)\times\Sigma(\Phi),\eta)\to(D^{2}\times SX,\xi_{a,\beta})\to(U,\xi_{\operatorname{std}})\qed

3. Partial Hyperbolicity

In this section, we review the theory of partially hyperbolic maps and blenders.

Remark 3.1.

This section contains extensive background aimed at non-experts in dynamics. We recommend that the reader look to Crovisier-Potrie [25], Hertz-Hertz-Ures [38], Hirsch-Pugh-Shub [39] or Bonatti-Diaz-Viana [10] for a more comprehensive treatment. We also recommend the excellent book of Fisher-Hasselblatt [32] for an accessible treatment of hyperbolic dynamics.

3.1. Fundamentals

Fix a compact smooth manifold $Y$ and a diffeomorphism

\Phi:Y\to Y

Definition 3.2 (Expansion/Contraction).

A sub-bundle $E\subset TY$ is uniformly expanding with respect to and a Riemannian metric $g$ on $Y$ if there are constants $C>0$ and $\lambda>1$ such that

|T{}^{n}(u)|\geq C\cdot\lambda^{n}\cdot|u|\qquad\text{for all }n\geq 0

Similarly, $E$ is uniformly contracting for and $g$ if it is uniformly expanding for ^-1 and $g$ .

Definition 3.3 (Domination).

A continuous splitting of $TY$ into continuous sub-bundles

TY=E_{1}\oplus\dots\oplus E_{k}

is dominated with respect to and a metric $g$ if there are constants $C>0$ and $\lambda>1$ such that

|T{}^{n}(u)|\geq C\cdot\lambda^{n}\cdot|T{}^{n}(v)|\qquad\text{for all }n\geq 0\text{ and any unit vectors }u\in E_{i+1}\text{ and }v\in E_{i}

The constant $\lambda$ is the constant of dilation and the splitting may also be called $\lambda$ -dominated.

Dominated splittings obey several fundamental properties (cf. [10, Appendix B] or [25]) that we now discuss briefly. First, dominated splittings are persistent with respect to $C^{1}$ -perturbation.

Theorem 3.4 ( $C^{1}$ -Persistence).

Let $\Phi:Y\to Y$ be a diffeomorphism with a $\lambda$ -dominated splitting

TY=E_{1}\oplus\dots\oplus E_{k}

Then for $\mu<\lambda$ , there is a $C^{1}$ -neighborhood $\mathcal{U}$ of such that every $\Psi\in\mathcal{U}$ has a $\mu$ -dominated splitting

TY=E_{1}(\Psi)\oplus\dots\oplus E_{k}(\Psi)\qquad\text{such that}\qquad\operatorname{dim}E_{i}(\Psi)=\operatorname{dim}E_{i}(\Phi)

Next, dominated splittings with an expanding (or contracting) factor possess a unique, expanding (or contracting) invariant foliation (cf. [39] and [10, Thm B.7]).

Theorem 3.5 (Foliations).

[39] Let $\Phi:Y\to Y$ be a diffeomorphism with a dominated splitting

TY=D\oplus E\quad\text{with $E$ uniformly expanding}

Then there is a unique Hölder foliation $F$ with smooth leaves such that

\text{$F$ is tangent to $E$}\quad\qquad{}_{*}F=F\qquad\text{and}\qquad F(P)\simeq{}^{\operatorname{rk}(E)}

Finally, the stable manifold theorem (c.f. Hirsch-Pugh-Shub [39, Thm 4.1]) asserts the existence of stable and unstable manifolds for hyperbolic invariant sets.

Theorem 3.6 (Invariant Manifolds).

Let be an invariant sub-manifold of a diffeomorphism $\Phi:Y\to Y$ with normal hyperbolic splitting $E^{u}\oplus T\Gamma\oplus E^{s}$ . Then there are unique invariant sub-manifolds

W^{s}(\Gamma,\Phi)\qquad\text{and}\qquad W^{u}(\Gamma,\Phi)\qquad\text{containing }\Gamma

that are locally invariant near , $C^{1}$ -robust and that satisfy

TW^{s}(\Phi,\Gamma)=E^{s}\oplus T\Gamma\qquad\text{and}\qquad TW^{u}(\Phi,\Gamma)=E^{u}\oplus T\Gamma\qquad\text{along }\Gamma

Notation 3.7 (Local Invariant Manifolds).

Given a normally hyperbolic invariant set and an open neighborhood $U$ of we will use the notation

W^{s}_{\operatorname{loc}}(\Gamma,\Phi;U)\qquad\text{and}\qquad W^{u}_{\operatorname{loc}}(\Gamma,\Phi;U)

to denote the respective components of $W^{s}(\Gamma,\Phi)\cap U$ and $W^{u}_{(}\Gamma,\Phi)\cap U$ that contain . We adopt analogous notation for the leaves of the foliations $F^{s}(\Phi)$ and $F^{u}(\Phi)$ when defined.

This paper will be entirely oriented towards the following class of diffeomorphisms.

Definition 3.8 (Partially Hyperbolic).

A diffeomorphism $\Phi:Y\to Y$ is partially hyperbolic if there is a splitting the tangent bundle into -invariant, continuous sub-bundles

TY=E^{s}(\Phi)\oplus E^{c}(\Phi)\oplus E^{u}(\Phi)

such that $E^{s}(\Phi)$ is uniformly contracting, $E^{u}(\Phi)$ is uniformly expanding and the splitting is dominated with respect to and some (or equivalently any) Riemannian metric $g$ .

Example 3.9.

Let $\Psi:\px@BbbR\times Y\to Y$ be an Anosov flow generated by a vector-field $V$ . Then for any $T$ , the time $T$ map

{}_{T}:Y\to Y

is partially hyperbolic. The stable and unstable bundles of _T are those of , while the central bundle is the span of $V$ .

Partially hyperbolic contactomorphisms have some special compatibility properties with the underlying contact structure. For instance, we have the following lemma (for use later).

Lemma 3.10.

Fix a contact $(2n+1)$ -manifold $(Y,\xi)$ and a partially hyperbolic contactomorphism

\Phi:(Y,\xi)\to(Y,\xi)

such that $\xi=E^{s}(\Phi)\oplus E^{u}(\Phi)$ and $E^{c}(\Phi)$ is transverse to $\xi$ . Then the stable and unstable foliations $F^{s}(\Phi)$ and $F^{u}(\Phi)$ have Legendrian leaves.

Proof.

The leaves of $F^{s}(\Phi)$ and $F^{u}(\Phi)$ are tangent to $\xi$ , and the dimensions of the leaves of $F^{s}(\Phi)$ and $F^{u}(\Phi)$ add to the rank of $\xi$ . It follows the dimension of the leaves $F^{s}(\Phi)$ and $F^{u}(\Phi)$ must be $n$ , so that the leaves are Legendrian. ∎

3.2. Holonomy

The holonomy of the stable and unstable foliations are a key tool in the analysis of partially hyperbolic maps, which we will need in Section 4. We next briefly review this concept.

Let $F$ be a transversely continuous foliation with smooth leaves on a manifold $Y$ . Let be a leaf and let $P$ be a point in . Recall that a a transversal to at $P$ is a sub-manifold $S\subset Y$ of dimension $\operatorname{codim}F$ tranverse to the leaves of $F$ and intersecting at $P$ .

Definition 3.11 (Holonomy).

[16, Ch 2] Let be a leaf of $F$ equipped with transversals $S$ and $T$ at points $P$ and $Q$ in . Fix a continuous path

\Gamma:[0,1]\to\Lambda\qquad\text{with}\qquad\Gamma(0)=P\text{ and }\Gamma(1)=Q

The holonomy is the unique correspondence assigning to $(\Gamma,S,T)$ the germ of a smooth map

\operatorname{Hol}_{F,\Gamma}:S\cap\operatorname{Nbhd}(P)\to T\cap\operatorname{Nbhd}(Q)\qquad\text{ with }\qquad\operatorname{Hol}_{F,\Gamma}(P)=Q

that satisfies the following properties.

•

The correspondence $\operatorname{Hol}$ respects path composition.
•

If $\Gamma,S,T$ are in a foliation chart, then $\operatorname{Hol}(s)$ is the point in $T$ in the same plaque as $s$ .

The holonomy only depends on up to homotopy relative to $P$ and $Q$ in [16, Prop 2.3.2]. Moreover, given a foliation $G$ such that $TF$ and $TG$ are $C^{0}$ -close, there is a continuation point

Q_{G}\in\operatorname{Nbhd}(Q)\cap T

on the leaf of $G$ through $P$ , converging to $Q$ as $G$ converges to $F$ . There is also a unique homotopy class of path $P$ to $Q_{G}$ corresponding to . Thus there is a holonomy map

\operatorname{Hol}_{G,\Gamma}:S\cap\operatorname{Nbhd}(P)\to T\cap\operatorname{Nbhd}(Q)

The holonomy $\operatorname{Hol}_{F,\Gamma}$ varies continuously with respect to the foliation $F$ .

By Theorem 3.5, the leaves of the stable foliation $F^{s}$ and unstable foliation $F^{u}$ of a partially hyperbolic diffeomorphism are contractible and so the holonomy is independent of the path . In this case, we denote the corresponding holonomies

\operatorname{Hol}^{s}=\operatorname{Hol}_{F^{s},\Gamma}\qquad\text{and}\qquad\operatorname{Hol}^{u}=\operatorname{Hol}_{F^{u},\Gamma}

The main property of holonomy that we will need is the following regularity result, which follows from (for instance) the uniform Hölder regularity results of Pugh-Shub-Wilkenson [55, Thm A].

Lemma 3.12 (Hölder Holonomy).

Let $\Phi:Y\to Y$ be a partially hyperbolic diffeomorphism. Fix points $P$ and $Q$ in an unstable leaf with transversals $S$ and $T$ . Then is exists

C^{1}

-neighborhood

\mathcal{U}

of a Hölder constant

\kappa\in(0,1)

and a neighborhod

\operatorname{Nbhd}(P)

so that the holonomy $\operatorname{Hol}^{u}:S\cap\operatorname{Nbhd}(P)\to T\cap\operatorname{Nbhd}(Q)$ of any diffeomorphism in $\mathcal{U}$ satisfies

(3.1)

\operatorname{dist}\big{(}\operatorname{Hol}^{u}(x),\operatorname{Hol}^{u}(y)\big{)}<\operatorname{dist}(x,y)^{\kappa}

3.3. Cone-Fields

There is an alternative formulation of dominated splittings in terms of cone-fields. Cone-fields will also be used extensively in the definition and construction of blenders.

Definition 3.13 (Cone Fields).

A continuous cone-field $K$ on a manifold $X$ is a bundle of the form

K=\big{\{}v\;:\;Q(v)\leq 0\big{\}}\qquad\text{for a continuous, non-degenerate quadratic form }Q:TX\to\px@BbbR

A diffeomorphism $\Phi:X\to Y$ induces a natural pushforward cone-field

{}_{*}K\qquad\text{with fiber}\qquad{}_{*}K_{x}=T{}_{x}(K_{{}^{-1}(x)})

The interior $\operatorname{int}K$ of a cone-field $K$ is the (fiberwise) cone over the interior of $K$ .

Example 3.14 (Metric Cones).

The cone-field $K_{\epsilon}E$ of width $\epsilon$ around a sub-bundle $E\subset TX$ of the tangent bundle of a Riemannian manifold $(X,g)$ is defined by

K_{\epsilon}E:=\big{\{}v\in TX\;:\;|v-\pi_{E}(v)|_{g}\leq\epsilon\cdot|\pi_{E}(v)|_{g}\big{\}}\qquad\text{where $\pi_{E}$ is orthogonal projection to $E$}

A cone-field $K$ has width less than $\epsilon$ with respect to $g$ if $K\subset K_{\epsilon}E$ for some linear sub-bundle $E\subset TX$ .

Definition 3.15 (Contraction/Dilation).

Fix a subset $A\subset X$ and a cone-field $K$ over $X$ . A diffeomorphism contracts $K$ over $A$ if

{}_{*}(K|_{A})\subset\operatorname{int}K

Given a Riemannian metric $g$ , we say that dilates $K$ over $A$ with constant of dilation $\lambda>0$ if

\lambda\cdot|v|_{g}\leq|{}_{*}v|_{g}\qquad\text{for every }v\in TX|_{A}

Theorem 3.16 (Invariant Cone-Fields).

(c.f. [25, §2.2]) Let $\Phi:Y\to Y$ be a diffeomorphism with a dominated splitting $TY=D\oplus E$ . Then for any Riemannian metric $g$ and any $\epsilon>0$ , there is an $N>0$ such that

{}^{N}\text{ contracts }K_{\epsilon}E\text{ for all }n\geq N

Moreover, if $E$ is uniformly expanding, then for any $\lambda>1$ , we may choose $N$ so that

{}^{N}\text{ dilates }K_{\epsilon}E\text{ with dilation factor $\lambda$ for all }n\geq N

3.4. Blenders

A blender is a type of robust hyperbolic set within a dynamical system, introduced by Bonatti-Diaz [7]. Blenders play a central role in the construction of robustly mixing maps in the partially hyperbolic setting.

In this section, we precisely define blenders (Definition 3.22) and associated structures. We start with the notion of a blender box, which is a chart where the blender structure will reside.

Definition 3.17 (Blender Box).

A blender box $B$ of type $(k,l)$ in an $n$ -manifold $X$ is a $C^{1}$ -embedded $n$ -manifold with corners $B$ with coordinates

(s,t,u):B\xrightarrow{\sim}D^{k}_{s}\times D^{1}_{t}\times D^{l}_{u}\subset{}^{n}

Here $D^{m}_{x}$ denotes a closed metric $m$ -ball of some (unspecified) radius in ${}^{m}_{x}$ , with coordinates $x_{i}$ . The boundary of $B$ has distinguished subsets that we denote as follows.

\partial^{s}B=\partial D^{k}\times D^{1}\times D^{l}\qquad\partial^{c}B=D^{k}\times\partial D^{1}\times D^{l}\qquad\partial^{u}B=D^{k}\times D^{1}\times\partial D^{l}

These are the stable, central and unstable boundaries, respectively. The boundary of $D^{1}$ is a union of two points, a negative point $\partial^{-}D^{1}$ and a positive point $\partial^{+}D^{1}$ . We also fix the notation

\partial^{l}B=D^{k}\times\partial^{-}D^{1}\times D^{l}\qquad\text{and}\qquad\partial^{+}B=D^{k}\times\partial^{+}D^{1}\times D^{l}

These subsets are the left-side and right-side of $B$ , respectively.

Next, we introduce the notion of compatible cone-fields and vertical/horizontal disks. These structures can be viewed as local (and more flexible) versions of the invariant bundles and foliations associated to a partially hyperbolic map.

Definition 3.18 (Compatible Cones).

A triple of smooth cone-fields $K^{s},K^{cu}$ and $K^{u}$ on $X$ are compatible with a blender box $B$ if there are inclusions

TD^{k}\times D^{1}\times D^{l}\subset K^{s}\qquad D^{k}\times TD^{1}\times TD^{l}\subset K^{cu}\qquad D^{k}\times D^{1}\times TD^{l}\subset K^{u}

We refer to these cone-fields as stable, central-unstable and unstable cones for $B$ , respectively.

Definition 3.19 (Vertical/Horizontal Disks).

An embedded $k$ -disk $D\subset B$ is called vertical with respect to an unstable cone-field $K^{u}$ if

TD\subset K^{u}\qquad\text{and}\qquad\partial D\subset\partial^{u}B

Similarly, an embedded $l$ -disk $D\subset B$ is called horizontal with respect to a stable cone-field $K^{s}$ if

TD\subset K^{s}\qquad\text{and}\qquad\partial D\subset\partial^{s}B

Given a horizontal disk $H\subset B$ that is disjoint from $\partial^{u}B$ , there are precisely two homotopy classes of vertical disk disjoint from $H$ , corresponding to the disks

D_{\operatorname{Right}}=0_{s}\times+1\times D^{l}\qquad\text{and}\qquad D_{\operatorname{Left}}=0_{s}\times-1\times D^{l}

A vertical disk $D$ is right of $H$ if it is homotopic to $D_{\operatorname{Right}}$ and left of $H$ if it is homotopic to $D_{\operatorname{Left}}$ .

It will be useful to record some basic properties of vertical disks in the following lemmas. These are entirely elementary and left to the reader (also see [7, §1] or [10]).

Lemma 3.20 (Vertical Manifolds).

Let $\Sigma\subset B$ be any smooth sub-manifold in a blender box $B$ with

T\Sigma\subset K^{u}\qquad\text{and}\qquad\partial\Sigma\subset\partial^{u}B

Then is diffeomorphic to an $l$ -disk, and therefore is a vertical disk.

Lemma 3.21 (Graphs).

Let $D$ be a vertical disk in a blender box $B$ with respect to an unstable cone-field $K^{u}$ of width $\epsilon$ . Then $D$ is the graph

D\subset D^{k}_{s}\times D^{1}_{t}\times D^{l}_{u}\qquad\text{of an $2\epsilon$-Lipschitz map }f:D^{l}_{u}\to D^{k}_{s}\times D^{1}_{t}

We are now prepared to introduce our operating definition of a blender (Definition 3.22). We warn the reader that this definition is quite cumbersome, since it is entirely functional and tailored to the specific application. We will discuss more motivation in Remark 3.23 below.

Definition 3.22 (Blender).

Let $\Phi:X\to X$ be a $C^{1}$ -diffeomorphism and let $B\subset X$ be a blender box (see Definition 3.17). The pair

(B,\Phi)

is a simple stable blender (or just a stable blender) if it satisfies the following properties.

(a)

There is a connected component $A$ of $B\cap\Phi(B)$ that is disjoint from

$\partial^{s}B\qquad\Phi(\partial^{c}B)\quad\text{and}\quad\Phi(\partial^{u}B)$
(b)

There is an $m\in\px@BbbN$ and a connected component $A^{\prime}$ of $B\cap{}^{m}(B)$ that is disjoint from

$\partial^{r}B\qquad\partial^{s}B\quad\text{and}\quad\Phi(\partial^{u}B)$

Moreover, there are constants $\mu>1>\epsilon$ and a compatible set of cone-fields $K^{s},K^{cu}$ and $K^{u}$ on $Y$ of width less than $\epsilon$ (with respect to the standard metric on $B$ ) such that

(c)

The cone-fields $K^{s}$ and $K^{u}$ are contracted and dilated (with constant $\mu$ ) by ^-1 and .

{}^{-1}_{*}K^{s}\subset\operatorname{int}K^{s}\qquad{}_{*}K^{u}\subset\operatorname{int}K^{u}\qquad{}^{-1}\text{ dilates }K^{s}\qquad\Phi\text{ dilates }K^{u}

(d)

The cone-field $K^{cu}$ is contracted and dilated (with constant $\mu$ ) by as follows.

{}_{*}K^{cu}\subset\operatorname{int}K^{cu}\quad\text{and}\quad\Phi\text{ dilates }K^{cu}\qquad\text{ over }{}^{-1}(A)

{}_{*}^{m}K^{cu}\subset\operatorname{int}K^{cu}\quad\text{and}\quad{}^{m}\text{ dilates }K^{cu}\qquad\text{ over }{}^{-m}(A^{\prime})

Finally, in any box $B$ equipped with such cone-fields $K^{\bullet}$ , has a unique hyperbolic fixed point $Q$ in the component $A$ of index $l$ (via [7, Lemma 1.6]), and the local stable manifold

W:=W^{s}_{\operatorname{loc}}(Q,\Phi;B)\subset W^{s}(Q,\Phi)\cap B

is a horizontal $k$ -disk in the blender box $B$ . We further assume that there are neighborhoods

U_{-}\text{ of }\partial^{l}B\qquad U_{+}\text{ of }\partial^{r}B\qquad U\text{ of }W

satisfying the following assumptions.

(e)

Every vertical disk $D$ through $B$ to the right of $W$ is disjoint from $U_{-}$ .
(f)
Every vertical disk $D$ through $B$ to the right of $W$ satisfies one of two possibilities.
1. (i)
  
  The intersection $\Phi(D)\cap A$ contains a vertical disk $D^{\prime}$ through $B$ to the right of $W$ and disjoint from $U_{+}$ .
2. (ii)
  
  The intersection ${}^{m}(D)\cap A^{\prime}$ contains a vertical disk $D^{\prime}$ through $B$ to the right of $W$ and disjoint from $U$ .

The pair $(B,\Phi)$ is an simple unstable blender if the pair $(B,{}^{-1})$ is a simple stable blender.

Remark 3.23 (Blender Property).

Definition 3.22 is best understood as being specifically tailored to enforce the following distinctive blender property (c.f. [7, Lem 1.8] or [10, Lem 6.6 and §6.2.2]):

(3.2)

Every vertical disk to the right of

W

must intersect the (global) stable manifold of

Q

This property has the following easy consequence ([7, Lem 1.8] or [10, Lem 6.8]): if $P$ is a hyperbolic periodic point with unstable manifold $W^{u}(P,\Phi)$ intersecting the blender box along a vertical disk to the right of $W$ , then

W^{s}(P,\Phi)\subset\operatorname{close}(W^{s}(Q,\Phi))

If $W^{s}(P,\Phi)$ has larger dimension than $W^{s}(Q,\Phi)$ , then this says that $W^{s}(Q,\Phi)$ is bigger than expecte. Since the distinctive property is robust, this can lead to robust transitivity properties of the global dynamics of [10, §7.1.3].

More contemporary accounts of blenders (c.f. [6] and [10, §6-7]) take the distinctive property (or a related property) as the definition of a blender. With this in mind, we highly encourage the reader to view Definition 3.22 as a concrete list of criteria leading to the distinctive property (3.2).

Remark 3.24 (Blender Variants).

Several alternative definitions have been introduced since [7] (cf. [10, §6.2.2]). Definition 3.22 matches the original definition of Bonatti-Diaz [7, p. 365] except that we replace the hypothesis [7, p. 365 H3] with the simplified hypothesis (c-d).

4. Blender Construction

In this section, we construct a contact version of the blenders used by Bonatti-Diaz [7] to demonstrate robust transitivity of certain perturbed partially hyperbolic maps.

Setup 4.1 (Blender Setup).

Let $(Y,\xi)$ be a contact $(2n+1)$ -manifold with contact form $\alpha$ and Reeb vector-field $R$ . Fix a strict contactomorphism

\Phi:Y\to Y

that satisfies the following properties.

(a)

The contactomorphism is partially hyperbolic with stable, central and unstable splitting

E^{s}(\Phi)\oplus E^{c}(\Phi)\oplus E^{u}(\Phi)\qquad\text{with}\qquad\xi=E^{s}(\Phi)\oplus E^{u}(\Phi)\quad\text{and}\quad E^{c}(\Phi)=\operatorname{span}(R)

(b)

There is a closed Reeb orbit $\Gamma\subset Y$ that is a normally hyperbolic set of fixed points of .
(c)

There is a neighborhood $V$ of and a smooth, integrable sub-bundle

$E_{\operatorname{sm}}^{s}(\Phi)\subset\xi\quad\text{with foliation}\quad F_{\operatorname{sm}}^{s}(\Phi)\qquad\text{ over }V$

that is uniformly contracted by and that agrees with $E^{s}(\Phi)$ on $W^{s}(\Gamma,\Phi)$ .
(d)

There are two points $P$ and $Q$ in such that the stable and unstable manifolds

$W^{s}(Q,\Phi)\qquad\text{and}\qquad W^{u}(P,\Phi)$

intersect cleanly along a heteroclinic orbit $\chi$ of orbit from $P$ to $Q$ .

Our goal over the following two sections (Sections 4 and 5) is to prove the following result. We will apply this result to prove the existence of robustly mixing diffeomorphisms in Section 6.

Theorem 4.2 (Heteroclinic Contact Blender).

For any contactomorphism $\Phi:Y\to Y$ as in Setup 4.1, there is an integer $N>0$ and a smooth family of contactomorphisms

\Psi:[0,1]\times Y\to Y\qquad\text{with}\qquad{}_{0}=\Phi

satisfying the following properties for all sufficiently small $r>0$ .

(a)

The points $P$ and $Q$ are hyperbolic fixed points of _r of index $n+1$ and $n$ , respectively.
(b)

There is a neighborhood $B_{r}(Q)$ of $Q$ such that $(B_{r}(Q),{}_{r}^{N})$ is a stable blender.
(c)

The intersections $W^{u}(P,{}_{r})\cap B_{r}(Q)$ contains a vertical disk $D_{Q}$ to the right of $W^{s}(Q,{}_{r})$ .

In this section (Section 4), we will construct the objects appearing in Theorem 4.2, and in the next section (Section 5), we will prove the blender properties (Definition 3.22). We will use the notation of Setup 4.1 and Theorem 4.2 throughout. We also use the following standing notation.

Notation 4.3 (Reeb Intervals).

Let $\gamma:[0,T]\to Y$ be an embedded segment of a Reeb trajectory with end-points $x=\gamma(0)$ and $y=\gamma(T$ . Then we use the shorthand

[x,y]=\gamma[0,T]\subset Y

4.1. Standard Chart

We will require a particular standard rectangular neighborhood of the segment of from $Q$ to $P$ . Our first task is to construct this chart carefully.

Lemma 4.4 (Standard Chart).

Let $V\subset Y$ be an open neighborhood of . Then there is a smoothly embedded cube $U\subset V$ , constants $L,\delta,\epsilon>0$ and local coordinates

(s,t,u):U\simeq[-3,3]^{n}_{s}\times[-\delta,L+\delta]_{t}\times[-\epsilon,\epsilon]^{n}_{u}

satisfying the following properties.

(a)

The contact form is given by $\alpha|_{U}=dt-(s_{1}du_{1}+\dots+s_{n}du_{n})$ .
(b)

The Reeb vector-field $R$ of $\alpha$ is given by $R|_{U}=\partial_{t}$ .
(c)

The points $P$ and $Q$ are given by $(0_{s},L_{t},0_{u})$ and $(0_{s},0_{t},0_{u})$ respectively.

(d)

The local stable and unstable manifolds $W_{\operatorname{loc}}^{s}(\Gamma,\Phi;U)$ and $W_{\operatorname{loc}}^{u}(\Gamma,\Phi;U)$ of $\Gamma\cap U$ are given by

[-3,3]^{n}_{s}\times[-\delta,L-\delta]_{t}\times 0_{u}\qquad\text{and}\qquad 0_{s}\times[-\delta,L-\delta]_{t}\times[-\epsilon,\epsilon]_{u}

(e)

The stable and unstable bundles $E^{s}(\Phi)$ and $F^{u}(\Phi)$ satisfy

TF^{s}(\Phi)=T{}^{n}_{s}\text{ on }W_{\operatorname{loc}}^{s}(\Gamma,\Phi;U)\qquad\text{and}\qquad TF^{u}(\Phi)=T{}^{n}_{u}\text{ on }W_{\operatorname{loc}}^{u}(\Gamma,\Phi;U)

(f)

The bundle $T{}^{n}_{s}=E^{s}_{\operatorname{sm}}(\Phi)$ is invariant under and uniformly contracted by on $U$ .
(g)

The heteroclinic orbit $\chi$ contains the point

$a=(1_{s},0_{t},0_{u})\qquad\text{where }1_{s}=(1,0,\dots,0)\in[-3,3]^{n}_{s}$
(h)

There is an integer $k>0$ such that

${}^{-k}(a)\notin U$

Proof.

We construct this chart in two steps: the construction of a nice transverse hypersurface and the construction of a good chart using that hypersurface.

Step 1: Transverse Hypersurface. In this step, we construct a nice embedded hypersurface. We shrink $V$ so that the smooth unstable bundle $E^{s}_{\operatorname{sm}}(\Phi)$ is defined on $V$ (see Setup 4.1). Consider the foliations $F^{u}(\Phi)$ and $F^{s}_{\operatorname{sm}}(\Phi)$ corresponding to the unstable bundle $F^{u}(\Phi)$ and the smooth stable bundle $E^{s}_{\operatorname{sm}}(\Phi)$ . These foliations are Legendrian (Lemma 3.10) and $F^{u}(\Phi)$ is Hölder with smooth leaves (Theorem 3.5).

Next, choose a ball _u in the leaf of $F^{u}(\Phi;Q)$ containing $Q$ . Since $F^{u}_{\operatorname{sm}}(\Phi)$ is smooth, we may choose an embedded codimension $1$ surface that is contained in the union of the leaves of $F^{s}_{\operatorname{sm}}(\Phi)$ intersecting _u. Denote this surface by

\Sigma\subset Y\qquad\text{with}\qquad{}_{u}\subset\Sigma

This surface is transverse to the Reeb vector-field, and is thus symplectic with symplectic form

d\alpha|\qquad\text{with primitive $\alpha|$ vanishing on the leaves of $F^{s}(\Phi)$ and $F^{u}(\Phi)$}

By flowing slightly by the Reeb flow, we acquire an embedding as follows (for small $\delta>0$ ).

(4.1)

\iota:(-\delta,\delta)_{t}\times\Sigma\to Y\qquad\text{with}\qquad\iota^{*}\alpha=dt+\alpha|

Next, apply the neighborhood theorem for Lagrangian foliations [58, Thm 7.1] to the smooth Legendrian foliation $F^{s}_{\operatorname{sm}}(\Phi)$ near the transverse smooth Lagrangian ${}_{u}\subset\Sigma$ . After shrinking , this yields a symplectic embedding

(4.2)

\jmath:\Sigma\to T^{*}F^{u}(\Phi;Q)\qquad\text{with}\qquad F^{s}_{\operatorname{sm}}(\Phi)\cap\Sigma=\jmath^{*}F_{\operatorname{std}}

Here $F_{\operatorname{std}}$ is the standard Lagrangian foliation of $T^{*}F^{u}(\Phi;Q)$ by cotangent fibers. Finally, consider the pullback $\lambda_{\operatorname{std}}$ of the standard Liouville form on $T^{*}{}_{u}$ by (4.2). Since $\lambda_{\operatorname{std}}$ and $\alpha|$ vanish on _s and _u, we can find a primitive $f$ such that

\lambda_{\operatorname{std}}-\alpha|=df

The primitives $\alpha|$ and $\lambda_{\operatorname{std}}$ vanish on $L_{u}$ and on the foliation $F^{s}(\Phi)\cap\Sigma$ , so $f$ is constant these manifolds. It follows that $f$ is constant on and so

(4.3)

\lambda_{\operatorname{std}}=\alpha|

Step 2: Cube Coordinates. In this step, we use to construct a rectangular chart and verify the requirements. Since $\chi$ is a homoclinic orbit asymptotic to $Q$ in the forward direction, we may choose a point $a\in\chi$ such that

a\in\Sigma\cap F^{s}_{\operatorname{sm}}(\Phi;Q)\subset F^{s}(\Phi;Q)

Here we use the fact that $Q\in\Gamma$ so that $F^{s}_{\operatorname{sm}}(\Phi;Q)=F^{s}(\Phi;Q)\subset W^{s}(\Gamma,\Phi)$ by Setup 4.1(c). Choose a smooth embedding of a cube of the form

(4.4)

[-\epsilon,\epsilon]_{u}^{n}\to{}_{u}\subset F^{u}(Q)\qquad\text{with}\qquad 0_{u}\mapsto Q

This extends naturally to a map of cotangent bundles, giving a Liouville embedding

\kappa:([-3,3]_{s}^{n}\times[-\epsilon,\epsilon]_{u}^{n},\lambda_{\operatorname{std}})\to(\Sigma,\alpha|)\qquad\text{where}\qquad\lambda_{\operatorname{std}}=-\sumop\slimits@_{i}s_{i}\cdot du_{i}

By shrinking around _s, we may assume that this map is a symplectomorphism. Using the Reeb flow as in (4.1), we may extend $\kappa$ to a local contactomorphism

\kappa:\big{(}[-3,3]_{s}^{n}\times[-\delta,L+\delta]_{t}\times[-\epsilon,\epsilon]_{u}\;,\;dt+\lambda_{\operatorname{std}}\big{)}\to(Y\;,\;\alpha)

The restriction of $\kappa$ to $0_{s}\times[-\epsilon,L+\epsilon]_{t}\times 0_{u}$ is a parametrization of a sub-arc of containing $Q$ and $P$ sending $0$ to $Q$ and $L$ to $P$ . Thus by choosing a sufficiently small surface and choosing $\delta,\epsilon$ small, we may guarantee that $\kappa$ is an embedding and the image of $\kappa$ lies in $V$ .

Now take $U$ to be the image of $\kappa$ and take $(x,s,u)$ as the coordinates induced by $\kappa$ . By construction $U\subset V$ . We now check that $U$ and $(s,t,u)$ satisfy (a-g). For (a), we note that

\kappa^{*}\alpha=dt+\lambda_{\operatorname{std}}=dt-(s_{1}du_{1}+\dots+s_{n}du_{n})

The requirements (b) follows immediately from (a). Requirements (c) and (f,g) follow trivially from the construction. To see property (d), note that the local stable manifold $W^{s}_{\operatorname{loc}}(\Gamma,\Phi;U)$ of is precisely the $R$ -orbit of the component of the local stable leaf $F^{s}_{\operatorname{loc}}(\Phi,Q;U)$ containing $Q$ . By construction, $\kappa$ maps $[-3,3]_{s}^{n}\times 0_{t}\times 0_{u}$ to $F^{s}_{\operatorname{loc}}(\Phi,Q;U)$ . It follows that $W^{s}(\Gamma,\Phi;U)$ is identified with the zero set

\{u=0\}=[-3,3]_{s}^{n}\times[-\epsilon,1+\epsilon]_{t}\times 0_{u}

An analogous discussion applies to $W^{u}_{\operatorname{loc}}(\Gamma,\Phi;U)$ , verifying (d). Requirement (e) follows from the same discussion, and the fact that $F^{s}(\Phi)$ and $F^{u}(\Phi)$ are $R$ -invariant. Finally, by shrinking the neighborhood $V$ in the construction, we may guarantee that the orbit $\chi$ contains a point that is not in $V$ or $U$ . It follows that ${}^{-k}(a)\notin U$ for some $k$ . This verifies (h).∎

We will require an enhancement of Lemma 4.4 that incorporates a set of invariant cone-fields.

Lemma 4.5 (Standard Chart With Cone-Fields).

For any $\mu>1$ and $\epsilon\in(0,1)$ , there exists

•

An integer $N\geq 1$ .
•

A cube $U\subset Y$ with coordinates $(s,t,u)$ as in Lemma 4.4.
•

A Riemannian metric $g$ on $Y$ that is compatible with the splitting $TY=E^{s}\oplus E^{c}\oplus E^{u}$
•

Continuous cone-fields $K^{s}$ and $K^{u}$ on $Y$ .

that satisfy the following properties (after possibly rescaling the contact form $\alpha$ ).

(a)

The cone-fields $K^{s}$ and $K^{u}$ are compatible with the blender box $U$ (see Definition 3.18).

$T{}^{n}_{s}\subset K^{s}\quad\text{and}\quad T{}^{n}_{u}\subset K^{u}$
(b)

The cone-fields $K^{s}$ is contracted by ^-N, and $K^{cu}$ and $K^{u}$ are contracted by ^N.

${}_{*}^{-N}(K^{s})\subset\operatorname{int}K^{s}\quad\text{and}\quad{}_{*}^{N}(K^{u})\subset\operatorname{int}K^{u}$
(c)

The cone-fields $K^{s}$ and $K^{u}$ are dilated by ^-N and ^N with contant of dilation $\mu$ , respectively.

$\mu\cdot|T{}^{N}(v)|<|v|\text{ if }v\in K^{u}\qquad\text{and}\qquad\mu\cdot|T{}^{-N}(v)|<|v|\text{ if }v\in K^{s}$
(d)

The cone-fields $K^{s}$ and $K^{u}$ are width less than $\epsilon$ for both $g$ and the standard metric on $U$ .

Proof.

Choose an auxilliary chart $U^{\prime}$ and coordinates $(s^{\prime},t^{\prime},u^{\prime})$ as in Lemma 4.4. Also choose a continuous metric $g$ on $Y$ that is compatible with the splitting $E^{s}\oplus E^{c}\oplus E^{u}$ . Finally, choose a $\delta<\epsilon$ sufficiently small such that the cone-fields

K^{s}=K_{\delta}(E^{s}(\Phi))\qquad K^{cu}=K_{\delta}(E^{c}(\Phi)\oplus E^{u}(\Phi))\qquad K^{u}=K_{\delta}(E^{u}(\Phi))

are width less than $\epsilon$ with respect to $g$ . By construction, these cone-fields satisfy (d). By Theorem 3.16 and Setup 4.1, we may choose an $N\geq 1$ such that satisfies (b) and (c).

To achieve (a), note that $E^{s}(\Phi),E^{c}(\Phi)\oplus E^{u}(\Phi)$ and $E^{u}(\Phi)$ agree with the tangent spaces $T{}^{n}_{s},T({}^{1}_{t}\times{}^{n}_{u})$ and $T{}^{n}_{u}$ along , respectively. Thus in a small neighborhood $V$ of , the chosen cone-fields satisfy (a). We may then rescale the coordinates $(s^{\prime},t^{\prime},u^{\prime})$ by taking

(s,t,u)=(c\cdot s^{\prime},c\cdot t^{\prime},c\cdot u^{\prime})\qquad\text{for $c$ large}

For $c$ sufficiently large, the cube with $(s,t,u)$ -coordinates $[2,-2]_{s}\times[-\delta,L+\delta]_{t}\times[-\epsilon,\epsilon]_{u}^{n}$ will lie within $V$ . Properties (a-b) are preserved after we scale the contact form by $c^{-1}$ . Properties (c-f,h) in Lemma 4.4 are preserved by this coordinate change. Property (g) in in Lemma 4.4 can be preserved by replacing $a$ with ${}^{j}(a)$ for large $j$ .∎

4.2. Family of Contactomorphisms

Our next task is to construct the family of contactomorphisms in Theorem 4.2 and discuss its basic properties. For the rest of the section, we fix

\text{constants $L$, $N$, $m$}\qquad U\subset Y\text{ with coordinates }(s,t,u)\qquad\text{metric $g$}\quad\text{and}\quad\text{cone-fields }K^{\bullet}

as in Lemmas 4.4 and 4.5. We can now begin the main construction of the family of maps. We will require two auxilliary contact Hamiltonians for the construction.

Construction 4.6 (Hamiltonian $H$ ).

Let $H:Y\to\px@BbbR$ be a contact Hamiltonian such that

H(s,t,u)=h(t)

Here $h:\px@BbbR\to\px@BbbR$ is a smooth function of the $t$ -variable satisfying the following constraints.

h(t)=t\quad\text{if}\quad t\leq L/3\qquad\text{and}\qquad h(t)=L-t\quad\text{if}\quad t\geq 2L/3

h(t)>0\quad\text{and}\quad|h^{\prime}(t)|<1/L\qquad\text{if}\quad 0<t<L

The contact Hamiltonian vector-field $V_{H}$ generating the flow of contactomorphisms ^H is

(4.5)

V_{H}=h(t)\cdot\partial_{t}-h^{\prime}(t)\cdot\sumop\slimits@_{i}s_{i}\cdot\partial_{s_{i}}\qquad\text{ in the chart }U

In particular, we have the following formulas for special ranges of $t$ .

(4.6)

V_{H}=t\partial_{t}-\sumop\slimits@_{i}s_{i}\partial_{s_{i}}\;\;\text{if $t\leq L/3$}\qquad\qquad V_{H}=(L-t)\partial_{t}+\sumop\slimits@_{i}s_{i}\partial_{s_{i}}\;\;\text{if $t\geq 2L/3$}

Using (4.5), it is a simple calculation to show that ^H takes the following general form on $U$ .

(4.7)

{}^{H}_{r}(s,t,u)=(f_{r}(t)\cdot s,\psi_{r}(t),u)\quad\text{where}\quad\partial_{r}\psi=h\circ\psi\quad\text{and}\quad\partial_{r}f=-h^{\prime}\cdot f

In particular, we have the following formulas for ^H on the ranges of $t$ appearing in (4.6).

(4.8)

{}^{H}_{r}(s,t,u)=(e^{-r}s,e^{r}t,u)\;\;\text{if $t\leq 1/3$}

(4.9)

{}^{H}_{r}(s,t,u)=(e^{r}s,L+e^{-r}(t-L),u)\;\;\text{if $t\geq 2L/3$}

Construction 4.7 (Hamiltonian $G$ ).

We define the (time-dependent) contact Hamiltonian

G:[0,1]_{r}\times Y\to\px@BbbR

as follows. Recall that the heteroclinic point $a$ in Lemma 4.4 is in the unstable manifold of $P$ . Moreover, the local unstable manifold of $P$ in $U$ is the set $0_{s}\times L_{t}\times[-\delta,\delta]^{n}_{u}$ . Thus we choose

(4.10)

m\geq 2\qquad\text{such that}\qquad{}^{-Nm}(a)=(0_{s},L_{t},x_{u})\text{ for some }x_{u}\in[-\delta,\delta]^{n}_{u}

Choose a neighborhood $W$ of ${}^{-Nm}(a)$ that is small enough so that the sets

({}^{H}_{r}\circ\Phi)^{j}(W)\qquad\text{for }j=0\dots m

are all disjoint for sufficiently small $r$ . Since ${}^{-Nm}(a)$ and $a$ are in $U$ , we may also assume that

W\subset U\qquad\text{and}\qquad({}^{H}_{r}\circ\Phi)^{Nm}(W)\subset U

for sufficiently small $r$ . Moreover, since ${}^{-k}(a)$ is not in ${}^{-1}(U)$ for some $k$ , we have that

({}^{H}_{r}\circ\Phi)^{Nm-k}(W)\cap{}^{-1}(U)=\emptyset

for sufficiently small $r$ and neighborhood $W$ . We may thus fix a pair of open sets $W^{\prime}\subset W^{\prime\prime}$ that have the following properties for sufficiently small $r$ .

({}^{H}_{r}\circ\Phi)^{Nm-k}(W)\subset W^{\prime}\qquad W^{\prime\prime}\cap{}^{-1}(U)=\emptyset

W^{\prime\prime}\cap({}^{H}_{r}\circ\Phi)^{j}(W)=\emptyset\quad\text{if}\quad 0\leq j\leq Nm\quad\text{and}\quad j\neq Nm-k

Now we define a smooth family of contactomorphism ${}^{K}_{r}:Y\to Y$ (generated by a parameter-dependent contact Hamiltonian $K$ ) implicitly by the following equation.

(4.11)

{}^{R}_{r}\circ{}^{Nm}\circ{}^{H}_{Nmr}=({}^{H}_{r}\circ\Phi)^{k}\circ G_{r}\circ({}^{H}_{r}\circ\Phi)^{Nm-k}

Here ^R is the Reeb flow of $Y$ . Note that ${}^{K}_{0}=\operatorname{Id}$ . We now define $G$ so that it satisfies

G_{r}=K_{r}\;\text{ on }\;W^{\prime}\qquad\text{and}\qquad G_{r}=0\;\text{ on }\;Y\setminus W^{\prime\prime}

Construction 4.8 (Family Of Maps).

We define the family of contactomorphisms

\Psi:[0,1]_{r}\times Y\to Y\qquad\text{as the composition}\qquad{}_{r}={}^{G}_{r}\circ{}^{H}_{r}\circ\Phi

Note that _r satisfies the following elementary properties for $W$ and $m$ as in Construction 4.7.

(4.12)

{}_{r}={}^{H}_{r}\circ\Phi\;\;\text{on}\;\;{}^{-1}(U)\qquad\text{and}\qquad{}_{r}^{Nm}(x)={}^{R}_{r}\circ{}^{Nm}\circ{}^{H}_{Nmr}\;\;\text{on}\;\;W

4.3. Properties Of Family

In this section, we prove several properties of the family in Construction 4.8. We will use these properties extensively in the proof of the blender properties.

We start by recording the effect of the maps on the coordinates in the standard chart $U$ .

Lemma 4.9 (Coordinate Projections).

Let $\pi_{W}$ and $\pi$ be the coordinate projections

\pi_{W}:U\to W^{u}_{\operatorname{loc}}(\Gamma,\Phi;U)=U\cap\{s=0\}\qquad\text{given by}\qquad\pi_{W}(s,t,u)=(0_{s},t,u)

\pi:U\to\Gamma\cap U=U\cap\{s,u=0\}\qquad\text{given by}\qquad\pi(s,t,u)=(0_{s},t,0_{u})

Then $\pi_{W}$ and $\pi$ commute with _r over $U\cap{}^{-1}(U)$ . In particular, if $\psi$ is the flow of $h\cdot\partial_{t}$ on _t then

t({}_{r}(x))=t(\pi\circ{}_{r}(x))=t({}_{r}\circ\pi(x))=\psi_{r}(t(x))

Proof.

On the set ${}^{-1}(U)$ , we have ${}_{r}={}^{H}_{r}\circ\Phi$ by the first identity in (4.12). By Lemma 4.4(d-f) and by examination of (4.7), both and ${}^{H}_{r}$ preserves the foliation

(4.13)

E^{s}_{\operatorname{sm}}(\Phi)=T{}^{n}_{s}\text{ on }U\qquad\text{and}\qquad T{}^{n}_{u}\text{ on }W^{u}_{\operatorname{loc}}(\Gamma,\Phi;U)

Moreover, both and ${}^{H}_{r}$ (and therefore _r) preserve the sets

W^{u}_{\operatorname{loc}}(\Gamma,\Phi;U)=U\cap\{s=0\}\qquad\text{and}\qquad\Gamma\cap U=U\cap\{s,t=0\}

This _r sends the leaf of $T{}^{n}_{s}$ through $x\in U\cap\{s=0\}$ to the leaf through ${}_{r}(x)$ . In other words, $\pi_{W}$ commutes with _r. Similarly, _r preserves the leaves of $T{}^{n}_{u}$ on $U\cap\{s=0\}$ and so

\pi\circ{}_{r}(x)={}_{r}\circ\pi(x)\qquad\text{if }x\in U\cap\{s=0\}

Since $\pi\circ\pi_{W}=\pi$ , this implies that $\pi$ commutes with _r in general. The final claim follows since $t\circ{}_{r}=\psi_{r}\circ t$ on $\Gamma\cap U$ , since fixes pointwise and $t\circ{}^{H}_{r}=\psi_{r}\circ t$ on $\Gamma\cap U$ . ∎

Lemma 4.10 (Coordinate Contraction).

Let $U^{\prime}=U\cap\dots\cap{}^{-N}(U)$ . Then for small $r$ , we have

\mu\cdot|s({}_{r}^{N}(x))|<|s(x)|\qquad\text{and}\qquad\mu\cdot|u(x)|<|u({}_{r}^{N}(x))|

Proof.

Note that $T{}^{n}_{s}$ is uniformly contracted by ^N by a factor of $\mu$ on $U$ and $T{}^{n}_{u}$ is uniformly expanded by ^N by a factor of $\mu$ on $W^{u}_{\operatorname{loc}}(\Gamma,\Phi;U)$ due to Lemma 4.5.

These properties are robust in the $C^{1}$ -topology, so the map _r also satisfies these properties for small $r$ . Since $x$ and $\pi(x)$ lie in the same fiber of $T{}^{n}_{s}$ , we can apply Lemma 4.10 to see that

|s({}^{N}_{r}(x))|=|s({}^{N}_{r}(x))-s({}^{N}_{r}(\pi_{W}(x))|\leq\mu^{-1}\cdot|s(x)-s(\pi_{W}(x))|=\mu^{-1}\cdot|s(x)|

Also, the maps $u,\pi$ and $\pi_{W}$ satisfy $u\circ\pi_{W}=u$ and $u\circ\pi=0$ . Thus by Lemma 4.10, we have

|u({}^{N}_{r}(x))|=|u\circ\pi_{W}({}^{N}_{r}(x))-u\circ\pi({}^{N}_{r}(x))|\geq\mu\cdot|u(\pi_{W}(x))-u(\pi(x))|\geq|u(x)|\qed

Next, note that the maps in the family is partially hyperbolic near . This follows from Theorems 3.4-3.5 and the partial hyperbolicity of .

Lemma 4.11 (Partial Hyperbolicity).

The maps _r (for sufficiently small $r$ ) has a dominated splitting

TY=E^{s}({}_{r})\oplus E^{c}({}_{r})\oplus E^{u}({}_{r})\quad\text{with stable/unstable foliations}\quad F^{s}({}_{r})\text{ and }F^{u}({}_{r})

The points $P$ and $Q$ become non-degenerate and hyperbolic fixed points of _r.

Lemma 4.12 (Fixed Points).

The points $Q$ and $P$ are hyperbolic fixed points of _r of index

\operatorname{ind}({}_{r};Q)=n\qquad\text{and}\qquad\operatorname{ind}({}_{r};P)=n+1

The local stable and unstable manifolds of $P$ and $Q$ in $U$ with respect to _r are given by

W^{s}_{\operatorname{loc}}(P,{}_{r};U)=[-3,3]^{n}_{s}\times(0,L+\epsilon]_{t}\times 0_{u}\qquad\quad W^{u}_{\operatorname{loc}}(P,{}_{r};U)=0_{s}\times L_{t}\times[-\delta,\delta]_{u}

\hskip 28.0ptW^{s}_{\operatorname{loc}}(Q,{}_{r};U)=[-3,3]^{n}_{s}\times 0_{t}\times 0_{u}\hskip 71.0ptW^{u}_{\operatorname{loc}}(Q,{}_{r};U)=0_{s}\times[-\epsilon,L)_{t}\times[-\delta,\delta]_{u}

Proof.

The points $P$ and $Q$ are fixed by due to Setup 4.1(d) and fixed by ${}^{H}_{r}$ due to (4.8-4.9). Thus (4.12) implies that $P$ and $Q$ are fixed by _r. To compute the index of $P$ , note that $T\Phi$ and $T{}^{H}_{r}$ at $P$ decomposes via the splitting $TY=E^{s}(\Phi)\oplus E^{c}(\Phi)\oplus E^{u}(\Phi)$ as follows.

T_{P}\Phi=T_{P}{}^{s}\oplus\operatorname{Id}_{c}\oplus T_{P}{}^{u}\qquad\text{and}\qquad T_{P}{}^{H}_{r}=e^{r}\operatorname{Id}_{s}\oplus\;e^{-r}\operatorname{Id}_{c}\oplus\operatorname{Id}_{u}

The linear maps $T_{P}{}^{s}$ and $T_{P}{}^{u}$ are the restrictions to $E^{s}(\Phi)$ and $E^{u}(\Phi)$ respectively. Since $P$ is in $U$ , the first formula in (4.12) implies that the differential of _r at $P$ is given by

T_{P}{}_{r}=e^{r}T_{P}{}^{s}_{r}\;\oplus\;e^{-r}\operatorname{Id}_{c}\;\oplus\;T_{P}{}^{u}_{r}

Since is a normally hyperbolic fixed set of by Setup 4.1, the maps $e^{r}T_{P}{}^{s}_{r}$ (for small $r$ ) and $T_{P}{}^{u}_{r}$ have real eigenvalues of norm bounded above by $1$ and below by $1$ , respectively. It follows that $T_{P}{}_{r}$ will be hyperbolic of index $n+1$ . The same analysis applies to the index of $Q$ .

To find the stable and unstable manifolds, fix $x\in U$ . By Lemma 4.9, we know that $t({}_{r}^{k}(x))=\psi^{k}_{r}(t(x))$ where $\psi_{r}$ of the vector-field $h\cdot\partial_{t}$ and $H$ is as in Construction 4.6. Moreover $h>0$ on $(0,L)$ and $h<0$ on $[-\epsilon,0)$ and $(L,L+\epsilon]$ . This implies that $t({}_{r}^{k}(x))$ converges to

L\text{ if }t(x)\in(0,L+\epsilon]\qquad 0\text{ if }t(x)=0

and that ${}_{r}^{k}(x)$ leaves $U$ if $t(x)\in[-\epsilon,0)$ . Now Lemma 4.10 implies that $|u({}^{k}_{r}(x))|$ diverges if $u(x)\neq 0$ , and so ${}^{k}_{r}(x)$ leaves $U$ . On the other hand, if $u(x)=0$ then $|s({}^{k}_{r}(x))|\to 0$ as $k\to\infty$ by Lemma 4.10. This shows that

W^{s}_{\operatorname{loc}}(P,{}_{r};U)=[-3,3]^{n}_{s}\times(0,L+\epsilon]_{t}\times 0_{u}\qquad W^{s}_{\operatorname{loc}}(Q,{}_{r};U)=[-3,3]^{n}_{s}\times 0_{t}\times 0_{u}

An identical analysis works for the unstable manifolds. ∎

Next, we have the following description of the local stable and unstable foliations, and the action of _r on the leaves in the chart $U$ .

Lemma 4.13 (Stable/Unstable Foliations).

The local stable and unstable foliations of _r satisfy

F^{s}({}_{r})=T{}^{n}_{s}\;\;\text{on}\;\;W_{\operatorname{loc}}^{s}(P,{}_{r};U)\qquad\text{and}\qquad F^{u}({}_{r})=T{}^{n}_{u}\;\;\text{on}\;W_{\operatorname{loc}}^{u}(Q,{}_{r};U)

Moreover, if ^s and ^u are leaves of $F^{s}({}_{r})$ and $F^{u}({}_{r})$ intersecting $\Gamma\cap U$ , respectively, then

{}_{r}({}^{s})\cap U={}^{H}_{r}({}^{s})\cap U\qquad\text{and}\qquad{}_{r}({}^{u})\cap U={}^{H}_{r}({}^{u})\cap U

Proof.

For the first claim, note that $T{}^{n}_{s}$ and $T{}^{n}_{u}$ are tangent to $W^{s}_{\operatorname{loc}}(P,{}_{r};U)$ and $W^{u}_{\operatorname{loc}}(Q,{}_{r};U)$ by Lemma 4.12. Moreover, $T{}^{n}_{s}$ and $T{}^{n}_{u}$ are uniformly contracted and expanded, respectively, on those sets via (4.12). The lemma then follows from the uniqueness of the strong stable and unstable foliations on the stable and unstable manifolds of a hyperbolic invariant set [39, §4]. The second claim follows from the first claim, the formula (4.12) and the fact that preserves the stable and unstable foliations in $U$ . ∎

Finally, we have the following computation of a certain important family of heteroclinics.

Lemma 4.14 (Heteroclinics).

Consider the Reeb segment $[b_{-r},b_{r}]$ containing the point $a$ , where

b_{\tau}=(1_{s},\tau,0_{u})\qquad\text{and we recall that}\qquad a=(1_{s},0,0_{u})

Then for $r>0$ sufficiently small, we have

(a)

The leaf of $F^{u}({}_{r})$ through $(0_{s},L-\delta,0_{u})$ meets $[b_{-r},b_{r}]$ at the point $(1_{s},r-\delta,0_{u})$ for $\delta\in[0,2r]$ .
(b)

The interval $(a,b_{r})$ is a connected component of the intersection of $W^{s}(P,{}_{r})$ and $W^{u}(Q,{}_{r})$ .
(c)

The points $a$ and $b_{r}$ are transverse homoclinic points of $Q$ and $P$ , respectively.

Proof.

Let $m$ and $W$ be the integer and fixed ( $r$ -independent) neighborhood in Construction 4.8. Note that $W$ is a fixed neighborhood of the point

{}^{-Nm}(a)=(0,L_{t},1_{u})

We may thus choose $r$ small enough so that we have the following inclusion for all $\delta\in[0,2r]$ .

{}_{-Nmr}^{H}(0_{s},L-\delta,1_{u})\in W

We first compute the image of $(0_{s},L-\delta,1_{u})$ under ${}_{r}^{Nm}\circ{}^{H}_{-Nmr}$ . By (4.12), we see that

{}_{r}^{Nm}\circ{}_{-Nmr}^{H}(0_{s},L-\delta,1_{u})={}^{R}_{r}\circ{}^{Nm}\circ{}^{H}_{Nmr}\circ{}^{H}_{-Nmr}(0_{s},L-\delta,1_{u})={}^{R}_{r}\circ{}^{Nm}(0_{s},L-\delta,1_{u})

Since the Reeb vector-field commutes with , the latter expression becomes

{}^{R}_{r}\circ{}^{Nm}(0_{s},L-\delta,1_{u})={}^{R}_{r}\circ{}^{Nm}\circ{}^{R}_{-\delta}(0_{s},L,1_{u})={}^{R}_{r-\delta}\circ{}^{m}(0_{s},L,1_{u})

Since $(1_{s},0,0_{u})={}^{Nm}(0_{s},L,1_{u})$ by (4.10) in Construction 4.8, we then acquire the equality

{}^{R}_{r-\delta}\circ{}^{Nm}(0_{s},L,1_{u})={}^{R}_{r-\delta}(1_{s},0,0_{u})=(1_{s},r-\delta,0_{u})

We thus acquire the following formula that we will shortly use to prove (a-c).

(4.14)

{}^{Nm}_{r}\circ{}^{H}_{-Nmr}(0_{s},L-\delta,1_{u})=(1_{s},r-\delta,0_{u})

Now we prove the claims above. For (a), note that by Lemma 4.13, ${}^{Nm}_{r}$ maps the leaf of $F^{u}({}_{r})$ containing ${}^{H}_{-Nmr}(0_{s},L-\delta,1_{u})$ to the leaf ${}^{Nm}_{r}(\Lambda)={}^{H}_{Nmr}(\Lambda)$ containing the points

{}^{Nm}_{r}\circ{}^{H}_{-Nmr}(0_{s},L-\delta,1_{u})=(1_{s},r-\delta,0_{u})

{}_{Nmr}^{H}\circ{}^{H}_{-Nmr}(0_{s},L-\delta,1_{u})=(0_{s},L-\delta,1_{u})

By Lemma 4.13, the leaf that contains $(0_{s},L-\delta,1_{u})$ also contains $(0_{s},L-\delta,0_{u})$ , proving (a). For (b), we note that by (a), the point $(0_{s},r-\delta,0_{u})$ is contained in the stable manifold $W^{s}(P,{}_{r})$ by Lemma 4.12. On the other hand, it is also contained in the leaf of in $F^{u}({}_{r})$ passing through $(1_{s},r-\delta,0_{u})$ by (a), and this leaf is contained in the unstable manifold $W^{u}(Q,{}_{r})$ by Lemma 4.12. This proves (b). For (c), we argue similarly to (b). By (a), the point $a=(1_{s},0_{t},0_{u})$ is contained in the leaf of $F^{u}({}_{r})$ containing $(0_{s},L-r,0_{u})$ , which is contained in the unstable manifold $W^{u}(Q,{}_{r})$ by Lemmas 4.12. On the other hand, $a$ is contained in $W^{s}(Q,{}_{r})$ (also by Lemma 4.12). Thus it is a homoclinic point for $Q$ . A similar discussion holds for $b_{r}$ and $P$ .∎

4.4. Family Of Boxes

In this section, we describe the families of smoothly embedded boxes

B_{r}(Q)

appearing in the blenders in Theorem 4.2. We start by introducing notation for several important points and quantities appearing in the description of the boxes.

Notation 4.15.

For $r\in[0,1]$ , let $P_{r}$ and $Q_{r}$ denote the points in $U$ given by

P_{r}=(0_{s},L-r,0_{u})\qquad\text{and}\qquad Q_{r}=(0_{s},r\cdot(1-e^{-8Nr}),0_{u})

Then we define the constant $m_{r}$ to be the unique integer satisfying

(4.15)

{}_{r}^{Nm_{r}}(Q_{r})\in[{}^{5N}_{r}(P_{r}),{}^{6N}_{r}(P_{r})]

Lemma 4.16.

The integers $m_{r}$ diverge to $\infty$ as $r\to 0$ . Precisely, there are constants $C,r_{0}>0$ such that

m_{r}>-Cr^{-1}\cdot\log(r)\qquad\text{for all }r\in(0,r_{0})

Proof.

Since _r restricts to the flow of $\psi_{r}$ on (see Construction 4.6 or Lemma 4.9), we may equivalently characterize $m_{r}$ by

\psi_{rNm_{r}}(r(1-e^{-8Nr}))\in[L-re^{-5Nr},L-re^{-6Nr}]

Here $\psi_{r}:{}_{t}\to{}_{t}$ is the flow of the vector-field $h\cdot\partial_{t}$ and $h$ is as in Construction 4.6. Briefly switch notation by letting $\psi_{r}(t)=\psi(r,t)$ , and let the quantities $T_{\operatorname{st}}(r),T_{\operatorname{mid}}$ and $T_{\operatorname{end}}(r)$ be the unique values such that

\psi(T_{\operatorname{st}}(r),r(1-e^{-8Nr}))=L/3\qquad\psi(T_{\operatorname{mid}},L/3)=2L/3\qquad\psi(T_{\operatorname{end}}(r),2L/3)=L-re^{-5Nr}

Note that $T_{\operatorname{mid}}$ is independent of $r$ . Moreover, using the formulas (4.8) and (4.9) on the intervals $[0,L/3]_{t}$ and $[2L/3,L]_{t}$ (see Construction 4.6), we can compute that

T_{\operatorname{st}}(r)=\log(L/3)-\log(r)-\log(1-e^{-8Nr})

T_{\operatorname{end}}(r)=-\log(3/2)-\log(r)+5Nr

Now we simply note that in the $r\to 0$ limit, we have

rNm_{r}\sim(T_{\operatorname{st}}(r)+T_{\operatorname{mid}}+T_{\operatorname{end}}(r))\geq-3\log(r)

Here $\sim$ denotes that the limit of the ratio is $1$ . This proves the result.∎

We can now introduce the definition of the blender box $B_{r}(Q)$ .

Construction 4.17 (Blender Box For ).

The blender box $B_{r}(Q)\subset U$ is defined as the set

B_{r}(Q)=D^{n}_{s}(2)\times D^{1}_{t}(t(Q_{r}))\times D^{n}_{u}(l\cdot\mu^{-m_{r}})

The constants in the formula are defined as follows.

•

The quantity $t(Q_{r})$ is the $t$ -coordinate $r\cdot(1-e^{-8Nr})$ of the point $Q_{r}$ .
•

The constant $l$ is a positive number such that $l/3$ is larger than the minimum radius of a ball in the unstable leaf $F^{u}({}_{r};P)$ containing $a$ .
•

The constant $\mu$ is an $r$ -independent constant of dilation for _r via Lemma 4.11.
•

The constant $m_{r}$ is the integer defined by the formula (4.15).

The stable, central, unstable, left and right boundaries are defined as in Definition 3.17.

We will also need an auxilliary Hölder continuous box constructed using unstable leaves. Let

(4.16)

S_{r}(Q)=D^{n}_{s}(2+\mu^{-m_{r}/2})\times D^{1}_{t}(t({}^{N}_{r}(Q_{r})))\times 0_{u}

The Hölder box $\tilde{B}_{r}(Q)$ is as the union of disks in the leaves of the unstable foliation $F^{u}({}_{r})$ that have boundary on the set $\{|u|=l\cdot\mu^{-m_{r}}\}$ and that intersect $S_{r}(Q)$ . That is

(4.17)

\tilde{B}_{r}(Q)=\big{\{}x\;:\;F^{u}({}_{r},x)\cap S_{r}(Q)\neq\emptyset\quad\text{and}\quad|u(x)|\leq l\cdot\mu^{-m_{r}}\big{\}}

There is a map $\pi:\tilde{B}_{r}(Q)\to S_{r}(Q)$ mapping a point $x$ to the intersection point $\pi(x)\in F^{u}({}_{r},x)\cap S_{r}(Q)$ . The following lemma allows us to replace $\tilde{B}_{r}(Q)$ with $B_{r}(Q)$ in some arguments.

Lemma 4.18.

The Hölder box $\tilde{B}_{r}(Q)$ contains the blender box $B_{r}(Q)$ for sufficiently small $r$ .

Proof.

Let $x\in B_{r}(Q)$ . By Construction 4.17 we know that $|u(x)|\leq l\cdot\mu^{-m_{r}}$ . Let $D\subset F^{u}({}_{r},x)$ be the disk given by $F^{u}({}_{r},x)\cap\{|u(x)|\leq l\cdot\mu^{-m_{r}}\}$ . This disk is tangent to the vertical cone-field $K^{u}$ in $U$ (see Lemma 4.5), and thus is the graph of a Lipschitz map

f:D_{u}^{n}(l\cdot\mu^{-m_{r}})\to{}^{n}_{s}\times{}^{1}_{t}

with a uniform Lipschitz constant $C$ independent of $r$ (see Lemma 3.21) and so the image of $f$ in ${}_{s}\times{}_{t}$ has diameter bounded by $C\cdot l\cdot\mu^{-m_{r}}$ . Thus if $y=f(0_{u})$ then

|s(y)-s(x)|<5C\cdot l\cdot\mu^{-m_{r}}\qquad\text{and}\qquad|t(y)-t(x)|<5C\cdot l\cdot\mu^{-m_{r}}

In particular, this implies that for sufficiently small $r$ , we have

|s(y)|\leq|s(x)|+5C\cdot l\cdot\mu^{-m_{r}}\leq 2+\mu^{-m_{r}/2}

|t(y)|\leq|t(x)|+5C\cdot l\cdot\mu^{-m_{r}}\leq t(Q_{r})+5C\cdot l\cdot\mu^{-m_{r}}\leq t({}_{r}(Q_{r}))

The last inequality follows from the fact that, for $r$ small, we have

t({}_{r}(Q_{r}))-t(Q_{r})=r(e^{9Nr}-e^{8Nr})\geq\frac{1}{2}r^{2}

Thus $D$ intersects $S_{r}(Q)$ at $y\times 0_{u}$ and $F^{u}({}_{r},x)\cap S_{r}(Q)$ is non-empty. In particular, $x$ is in $S_{r}(Q)$ and $B_{r}(Q)\subset\tilde{B}_{r}(Q)$ . ∎

5. Proof Of Blender Axioms

In the previous section, we constructed the family of maps and an accompanying family of blender boxes. Our objective in this section is to prove that the pair

({}_{r}^{N},B_{r}(Q))\qquad\text{for sufficiently small }r

satisfies the axioms of a stable blender given in Definition 3.22.

Construction 5.1 (Useful Holonomy).

The following holonomy map will be useful in the proofs below. Recall that $a$ and $P$ lie on a leaf of the unstable foliation $F^{u}(\Phi)$ of , with transversals

(5.1)

S=T=[-2,2]^{n}_{s}\times[-\epsilon,L+\epsilon]_{t}\times 0_{u}

By Lemma 3.12, we can choose a neighborhood $\operatorname{Nbhd}(P)$ and a Hölder constant $\kappa$ so that the corresponding holonomy maps $\operatorname{Hol}_{{}_{r}}$ from $\operatorname{Nbhd}(P)\cap S$ to $T$ are Hölder with Hölder constant $\kappa$ for sufficiently small $r$ . For small $r$ , this restricts to a holonomy map

(5.2)

\operatorname{Hol}_{{}_{r}}:D^{n}_{s}(2\cdot\mu^{-m_{r}})\times[L-2r,L]_{t}\times 0_{u}\to[-2,2]^{n}_{s}\times[-\delta,L+\delta]_{t}\times 0_{u}

Lemma 4.14(a) can be restated as the following formula.

(5.3)

\operatorname{Hol}_{{}_{r}}(0_{s},L-\delta,0_{u})=(1_{s},r-\delta,0_{u})\qquad\text{for any }\delta\in[0,2r]

5.1. Axiom A

We start by defining the subset $A$ and proving the axiom in Definition 3.22(a).

Definition 5.2 (Blender Set $A$ ).

We let $A_{r}(Q)\subset B_{r}(Q)$ be the connected component of the intersection $B_{r}(Q)\cap{}^{N}(B_{r}(Q))$ that contains the point $Q=(0_{s},0_{t},0_{u})$ .

Lemma 5.3 (Blender Axiom A).

The intersection $B_{r}(Q)\cap{}_{r}^{N}(B_{r}(Q))$ is disjoint from

\partial^{s}B_{r}(Q)\qquad{}^{N}_{r}(\partial^{c}B_{r}(Q))\quad\text{and}\quad{}_{r}^{N}(\partial^{u}B_{r}(Q))\qquad\text{for sufficiently small }r

In particular, the connected component $A_{r}(Q)$ of $B_{r}(Q)\cap{}_{r}^{N}(B_{r}(Q))$ satisfies these properties.

Proof.

We start by noting that for sufficiently small $r$ , we have

(5.4)

B_{r}(Q)\subset U\cap{}_{r}^{-1}(U)\cap\dots\cap{}_{r}^{-N}(U)

Now fix points $x\in\partial^{s}B_{r}(Q)$ and $y\in{}^{N}_{r}(\partial^{u}B_{r}(Q))$ . By Lemma 4.10 and Construction 4.17, we know that the coordinates of $x$ and $y$ satisfy

|s(x)|=2\qquad\text{and}\qquad|u(y)|>\mu\cdot l\cdot\mu^{-m_{r}}

Also by Lemma 4.10 and Construction 4.17) we know that any $z\in B_{r}(Q)\cap{}^{N}_{r}(B_{r}(Q))$ satisfies

|s(z)|<2\cdot\mu^{-1}\qquad\text{and}\qquad|u(z)|=l\cdot\mu^{-m_{r}}

It follows that $B_{r}(Q)\cap{}^{N}_{r}(B_{r}(Q))$ is disjoint from $\partial^{s}B_{r}(Q)$ and ${}^{N}_{r}(\partial^{u}B_{r}(Q))$ . For the central boundary, note that $B_{r}(Q)$ consists of points where $t\leq L/3$ for small $r$ . Thus by Lemma 4.10

|t({}^{N}_{r}(w))|=|\psi_{Nr}(t(w))|=e^{Nr}\cdot r\cdot(1-e^{-8Nr})\qquad\text{ for any }w\in\partial^{c}B_{r}(Q)

On the other hand, any $z\in B_{r}(Q)$ has $|t(z)|\leq r\cdot(1-e^{-8Nr})$ by Construction 4.17. Thus $B_{r}(Q)$ and ${}^{N}_{r}(\partial^{c}B_{r}(Q))$ are disjoint, and the proof is finished. ∎

5.2. Axiom B

Next, we introduce the subset $A^{\prime}$ appearing in Definition 3.22 and prove the axiom in Definition 3.22(b). We require the following lemma for the definition of $A^{\prime}$ .

Lemma 5.4.

The heteroclinic point $a\in\chi$ is contained in the intersection

B_{r}(Q)\cap{}^{Nm_{r}}(B_{r}(Q))

Proof.

By the construction of the integer $m_{r}$ (see Notation 4.15), we know that

{}^{Nm_{r}}_{r}(Q_{r})\in\big{[}{}_{r}^{5}(P_{r}),{}_{r}^{6}(P_{r})\big{]}\qquad\text{(using Notation \ref{not:Reeb_intervals})}

Since _r and ${}^{H}_{r}$ agree on the Reeb segment $\Gamma\cap U$ , and $({}^{H}_{r})^{Nm_{r}}$ maps the interval $\Gamma\cap B_{r}(Q)$ to an interval in containing $[0,{}^{5}_{r}(P_{r})]$ , we thus deduce that

P_{r}\in{}^{Nm_{r}}([-Q_{r},Q_{r}])\qquad\text{where}\qquad\pm Q_{r}=0_{s}\times\pm r(1-e^{-8Nr})\times 0_{u})

Let $x\in[-Q_{r},Q_{r}]$ be the point with ${}^{Nm_{r}}(x)=P_{r}$ and let $D\subset B_{r}(Q)$ be the disk

D=0_{s}\times x\times D^{n}_{u}(l\cdot\mu^{-m_{r}})\qquad\text{contained in}\qquad F^{u}({}_{r},x)\cap B_{r}(Q)

This disk is a disk of radius $l\cdot\mu^{-m_{r}}$ in $F^{u}({}_{r},x)$ . Since ${}^{N}_{r}$ uniformly expands distances in leaves of $F^{u}$ (see Lemma 4.5), the disk ${}^{Nm_{r}}(D)$ has radius larger than $l$ in $F^{u}({}_{r},P_{r})$ . It follows from the definition of $l$ that

a\in{}^{Nm_{r}}(D)\subset{}^{Nm_{r}}(B_{r}(Q))\qed

Definition 5.5 (Blender Set $A^{\prime}$ ).

The sets $A^{\prime}_{r}(Q)$ and $\tilde{A}_{r}(Q)$ are the components of the intersections

B_{r}(Q)\cap{}^{Nm_{r}}(B_{r}(Q))\qquad\text{and}\qquad\tilde{B}_{r}(Q)\cap{}^{Nm_{r}}(\tilde{B}_{r}(Q))

that contain the point $a=(1_{s},0_{t},0_{u})$ , respectively. Note that $A_{r}(Q)\subset\tilde{A}_{r}(Q)$ .

We next begin working towards the proof of the corresponding axiom, Definition 3.22(b). We need some technical lemmas about the holonomy map (see Construction 5.1). Consider the set

C_{r}=D^{n}_{s}(5\cdot\mu^{-m_{r}})\times[L-2r,L]_{t}\times 0_{u}

The holonomy from Construction 5.1 is well-defined on $C_{r}$ for small $r$ , yielding a smooth map

\operatorname{Hol}_{{}_{r}}:C_{r}\to[-3,3]^{n}_{s}\times[-\epsilon,L+\epsilon]_{t}\times 0_{u}\qquad\text{for small }r

Our first goal is to analyze the image of $C_{r}$ under the holonomy map.

Lemma 5.6 (Holonomy Estimates).

There is a $\kappa>0$ independent of $r$ with the following property. Fix

x\in C_{r}\qquad\text{and}\qquad y=\operatorname{Hol}_{{}_{r}}(x)

Then for sufficiently small $r$ , the coordinates of $y$ satisfy

|s(y)|\leq\mu^{-\kappa m_{r}}\qquad\text{and}\qquad|t(y)-(r+t(x)-L)|\leq\mu^{-\kappa m_{r}}

Proof.

If $x$ has $s(x)=0$ , then (5.3) says that $s(y)=0$ and $t(y)=r+t(x)-L$ . In general, let $x^{\prime}=\pi(x)$ be the projection of $x$ to the $t$ -axis and let $y^{\prime}=\operatorname{Hol}_{{}_{r}}(x^{\prime})$ . Then by Construction 5.1

\operatorname{dist}(y^{\prime},y)\leq\operatorname{dist}(x^{\prime},x)^{\kappa}=|s(x)|^{\kappa}\leq(5l\mu^{-m_{r}})^{\kappa}

Here $\kappa$ is the Hölder constant in Construction 5.1. The implies that

|s(y)|=|s(y)-s(y^{\prime})|\leq(5l\mu^{-m_{r}})^{\kappa}\qquad\text{and}\qquad|s(y)-(r+t(x)-L)|\leq(5l\mu^{-m_{r}})^{\kappa}

Since $m_{r}$ grows faster than $1/r$ , we can eliminate the factor of $5l$ by shrinking $\kappa$ . ∎

Next let _r be the union of the disks $D$ in the unstable foliation satisfying the following properties.

\partial D\subset\{|u|\leq l\cdot\mu^{-m_{r}}\}\qquad\text{and}\qquad D\cap\{u=0\}\in\operatorname{Hol}_{{}_{r}}(\partial C_{r})

Lemma 5.7.

The set _r is disjoint from $\tilde{B}_{r}(Q)\cap{}^{Nm_{r}}(\tilde{B}_{r}(Q))$ .

Proof.

Fix $z\in{}_{r}$ and let $D$ be the corresponding unstable disk of radius $l\cdot\mu^{-m_{r}}$ , centered at $y=\operatorname{Hol}_{{}_{r}}(x)$ where $x\in\partial C$ . Recall that $S_{r}(Q)$ is the intersection of $\tilde{B}_{r}(Q)$ with $\{u=0\}$ , given by

S_{r}(Q)=D^{n}_{s}(2+\mu^{-m_{r}/2})\times D^{1}_{t}(t({}_{r}^{N}(Q_{r})))\times 0_{u}

Note that $P$ is connected to $\operatorname{Hol}(a)$ by a path of length less than $l/3$ in the unstable leaf $F^{u}(\Phi,P)$ . Moreover, $\operatorname{Hol}_{{}_{r}}$ and $C_{r}$ converge to $\operatorname{Hol}$ and $P$ as $r\to 0$ . Therefore $\operatorname{Hol}_{{}_{r}}(x)$ and $x$ are connected by a path of length less than $l/2$ in $F^{u}({}_{r},x)$ for small $r$ , and $z$ is connected to $x$ by a path in of length less than $l$ . Since ${}^{Nm_{r}}(\tilde{B}_{r}(Q))$ contains the unstable disks of radius $l$ around ${}^{Nm_{r}}(S_{r}(Q))$ , it follows that

z\in{}^{Nm_{r}}(\tilde{B}_{r}(Q))\qquad\text{if and only if}\qquad x\in{}^{Nm_{r}}(S_{r}(Q))

We now claim that the intersection ${}^{Nm_{r}}(S_{r}(Q))\cap\partial C_{r}$ is contained in the set

(5.5)

D_{s}^{n}(5\cdot l\cdot\mu^{-m_{r}})\times\{L-2r\}_{t}\times 0_{u}

Indeed, any point in $S_{r}(Q)$ satisfies $|s(x)|\leq 2+\mu^{-m-r/2}$ and $t(x)\leq t({}_{r}^{N}(Q_{r}))$ . Therefore it follows from Lemmas 4.9-4.10 and the construction of $m_{r}$ that

|s({}^{Nm_{r}}(x))|\leq(2+\mu^{-m_{r}/2})\cdot\mu^{-m_{r}}<5\mu^{-m_{r}}

t({}^{Nm_{r}}(x))\leq t({}^{N(m_{r}+1)}(Q_{r}))\leq t({}^{7N}(P_{r}))=L-re^{7Nr}<L

By the definition of $C_{r}$ , this implies that $x$ is in ${}^{Nm_{r}}(S_{r}(Q))\cap\partial C_{r}$ only if $x$ is in the set (5.5).

Finally, we claim that if $x$ is in (5.5), then $z$ is disjoint from $\tilde{B}_{r}(Q)$ for small $r$ . Indeed, by Lemma 5.6, we know that

t(y)\leq r+t(x)-L+\mu^{-\kappa m_{r}}=L-r+\mu^{-\kappa m_{r}}<-r(1-r^{-9Nr})=-t({}_{r}^{N}(Q_{r}))

On the other hand, if $z$ is in $\tilde{B}_{r}(Q)$ , then $y$ must be contained in $S_{r}(Q)$ which only consists of points $y$ with $t(y)\geq-t({}^{N}_{r}(Q_{r}))$ . This shows that $z$ must be disjoint from $\tilde{B}_{r}(Q)\cap{}^{Nm_{r}}(\tilde{B}_{r}(Q))$ , concluding the proof. ∎

We now apply Lemmas 5.6 and 5.7 to acquire some estimates on $A^{\prime}_{r}(Q)$ and $\tilde{A}_{r}(Q)$ .

Lemma 5.8 (Bounds On $\tilde{A}$ ).

There is a $\kappa>0$ so that for any point $z$ in $\tilde{A}_{r}(Q)$ and small $r$ , we have

(5.6)

|s(z)-1_{s}|\leq\mu^{-\kappa m_{r}}\qquad-t(Q_{r})\leq t(z)\leq r(1-e^{-6Nr})+\mu^{-\kappa m_{r}}\qquad|u(z)|\leq l\cdot\mu^{-m_{r}}

Proof.

The bound on $u(z)$ and the lower bound on $t$ follow since $\tilde{A}_{r}(Q)\subset\tilde{B}_{r}(Q)$ .

For the remaining bounds, let $V_{r}\subset U$ denote the tube $[-2,2]^{n}_{s}\times[-\delta,L+\delta]\times D^{n}_{u}(l\cdot\mu^{-m_{r}})$ . Note that ${}_{r}\subset V_{r}$ and $V_{r}\setminus{}_{r}$ consists of two components $V^{\operatorname{in}}_{r}$ and $V^{\operatorname{out}}_{r}$ where

V^{\operatorname{in}}_{r}=\big{\{}z\in D\;:\;D\text{ is unstable disk with $D\cap S_{r}(Q)\in\operatorname{Hol}_{{}_{r}}(\operatorname{int}(C_{r}))$ and $\partial D\subset\{|u|=l\cdot\mu^{-m_{r}}\}$}\big{\}}

We will not need a description of $V^{\operatorname{out}}_{r}$ . Since $\tilde{B}_{r}(Q)\subset V_{r}$ by construction, $\tilde{A}_{r}(Q)\subset V_{r}$ as well. By Lemma 5.7, $\tilde{A}_{r}(Q)$ must be in one of the components $V^{\operatorname{in}}_{r}$ and $V^{\operatorname{out}}_{r}$ . A direct computation gives $a\in V_{r}^{\operatorname{in}}$ , so it follows from Definition 5.5 that $\tilde{A}_{r}(Q)\subset V^{\operatorname{in}}_{r}$ .

It therefore suffices to prove the remaining bounds for a point $z$ in $V^{\operatorname{in}}_{r}$ . Let $D$ be the unstable disk through $z$ and let $y=\operatorname{Hol}_{{}_{r}}(x)$ be the intersection of $D$ with $D_{r}(Q)$ . By Lemma 5.6

|s(y)|\leq\mu^{-\kappa m_{r}}\qquad\text{and}\qquad t(y)\leq L-r+t(x)\leq L-r+t({}^{7}_{r}(P_{r}))=r(1-e^{7Nr})

The point $z$ lies on the vertical disk $D$ , which is a graph of a $2\epsilon$ -Lipschitz graph from $D^{n}_{u}(l\cdot\mu^{-m_{r}})$ . It follows that $\operatorname{dist}(z,y)\leq C\mu^{-m_{r}}$ for some $C$ independent of $r$ , and so

|s(x)|\leq\mu^{-\kappa m_{r}}+C\mu^{-m_{r}}\qquad t(y)\leq r(1=e^{7Nr})+C\mu^{-m_{r}}

The remaining estimates follow by taking $r$ small and possibly shrinking $\kappa$ . ∎

The axiom in Definition 3.22(b) is now an easy consequence of Lemma 5.8.

Lemma 5.9 (Blender Axiom B).

The regions $\tilde{A}_{r}(Q)$ and $A^{\prime}_{r}(Q)$ is disjoint from

\partial^{r}B_{r}(Q)\qquad\partial^{s}B_{r}(Q)\quad\text{and}\quad{}_{r}^{N}(\partial^{u}B_{r}(Q))

Proof.

Disjointness from ${}^{N}(\partial^{u}B_{r}(Q))$ follows from the same argument as in Lemma 5.3. For the other two boundary regions, note that by Lemma 5.8 we have

|s(x)-1_{s}|\mu^{-\kappa m_{r}}\quad\text{and}\quad t(x)\leq r(1-e^{-7Nr})+\mu^{-\kappa m-r}\quad\text{for all }x\in\tilde{A}_{r}(Q)

This implies that for sufficiently small $r$ , the set $\tilde{A}_{r}(Q)$ is disjoint from the sets

\partial^{s}B_{r}(Q)=B_{r}(Q)\cap\{|s|=2\}\qquad\text{and}\qquad\partial^{r}B_{r}(Q)=B_{r}(Q)\cap\{t=r\cdot(1-e^{-8Nr})\}\qed

5.3. Axioms C And D

Next, we prove the blender axioms related to cone-fields (see Definition 3.22(c-d)). The first of these axioms is relatively straightforward.

Lemma 5.10 (Blender Axiom C).

There are compatible cone-fields $K^{s}$ and $K^{u}$ for $B_{r}(Q)$ of width less than $\epsilon$ (with respect to the standard metric) that are contracted and dilated (with constant $\mu$ ) as follows.

({}_{r}^{-N})_{*}K^{s}\subset\operatorname{int}K^{s}\qquad({}_{r}^{N})_{*}K^{u}\subset\operatorname{int}K^{u}\qquad{}_{r}^{-N}\text{ dilates }K^{s}\qquad{}_{r}^{N}\text{ dilates }K^{u}

Proof.

Let $K^{s}$ and $K^{u}$ be the cone-fields in Lemma 4.5. Note that ${}_{r}\to\Phi$ in the $C^{\infty}$ -topology. Moreover, contraction and dilation (with constant $\mu$ ) of a given cone-field are $C^{1}$ -robust properties. Thus, Lemma 4.5 implies this axiom for our choice of constants $N,\mu,\epsilon$ and small $r$ . ∎

The second cone-field related axiom (regarding the central-unstable cone-field) is more difficult and will require some preliminary results. To start, choose a Riemannian metric

g\text{ on }TY\qquad\text{such that the splitting }E^{u}({}_{r})\oplus T{}^{1}_{t}\oplus T{}^{n}_{s}\text{ is orthogonal on $U$}

We consider the following cone-fields of width $\epsilon$ with respect to $g$ .

K^{us}_{\epsilon}=K_{\epsilon}E^{u}({}_{r})\cap(E^{u}({}_{r})\oplus T{}^{n}_{s})=\{u+v\;:\;u\in E^{u}({}_{r})\text{ and }v\in T{}^{n}_{s}\text{ with }|u|\geq\epsilon\cdot v\}

K^{cs}_{\delta}=K_{\delta}(T{}^{1}_{t})\cap(T{}^{1}_{t}\oplus T{}^{n}_{s})=\{u+v\;:\;u\in T{}^{1}_{t}\text{ and }v\in T{}^{n}_{s}\text{ with }|u|\geq\delta\cdot v\}

These cone-fields are well-defined over $U$ (i.e. wherever $(s,t,u)$ -coordinates are well-defined). Finally, we define the fiberwise sum of cones

K^{cu}_{\delta,\epsilon}=K^{us}_{\epsilon}+K^{cs}_{\delta}

The following key lemma describes choices of the parameters of the cone $K^{cu}$ that guarantee contraction and dilation.

Lemma 5.11 (Stretching $K^{cu}$ ).

Fix a positive integer $k$ , positive constants $\mu,\epsilon,\nu,\eta$ , and a subset $V\subset U$ with the following properties.

•

The constants satisfy $\mu^{2}>1+\epsilon^{2}$ and $\nu>1>\eta$ .
•

$T{}^{k}_{r}$ dilates $E^{u}({}_{r})$ with constant $\mu$ and $T{}^{k}_{r}$ dilates $T{}^{n}_{s}$ with constant $\mu^{-1}$ over $V$ .
•

$T{}^{k}_{r}(R)=\nu\cdot R+w$ where $w\in E^{s}(\Phi)=T{}^{n}_{s}$ and $|w|\leq\eta\cdot|R|$ over the subset $V$ , where $R=\partial_{t}$ .

Then $K^{cu}_{\epsilon,\delta}$ is contracted and uniformly dilated by ${}_{r}^{k}$ over the susbet $V$ for

\frac{\eta}{1-\mu^{-1}}<\delta<\sqrt{\nu^{2}-1}

Proof.

To prove that $K^{cu}_{\epsilon,\delta}$ is uniformly dilated with some positive constant of dilation, we write an arbitrary vector $v$ in $K^{cu}_{\epsilon,\delta}$ as follows.

v=(v^{u}+v^{s})+(aR+w^{s})\qquad\text{where}\qquad|v^{u}|\geq\epsilon|v^{s}|\text{ and }|aR|=a\geq\delta|w^{s}|

We then compute the norm of the image of $v$ under ^Nk.

|{}^{k}_{r}(v)|^{2}=|{}^{k}_{r}(v^{u})|^{2}+|{}^{k}_{r}(aR)|^{2}+|{}^{k}_{r}(v^{s}+w^{s})|^{2}

\geq\mu^{2}\cdot|v^{u}|^{2}+\nu^{2}\cdot|aR|^{2}+|aw|^{2}+\mu^{-2}\cdot|v^{s}+w^{s}|^{2}\geq\mu^{2}\cdot|v^{u}|^{2}+\nu^{2}\cdot|aR|^{2}

Now since $|v^{u}|\geq\epsilon|v^{s}|$ and $|aR|\geq\delta|w^{s}|$ , we see that

(1+\epsilon^{2})\cdot|v^{u}|^{2}\geq|v^{u}+v^{s}|^{2}\qquad\text{and}\qquad(1+\delta^{2})\cdot|aR|^{2}\geq|aR+w^{s}|^{2}

Finally, we calculate that

|{}^{Nk}_{r}(v)|^{2}\geq\frac{\mu}{1+\epsilon^{2}}\cdot|v^{u}+v^{s}|^{2}+\frac{\nu^{2}}{1+\delta^{2}}\cdot|aR|^{2}\geq\operatorname{min}(\frac{\mu}{1+\epsilon^{2}},\frac{\nu^{2}}{1+\delta^{2}})\cdot|v|^{2}

Since $\mu^{2}>1+\epsilon^{2}$ and $\nu^{2}>1+\delta^{2}$ by assumption, we thus find that $v$ is uniformly expanded.

To prove that $K^{cu}_{\delta,\epsilon}$ is contracted, we argue as follows. First, note that for any $v=v^{u}+v^{s}$ in $K^{us}_{\epsilon}$ with $|v^{u}|\geq\epsilon|v^{s}|$ , we have

{}^{k}_{r}(v)={}^{k}_{r}(v^{u})+{}^{k}_{r}(v^{s})\qquad\text{where}\qquad{}^{k}_{r}(v^{u})\in E^{u}({}_{r})\text{ and }{}^{k}_{r}(v^{s})\in T{}^{n}_{s}

By our hypothesis on the dilation of $T{}_{r}^{k}$ , we know that

|T{}^{k}_{r}(v^{u})|\geq\mu^{2}\cdot\epsilon\cdot|T{}^{k}_{r}(v^{s})|>\epsilon\cdot|T{}^{k}_{r}(v^{s})|

Therefore $T{}_{r}^{k}(v)$ is strictly contracted by $T{}_{r}^{k}$ . Likewise, take any vector $v=aR+v^{s}$ in $K^{us}_{\delta}$ with $|aR|\geq\delta\cdot|v^{s}|$ . Then

T{}^{k}_{r}(v)=a\nu R+(aw+{}^{k}_{r}(w^{s})

Now we note that we have the following estimate.

|aw+{}^{k}_{r}(w^{s})|\leq|aw|+|{}^{k}(w^{s})|\leq\eta|aR|+\epsilon\cdot\mu^{-1}\cdot|aR|\leq(\eta+\delta\cdot\mu^{-1})\cdot|aR|

By assumption we have $\eta+\delta\cdot\mu^{-1}<\delta$ and thus $T{}^{k}_{r}$ contracts the cone $K^{us}_{\delta}$ . We have thus proven that

T{}^{k}_{r}(K^{cu}_{\delta,\epsilon})\subset\operatorname{int}K^{cu}_{\delta,\epsilon}\qed

We next verify the third criterion in Lemma 5.11 in the cases relevant to our axiom. Recall that we have fixed constants $N,m$ (see the beginning of Section 4.2).

Lemma 5.12 (Axiom D, Part 1).

For sufficiently small $r$ , we have

T{}^{N}_{r}(R)=e^{Nr}\cdot R\qquad\text{over the subset }A_{r}(Q)

Proof.

For sufficiently small $r$ , we have

{}^{j}_{r}(B_{r}(Q))\subset[-2,2]_{s}\times[-\delta,L/3]_{t}\times[-\epsilon,\epsilon]\text{ for all }j=0,\dots,N

Here $\delta,\epsilon,L$ are the parameters of the chart in Lemma 4.4. In this region, we know that ${}_{r}={}^{H}_{r}\circ\Psi$ . Therefore by Construction 4.6 (and more specifically (4.8)) we find that

T{}^{H}_{r}(R)=e^{r}R\quad\text{and}\quad T\Psi(R)=R\qquad\text{and therefore}\quad T{}_{r}^{N}(R)=e^{Nr}\cdot R\qed

For the other part of Axiom D, we require the following lemma tracking the behavior of the set $A^{\prime}_{r}(Q)$ under ${}^{-1}_{r}$ .

Lemma 5.13 ( $\tilde{A}$ Stays In $U$ ).

Let $m$ and $W$ be as in Construction 4.7 and (4.10). Then

\tilde{A}_{r}(Q)\subset{}^{Nm}_{r}(W)\qquad\text{and}\qquad{}^{-j}(\tilde{A}_{r}(Q))\subset U\text{ for all }j=Nm,\dots,m_{r}\qquad\text{for small $r$}

Proof.

For the first claim, note that $N$ and $m$ are independent of $r$ and ${}_{r}\to\Phi$ in $C^{\infty}$ as $r\to 0$ . Thus we may choose an $r$ -independent open neighborhood $V\subset{}^{Nm}_{r}(W)$ with $a\in V$ . By Lemma 5.8, we know that $\tilde{A}_{r}(Q)$ is contained in the region

B_{r}(Q)\cap\{|s-1_{s}|\leq\beta\cdot l\cdot\mu^{-\kappa m_{r}}\}\qquad\text{for $\beta,\kappa>0$}

It follows from Construction 4.17 that this region is contained in a ball of radius bounded by $r$ around $a$ . Thus for small $r$ , this region is contained in $V$ and the first claim is proven.

For the second claim, we require some preliminary observations. Consider the subsets $\Xi\subset\Gamma$ and $\Sigma\subset W_{\operatorname{loc}}^{s}(\Phi;U)$ given by

\Xi=0_{s}\times[0,t(Q_{r})]_{t}\times 0_{u}=[Q,Q_{r}]\qquad\text{and}\qquad\Sigma=D^{n}_{s}(5/2)\times[0,t(Q_{r})]_{t}\times 0_{u}

Note that $\tilde{B}_{r}(Q)\cap\{u=0,t\geq 0\}$ is contained in by Construction 4.17 and (4.16). Let $\psi_{r}$ be the flow of $h\cdot\partial_{t}$ (see Lemma 4.10). Then $\psi_{r}^{j}(0)=0$ for all $j$ and by construction of $m_{r}$ , we know that

\psi_{r}^{j}(t(Q_{r}))\in[-\delta,L+\delta]_{t}\quad\text{and}\quad{}_{r}(Q_{r})\in\Gamma\cap U\qquad\text{for all }j=0,\dots,Nm_{r}

Moreover, since ${}_{r}(\Xi)=0_{s}\times[0,\psi_{r}(t(Q_{r}))]_{t}\times 0_{u}$ and _r contracts the $s$ -coordinate (see Lemma 4.10), this implies that

{}_{r}^{j}(\Xi)\subset\Gamma\cap U\quad\text{and}\quad{}_{r}^{j}(\Sigma)\subset\{u=0\}\cap U\qquad\text{for all }j=0,\dots,Nm_{r}

Now we prove the second claim. By the first claim and the definition of $\tilde{A}_{r}(Q)$ , we know that

(5.7)

{}^{-Nm}(\tilde{A}_{r}(Q))\subset W\subset U\qquad\text{and}\qquad{}^{-Nm}_{r}(\tilde{A}_{r}(Q))\subset{}^{Nm_{r}-Nm}_{r}(\tilde{B}_{r}(Q))

From these two inclusions, it follows that ${}^{-Nm}_{r}(\tilde{A}_{r}(Q))$ is included in the set

V^{\prime}=\{x\in F^{u}({}_{r};y)\;:\;y\in{}^{Nm_{r}-Nm}_{r}(\Sigma)\quad\text{and}\quad|u(x)|\leq 2\}

Here we choose $W$ in Construction 4.6 so that $W$ is contained in $\{|u|\leq 2\}$ . Finally, we note that

{}_{r}^{-j}(V^{\prime})\subset U\qquad\text{for all }j=0,\dots,Nm_{r}-Nm

Indeed, the intersection of ${}_{r}^{-j}(V^{\prime})$ with $\{u=0\}$ is ${}^{Nm_{r}-Nm-j}(\Sigma)$ and the union $V^{\prime\prime}$ of unstable disks intersecting ${}^{Nm_{r}-Nm-j}(\Sigma)$ with boundary on $\{u=3/2\}$ is contained in $U$ . On the other hand, since ${}^{-1}_{r}$ contracts distances in $U$ , ${}_{r}^{-j}(V^{\prime})\subset V^{\prime\prime}$ . This proves the second claim. ∎

Lemma 5.14 (Axiom D, Part 2).

For sufficiently small $r$ , we have

T{}^{Nm_{r}}(R)=\lambda_{r}\cdot R+v\qquad\text{over}\quad{}^{-Nm_{r}}(A^{\prime}_{r}(Q))

Here $\lambda_{r}\geq r^{-1/2}$ and $|v|\leq r^{2}\cdot\mu^{-\log(r)/r}$ for small $r$ .

Proof.

Fix an arbitrary point $x$ in ${}^{-Nm_{r}}(\tilde{A}_{r}(Q))$ . We let $n^{\operatorname{st}}_{r}$ and $n^{\operatorname{mid}}_{r}$ denote the quantities

n_{r}^{\operatorname{st}}=\operatorname{min}\big{\{}j\>:\;t({}_{r}^{Nj}(x))\geq L/3\big{\}}\qquad\text{and}\qquad n_{r}^{\operatorname{mid}}=\operatorname{min}\big{\{}j-n^{\operatorname{st}}_{r}\>:\;t({}_{r}^{Nj}(x))\geq 2L/3\big{\}}

Finally, let $m$ and $W$ be as in Construction 4.7 and (4.10), and let

n^{\operatorname{end}}_{r}=m_{r}-Nm-n^{\operatorname{st}}_{r}-n^{\operatorname{mid}}_{r}

By essentially identical analysis to Lemma 4.16, we have the following asymptotic behavior for these quantities.

(5.8)

n^{\operatorname{st}}_{r}\sim-2r^{-1}\cdot\log(r)\qquad n^{\operatorname{mid}}_{r}\sim r^{-1}\quad\text{and}\quad n^{\operatorname{end}}_{r}\sim-r^{-1}\cdot\log(r)

Here $\sim$ means the limit of the ratio is $1$ as $r\to 0$ . We will analyze the sequence of terms $T{}^{Nj}_{r,x}(R)$ starting at the point $x$ in four regimes, corresponding to the following intervals for the index $j$ .

[0,n^{\operatorname{st}}_{r}]\qquad[n^{\operatorname{st}}_{r},n^{\operatorname{st}}_{r}+n^{\operatorname{mid}}_{r}]\qquad[n^{\operatorname{st}}_{r}+n^{\operatorname{mid}}_{r},m_{r}-Nm]\qquad[m_{r}-Nm,m-r]

We will call these the starting regime, middle regime, ending regime and extra regime. Crucially, by Lemma 5.13, we know that ${}^{Nj}_{r}(x)$ stays in $U$ for the first three periods.

Step 1: Starting Regime. Start by assuming that $j$ is in the interval $[0,n^{\operatorname{st}}_{r}]$ . In this regime, $t({}_{Nj}(r))\leq L/3$ . It follows from (4.12) and Construction 4.6 (see (4.8)) that

(5.9)

T{}^{Nj}_{r}(R)=(T{}^{H}_{r}\circ T\Phi)^{Nj}(R)=(T{}^{H}_{r})^{Nj}(R)=e^{rNn^{\operatorname{st}}_{r}}\cdot R

Step 2: Middle Regime. Next, assume that $j$ is in the middle interval $[n^{\operatorname{st}}_{r},n^{\operatorname{st}}_{r}+n^{\operatorname{mid}}_{r}]$ . In this case, it follows from the general form for ${}^{H}_{r}$ in (4.7) that for any $y$ in $U$ with $t(y)\in[L/3,2L/3]$ , we have

(5.10)

T{}^{H}_{r,y}(R)=\partial_{t}\psi_{r}(t(y))\cdot R+\partial_{t}f_{r}(t(y))(s(y)\cdot\partial_{s})

under the splitting $TU=T{}^{n}_{s}\oplus T{}^{1}_{t}\oplus T{}^{n}_{u}$ . Here $\psi$ is given in (4.7) and satisfies

\psi_{0}(t)=t\qquad\partial_{r}\psi=h\circ\psi

By the assumption that $|h^{\prime}|\leq 1/L$ on the interval $[0,L]$ , we know that $|h|\leq 1$ and so

(5.11)

|\partial_{t}\psi_{r}(t(y))|\geq e^{-r}\qquad\text{for small }r

Next note that $|\partial_{t}f_{r}(t(y))|$ is bounded by some constant $C>0$ for small $r$ . Finally, ${}_{r}^{N}$ and ^N both preserve $T{}^{n}_{s}$ and contract $T{}^{n}_{s}$ by a factor of $\mu^{-1}$ for small $r$ (see Lemmas 4.5 and 4.11).

Now let $x_{\operatorname{st}}$ and $R_{\operatorname{st}}$ denote the point and vector that is left after we exit the early regime.

x_{\operatorname{st}}={}^{Nn^{\operatorname{st}}_{r}}_{r}(x)\qquad\text{and}\qquad R_{\operatorname{st}}=T{}_{r,x}^{Nn^{\operatorname{st}}_{r}}(R)\in\operatorname{span}(R)

It follows from the discussion above and (5.10) that we can write the following expansion.

(5.12)

T{}^{Nn^{\operatorname{mid}}_{r}}_{r,x_{\operatorname{st}}}(R_{\operatorname{st}})=T({}^{H})^{Nn^{\operatorname{mid}}_{r}}_{r,x_{\operatorname{st}}}(R_{\operatorname{st}})+\sumop\slimits@_{k=0}^{Nn^{\operatorname{mid}}_{r}}w_{k}

Here the vectors $w_{k}$ can be written as follows.

w_{k}=T{}_{r}^{Nn^{\operatorname{mid}}_{r}-k}(c_{k}\cdot(s({}^{k}_{r}(x_{\operatorname{st}}))\cdot\partial_{s}))\in T{}^{n}_{s}\qquad\text{where}\qquad c_{k}=\partial_{r}f_{r}(t({}^{k}_{r}(x_{\operatorname{st}})))

Now we estimate the terms appearing in (5.12). First, by (5.11) we know that

|T({}^{H})^{Nn^{\operatorname{mid}}_{r}}_{r,x_{\operatorname{st}}}(R_{\operatorname{st}})|\geq e^{-Nn^{\operatorname{mid}}_{r}}\cdot|R_{\operatorname{st}}|\qquad\text{and}\qquad T({}^{H})^{Nn^{\operatorname{mid}}_{r}}_{r,x_{\operatorname{st}}}(R_{\operatorname{st}})\in\operatorname{span}(R)

Next we estimate the norm of $w_{k}$ . As noted previously, $|c_{k}|\leq C$ for some $C$ independent of $k$ and $r$ small. Since ${}^{k}_{r}(x_{\operatorname{st}})$ is in the image of $U$ under the $(Nn_{r}^{\operatorname{st}}+k)$ -th power of _r and _r uniformly contracts $T{}^{n}_{s}$ by a factor of $\mu^{-1}$ , we know that the $s$ -vector $s({}^{k}(x_{\operatorname{st}}))\cdot\partial_{s}$ is bounded as follows.

|s({}^{k}(x_{\operatorname{st}}))\cdot\partial_{s}|\leq 2\cdot\mu^{-n^{\operatorname{st}}_{r}-\lfloor k/N\rfloor}

Finally, $T{}^{N}_{r}$ uniformly contracts vectors in $T{}^{n}_{s}$ by a factor of $\mu^{-1}$ . Combining these estimates, we find that for some constant $C^{\prime}>0$ independent of $x$ and small $r$ , we have

|w_{k}|\leq C^{\prime}\cdot\mu^{-(n^{\operatorname{st}}_{r}+n^{\operatorname{mid}}_{r})}\qquad\text{and}\qquad\Big{|}\sumop\slimits@_{k=0}^{Nn^{\operatorname{mid}}_{r}}w_{k}\Big{|}\leq C^{\prime}\cdot Nn^{\operatorname{mid}}_{r}\mu^{-(n^{\operatorname{st}}_{r}+n^{\operatorname{mid}}_{r})}\leq C^{\prime\prime}\cdot\frac{1}{r}\cdot\mu^{-2\log(r)/r}

The outcome of this analysis of the middle regime is the following formula.

(5.13)

R_{\operatorname{mid}}=T{}_{r,x}^{N(n^{\operatorname{st}}_{r}+n^{\operatorname{mid}}_{r})}(R)=e^{rNn^{\operatorname{st}}_{r}}\cdot T{}_{r,x}^{Nn^{\operatorname{mid}}_{r}}(R)=A_{\operatorname{mid}}\cdot e^{rNn^{\operatorname{st}}_{r}}\cdot R+w

Here $A_{\operatorname{mid}}$ is some constant bounded by $e^{-Nn^{\operatorname{mid}}_{r}}$ and $w$ is a vector in $T{}^{n}_{s}$ with norm bounded by the quantity $C^{\prime\prime}\cdot r^{-1}\cdot\mu^{-2\log(r)/r}$ .

Step 3: Ending Regime. We next examine the ending regime, where $j$ is in the interval $[n^{\operatorname{st}}_{r}+n^{\operatorname{mid}}_{r},m_{r}-Nm]$ . In this regime, $t({}_{Nj}(r))\geq 2L/3$ . By using (4.12) and Construction 4.6 (and specifically (4.9)), we see that we have

T{}^{Nn^{\operatorname{end}}_{r}}_{r}(R_{\operatorname{mid}})=A_{\operatorname{mid}}\cdot e^{rNn^{\operatorname{st}}_{r}}\cdot T{}^{Nn^{\operatorname{end}}_{r}}_{r}(R)+T{}^{Nn^{\operatorname{end}}_{r}}_{r}(w)=A_{\operatorname{mid}}\cdot e^{rN(n^{\operatorname{st}}_{r}-n^{\operatorname{end}}_{r})}R+T{}^{Nn^{\operatorname{end}}_{r}}_{r}(w)

Focusing on the first term, the lower bound $A_{\operatorname{mid}}\geq e^{-rNn^{\operatorname{mid}}_{r}}$ and the asymptotic formula (5.8) imply that

A_{\operatorname{mid}}\cdot e^{rN(n^{\operatorname{st}}_{r}-n^{\operatorname{end}}_{r})}\geq e^{-r\cdot\log(r)/r}\geq r^{-1/2}\qquad\text{for small }r

Moreover, ^N uniformly contracts $T{}^{n}_{s}$ by a factor of $\mu^{-1}$ , and we thus see that

w^{\prime}=T{}^{Nn^{\operatorname{end}}_{r}}_{r}(w)\qquad\text{satisfies}\qquad|w^{\prime}|\leq N\cdot n^{\operatorname{mid}}_{r}\cdot\mu^{-(n^{\operatorname{st}}_{r}+n^{\operatorname{mid}}_{r}+n^{\operatorname{end}}_{r})}\leq N\cdot m_{r}\cdot\mu^{-m_{r}+Nm}

Combining all of the analysis up to now, we have proven that

(5.14)

T{}^{N(m_{r}-m)}_{r}(R)=\lambda_{r}\cdot R+v\qquad\text{over }{}^{-Nm_{r}}(A^{\prime}_{r}(Q))

where $\lambda_{r}\geq r^{-1/2}$ and $v\in T{}^{n}_{s}$ satisfies $|v|\leq r^{2}\cdot\mu^{-\log(r)/r}$ for small $r$ .

Step 4: Extra Regime. Finally, we consider the last regime. By Lemma 5.13, we know that

{}^{N(m_{r}-m)}_{r}(x)\subset W

By the construction of _r, or more precisely (4.12), we know that

T{}^{Nm}_{r}=T{}^{R}_{r}\circ T{}^{Nm}\circ T{}^{H}_{Nmr}\qquad\text{on }W

Note that: $T{}^{R}_{r}$ preserves $R$ and $T{}^{n}_{s}$ ; $T{}^{Nm}$ preserves $R$ and shrinks $T{}^{n}_{s}$ by a factor of $\mu^{-1}$ ; and $T{}^{H}_{Nmr}$ preserves $R$ and expands $T{}^{n}_{s}$ by at most a factor of $e^{Nmr}$ . It follows that

T{}^{Nm_{r}}_{r}(R)=\lambda_{r}\cdot R+v

where $\lambda_{r}$ and $v$ satisfy the same estimates as in (5.14). This concludes the proof.∎

We are (finally) ready to prove the second cone axiom. Thanks to the onerous work of Lemmas 5.12 and 5.14, this will be a simple application of Lemma 5.11.

Lemma 5.15 (Blender Axiom D).

There is a compatible cone-field $K^{cu}$ for $B_{r}(Q)$ of width less than $\epsilon$ (with respect to the standard metric) that is contracted and dilated uniformly as follows.

({}_{r}^{N})_{*}K^{cu}\subset\operatorname{int}K^{cu}\quad\text{and}\quad{}^{N}_{r}\text{ dilates }K^{cu}\qquad\text{ over }{}^{-N}(A_{r}(Q))

({}^{Nm_{r}}_{r})_{*}K^{cu}\subset\operatorname{int}K^{cu}\quad\text{and}\quad{}^{Nm_{r}}_{r}\text{ dilates }K^{cu}\qquad\text{ over }{}_{r}^{-Nm_{r}}(A^{\prime}_{r}(Q))

Proof.

Fix constants $N,\mu,\epsilon$ as in Lemma 4.5 such that $\mu>1+\epsilon^{2}$ . By Lemma 5.13, we may take $r$ small enough so that $T{}^{N}_{r}$ dilates $E^{u}({}_{r})$ by constant $\mu$ and dilates $T{}^{n}_{s}$ by $\mu^{-1}$ over $U$ . We take

K^{cu}=K^{cu}_{\epsilon,\delta}

for a judiciously chosen $\delta$ . To choose $\delta$ appropriately, note that by Lemma 5.12, we have

T{}^{N}_{r}(R)=e^{Nr}\cdot R\qquad\text{ over the subset }{}^{-N}_{r}(A_{r}(Q))\subset U

It follows by Lemma 5.11 that $K^{cu}$ is contracted and uniformly dilated by ${}^{N}_{r}$ as long as

(5.15)

0\leq\delta\leq\sqrt{e^{Nr}-1}

Likewise, by Lemma 5.14, we know that for large $r$

T{}^{Nm_{r}}(R)=\lambda_{r}\cdot R+v\qquad\text{ over the subset }{}^{-Nm_{r}}_{r}(A_{r}(Q))\subset U

where $|\lambda_{r}|\geq r^{-1/2}$ and $|v|\leq r^{2}\cdot\mu^{-\log(r)/r}$ . It follows by Lemma 5.11 that $K^{cu}$ is contracted and uniformly dilated by ${}^{Nm_{r}}_{r}$ as long as

(5.16)

\frac{\mu^{-\log(r)/r}}{1-\mu^{-1}}\leq\delta\leq\sqrt{r^{-1}-1}\leq\sqrt{\lambda_{r}^{2}-1}

On the other hand, for every small $r$ we know that

\frac{\mu^{-\log(r)/r}}{1-\mu^{-Nm_{r}}}\leq r^{2}\leq\sqrt{e^{Nr}-1}

Thus we may choose $\delta$ to satisfy both (5.15) and (5.16). The result now follows by Lemma 5.11.∎

5.4. Axioms E And F

Finally, we prove the blender axioms regarding vertical disks, Definition 3.22(e-f). The first such axiom is relatively straightforward.

Lemma 5.16 (Blender Axiom E).

Let $D$ be a vertical disk through $B_{r}(Q)$ to the right of the local unstable manifold of $Q$ with respect to _r. Then

\operatorname{dist}(D,\partial^{l}B_{r}(Q))>r^{3}\qquad\text{for small }r

In particular, there is a neighborhood $U_{-}$ of $\partial^{l}B_{r}(Q)$ that is disjoint from all such disks $D$ .

Proof.

By Lemma 4.12, the local unstable manifold $W$ of $Q$ with respect to _r in $B_{r}(Q)$ is given by $D^{n}_{s}(2)\times 0_{t}\times 0_{u}$ . It follows that a vertical disk $D$ to the right of $W$ must intersect the manifold

W^{\prime}=D^{n}_{s}(2)\times D^{1}_{t}(t(Q_{r}))\times 0_{u}

at a point $x$ with $t(x)>0$ and $u(x)=0$ . By Lemma 3.21, $D$ is the graph of a $2\epsilon$ -Lipschitz map

f:D^{n}_{u}(l\cdot\mu^{-\mu_{r}})\to D^{n}_{s}(2)\times D^{1}_{t}(t(Q_{r}))

The image of $f$ , or equivalently the projection of $D$ to $D^{n}_{s}(2)\times D^{1}_{t}(t(Q_{r}))$ , is contained in a ball of radius $2\epsilon\cdot l\cdot\mu^{-\mu_{r}}$ and $t$ -coordinate of $D$ is lower bounded by

t_{0}-2\epsilon\cdot l\cdot\mu^{-m_{r}}>-2\epsilon\cdot l\cdot\mu^{-m_{r}}

By Construction 4.17, the left boundary $\partial^{l}B_{r}(Q)$ consists of points in $B_{r}(Q)$ with $t$ -coordinate $-t(Q_{r})$ . Finally, note that $m_{r}$ diverges faster than $1/r$ as $r\to 0$ by Lemma 4.16. Therefore

\operatorname{dist}(D,\partial^{l}B_{r}(Q))>r\cdot(1-e^{-8Nr})-2\epsilon\cdot l\cdot\mu^{-m_{r}}>r^{3}\qquad\text{for small $r$ }\qed

The final blender axiom is more difficult. We prove the following version.

Lemma 5.17 (Blender Axiom F).

Let $D$ be a vertical disk through $B_{r}(Q)$ to the right of the local unstable manifold $W$ of $Q$ with respect to _r. Consider the intersection point

x=(s,t,0_{u})=D\cap W^{\prime}\qquad\text{where}\qquad W^{\prime}:=D^{n}_{s}\times D^{1}_{t}\times 0_{u}

Then the following two alternatives hold for $D$ .

(a)

If $0<t(x)\leq t({}^{-3N}(Q_{r}))$ then the intersection ${}_{r}^{N}(D)\cap A_{r}(Q)$ contains a disk $\tilde{D}$ through $B_{r}(Q)$ to the right of $W$ , such that

$\operatorname{dist}(\tilde{D},\partial^{r}B_{r}(Q))>r^{3}$
(b)

Otherwise, if $t({}^{-3N}(Q_{r}))\leq t(x)$ then the intersection ${}_{r}^{Nm_{r}}(D)\cap A^{\prime}_{r}(Q)$ contains a disk $\tilde{D}$ through $B_{r}(Q)$ to the right of $W$ such that

$\operatorname{dist}(\tilde{D},W)>r^{3}$

Thus there are neighborhoods $U_{+}$ of $\partial^{r}B_{r}(Q)$ and $U$ of $W$ such that $\tilde{D}$ is disjoint from either $U$ or $U_{+}$ .

Proof.

We proceed in several steps. First, we address (a) which is easy. Second, we

Step 1. We prove (a) first. Note that the $t$ -coordinate of ${}_{r}^{N}(x)$ satisfies

0<t({}^{N}_{r}(x))=e^{Nr}\cdot t(x)\leq e^{-2Nr}\cdot t(Q_{r})

Take the disk $\tilde{D}$ to be the connected component of ${}_{r}^{N}(D)$ containing ${}_{r}^{N}(x)$ . Then $\tilde{D}$ is a vertical disk to the right of $W$ since it intersects $W^{\prime}$ at a point with positive $t$ -coordinate. By Lemma 3.21, it is the graph of a $2\epsilon$ -Lipschitz map

f:D^{n}_{u}(l\cdot\mu^{-m_{r}})\to D^{n}_{s}(2)\times D^{1}_{t}(t(Q_{r}))

As in Lemma 5.16, this implies that the $t$ -coordinate is bounded by

e^{-2Nr}\cdot t(Q_{r})+2\epsilon\cdot l\cdot\mu^{-m_{r}}<e^{-Nr}\cdot t(Q_{r})\qquad\text{for small }r

The right side $\partial^{r}B_{r}(Q)$ is the set of points with $t$ -coordinate $t(Q_{r})$ , by Definition 4.17. Therefore

\operatorname{dist}(\partial^{r}B_{r}(Q),\tilde{D})>t(Q_{r})-e^{-Nr}\cdot t(Q_{r})=e^{-Nr}\cdot(1-e^{-Nr})\cdot(1-e^{-8Nr})>r^{3}\qquad\text{for small }r

Step 2. In this step, we consider a useful special case of (b). Let $C^{\prime}_{r}\subset W^{\prime}$ denote the set

C^{\prime}_{r}=D^{n}_{s}(2+\mu^{-m_{r}/2})\times[t({}^{-4N}_{r}(Q_{r})),t({}^{N}_{r}(Q_{r}))]_{t}\times 0_{u}

Given a point $w\in C^{\prime}_{r}$ , we let $D_{w}$ denote the following vertical disk

D_{w}=\big{\{}(s,t,u)\in F^{u}({}_{r},w)\;:\;|u|\leq l\cdot\mu^{-m_{r}}\big{\}}

Note that by Lemmas 4.10 and 4.9, we have

{}^{Nm_{r}}_{r}(C^{\prime}_{r})\subset D^{n}_{s}(2\cdot\mu^{-m_{r}})\times[t({}^{N(m_{r}-4)}_{r}(Q_{r})),t({}^{N(m_{r}+1)}_{r}(Q_{r}))]

By the construction of $m_{r}$ (see Notation 4.15) we know that

t({}_{r}^{N(m_{r}-4)}(Q_{r}))\geq t({}_{r}^{N}(P_{r}))\qquad\text{and}\qquad t({}_{r}^{N(m_{r}+1)}(Q_{r}))\geq t({}_{r}^{7N}(P_{r}))

Therefore we have the following inclusion

{}^{Nm_{r}}_{r}(C^{\prime}_{r})\subset D^{n}_{s}(2\cdot\mu^{-m_{r}})\times[t({}^{N}_{r}(P_{r})),t({}^{7}_{r}(P_{r}))]

For sufficiently small $r$ , Construction 5.1 yields a well-defined holonomy map

\operatorname{Hol}_{{}_{r}}:D^{n}_{s}(2\cdot\mu^{-m_{r}})\times[t({}^{N}_{r}(P_{r})),t({}^{7N}_{r}(P_{r}))]\to D^{n}_{s}(3)\times[-\delta,L+\delta]_{t}\times 0_{u}

Thus the point $z={}^{Nm_{r}}_{r}(w)$ has well-defined holononomy. By Lemma 5.6, we have

|\operatorname{Hol}_{{}_{r}}(z)-(r+t(z)-L)|\leq\mu^{-\kappa m_{r}}

In particular, this implies that for small $r$ , we have the following inequality.

(5.17)

t(\operatorname{Hol}_{{}_{r}}(z))\geq r+t({}^{N}_{r}(P_{r}))-L-\mu^{-\kappa m_{r}}=r(1-e^{-Nr})-\mu^{-\kappa m_{r}}>\frac{1}{2}r^{2}

Moreover, $\operatorname{Hol}_{{}_{r}}(z)$ lies on the unstable disk ${}^{Nm_{r}}(D_{w})$ through ${}_{r}^{Nm_{r}}(w)$ , since it contains all points in the unstable leaf of distance less than $l$ from ${}_{r}^{Nm_{r}}(w)$ . We thus acquire a point

x_{w}={}_{r}^{-Nm_{r}}(\operatorname{Hol}_{{}_{r}}(z))\quad\text{with}\quad{}^{Nm_{r}}_{r}(x_{w})\in\tilde{A}_{r}(Q)\cap\{u=0\}\text{ and }t({}^{Nm_{r}}_{r}(x_{w}))>\frac{1}{2}r^{2}

Note that $x_{w}$ varies continuously with $w$ .

Step 3. In this step we discuss the general case of (b). Fix a vertical disk $D\subset B_{r}(Q)$ as in (b), with $t({}^{-3N}(Q_{r}))<t(x)$ where $x=D\cap W^{\prime}$ . Let

\pi:U\to[-3,3]^{n}_{s}\times[-\epsilon,L+\epsilon]_{t}\times 0_{u}

be projection to the $(s,t)$ -plane and let $\Sigma\subset W^{\prime}$ be the union of all projections $\pi(D_{w})$ where $D_{w}$ intersects $\pi(D)$ . The disks $D$ and $D_{w}$ are $2\epsilon$ -Lipschitz graphs over $D^{n}_{u}(l\cdot\mu^{-m_{r}})$ (Lemma 3.21). Therefore these projections are all contained in balls of radius $2\epsilon\cdot l\cdot\mu^{-m_{r}}$ in $W^{\prime}$ (see also Step 2). It follows that there is a $C>0$ independent of $r$ such that

\operatorname{dist}(x,y)\leq C\cdot\mu^{-m_{r}}\qquad\text{for all }y\in\Sigma

This implies that $\Sigma\subset C^{\prime}_{r}$ for sufficiently small $r$ , and therefore that

D\subset V\qquad\text{where}\qquad V=\{z\in D_{w}\;:\;w\in C^{\prime}_{r}\}

since ^′ contains the $(s,t)$ -ball of radius $C\cdot\mu^{-m_{r}}$ around $x$ for small $r$ . We let $\pi^{\prime}:V\to C^{\prime}_{r}$ be the obvious projection mapping $z\in D_{w}$ to $w$ .

By Step 2, the projection $\pi^{\prime}:V\to C^{\prime}_{r}$ has a natural continuous section

\sigma:C^{\prime}_{r}\to V\qquad\text{given by}\qquad\sigma(w)=x_{w}

Since $D$ is vertical, it must necessarily intersect one point $x\in D$ in the image of $\sigma$ . By Step 2

t({}^{Nm_{r}}(x))>\frac{1}{2}r^{2}\qquad\text{and}\qquad{}^{Nm_{r}}(x)\in\tilde{A}_{r}(Q)\cap W^{\prime}

Note that $\tilde{A}_{r}(Q)\subset B_{r}(Q)$ for small $r$ . We let $\tilde{D}$ be the component of ${}_{r}^{Nm_{r}}(D)\cap B_{r}(Q)$ containing $x$ . This is a vertical disk containing $x$ , so by the usual considerations we have

\operatorname{dist}(x,z)<C\cdot\mu^{-m_{r}}\qquad\text{for all }z\in\tilde{D}

It follows that for sufficiently small $r$ , we have the lower bound

\operatorname{dist}(\tilde{D},W)\geq\operatorname{dist}(x,W)-C\mu^{-m_{r}}=t(x)-C\mu^{-m_{r}}>\frac{1}{2}r^{2}-C\mu^{-m_{r}}>r^{3}

This constructs the required disk and concludes the proof.∎

5.5. Proof Of Theorem 4.2

We are finally ready to conclude the proof of Theorem 4.2. We need a final lemma, demonstrating Theorem 4.2(c).

Lemma 5.18.

The intersections $W^{u}(P,{}_{r})\cap B_{r}(Q)$ contains a vertical disk $D_{Q}$ to the right of $W^{s}(Q,{}_{r})$ .

Proof.

We demonstrate this for $B_{r}(Q)$ . By Lemma 4.14, $W^{u}(P,{}_{r})$ contains a disk $D_{r}\subset U$ in $F^{u}({}_{r})$ centered at the homoclinic point

b_{r}=(1_{s},r,0_{u})\quad\text{of}\quad P

Let $c_{r}=(0_{s},r,0_{u})$ be the intersection point $F^{s}({}_{r},b_{r})\cap\Gamma$ and let $D^{\prime}_{r}$ be a disk in $F^{u}({}_{r})\cap U$ centered at $c_{r}$ . Note that we may take the radius of $D^{\prime}_{r}$ and $D_{r}$ to be lowerbounded by $A>0$ independent of $r$ . Let $l_{r}$ denote the unique integer such that

{}_{r}^{l_{r}}(c_{r})\in({}_{r}(P_{r}),{}_{r}^{2}(P_{r})]\quad\text{or equivalently}\quad t({}_{r}^{l_{r}}(c_{r}))\in(L-re^{-r},L-re^{-2r}]

As in Lemma 4.16, we know that $l_{r}>1/r$ if $r$ is small. Now note that ${}_{r}^{N}$ uniformly expands $F^{u}({}_{r})$ with constant of dilation (greater than) $\mu$ , for small $r$ . Therefore ${}_{r}^{l_{r}}(D^{\prime}_{r})$ contains the disk in $F^{u}({}_{r})$ of radius greater than $2\cdot l$ around ${}^{l_{r}}_{r}(c_{r})$ for small $r$ . In particular, by Lemma 4.14 and the definition of $l$ (Construction 4.17), there is a sub-disk

D^{\prime\prime}_{r}\subset{}^{l_{r}}(D^{\prime}_{r})\qquad\text{with a point $x=(1_{s},t(x),0_{u})$ with $t(x)\in[r(1-e^{-r}),r(1-e^{-2r})]$}

Thus the disk is to the right of $W^{s}_{\operatorname{loc}}(Q,{}_{r})\cap B_{r}(Q)$ . The disk must also be contained in a unstable disk fiber of $\tilde{B}_{r}(Q)$ (see Construction 4.17). Therefore every point in $D^{\prime\prime}_{r}$ is within distance $2\epsilon\cdot l\cdot\mu^{-m_{r}}$ of $x$ and so

t(y)\leq t(x)+2\epsilon\cdot l\cdot\mu^{-m_{r}}\leq r(1-e^{-3r})\leq 4r^{2}\qquad\text{for every $y\in D^{\prime\prime}_{r}$ and small $r$}

Similarly, $t(y)\geq r^{2}$ . Finally, since $b_{r}$ and $c_{r}$ are on the same stable disk in $U$ , we have

\operatorname{dist}({}^{m_{r}}_{r}(c_{r}),{}^{m_{r}}_{r}(b_{r}))\leq\mu^{-m_{r}}\leq\mu^{-1/r}

By taking a path $\gamma$ from ${}^{m_{r}}_{r}(c_{r})$ to $x$ in $F^{u}({}_{r})$ and using the holonomy map of $F^{u}$ (see Construction 5.1) we get a small unstable disk $D_{Q}$ in $F^{u}({}_{r})$ centered at a point $x_{Q}=\operatorname{Hol}_{{}_{r}}(x)$ with

\operatorname{dist}(x_{Q},x)\leq\mu^{-\kappa/r}

Here $\beta$ is the uniform Hölder constants in Construction 5.1. It follows, as with $D^{\prime\prime}$ , that $D_{Q}$ is a vertical disk to the right of $W^{s}_{\operatorname{loc}}(Q,{}_{r})\cap B_{r}(Q)$ .∎

Theorem 4.2 is an immediate consequence of Lemma 4.12 for Theorem 4.2(a), the Lemmas 5.3, 5.9, 5.10, 5.15, 5.16 and 5.17 for Theorem 4.2(b) and the Lemma 5.18 for Theorem 4.2(c).

6. Robustly Mixing Contactomorphisms

In the final section of this paper, we prove Theorem 11. The proof is a small modification of the proof of Theorem A of [7] in [7, Section 4.C].

Theorem 6.1 (Theorem 11).

Let $(Y,\xi)$ be a closed contact manifold admitting an Anosov Reeb flow with $C^{\infty}$ stable and unstable foliations. Then the $C^{1}$ -open set of robustly mixing contactomorphisms

\operatorname{Cont}_{\operatorname{RM}}(Y,\xi)\subset\operatorname{Cont}(Y,\xi)

is non-empty. Moreover, if $T$ is the period of a closed Reeb orbit of , then _T is in the closure of this set.

Proof.

Since is Reeb Anosov, it must be a transitive by the Plante alternative [32, Thm 8.1.3 and 8.1.4]. A transitive Anosov flow must also have a dense set of closed orbits [32, Thm 6.2.10].

Fix a closed orbit of period $T$ and an open neighborhood $U$ of . Also pick an auxilliary orbit with a neighborhood $V$ . Let $\alpha$ denote the contact form with Reeb vector-field generating the Reeb flow and choose a $1$ -parameter family of contact forms $\alpha_{s}$ on $Y$ such that

\alpha_{0}=\alpha\qquad\alpha_{s}|_{U}=\alpha\quad\text{and}\quad\alpha_{s}|_{V}=(1+s)\cdot\alpha

Let ^s denote the Reeb flow of $\alpha_{s}$ . Anosov flows are $C^{1}$ -structurally stable ([1] or [32, Thm 5.4.22]), and thus _s is a smooth Anosov Reeb flow for sufficiently small $s$ . Moreover, and are orbits of ^s for all $s$ .

Thus, choose an $s$ such that ^s is Reeb Anosov and such that has a period that is not a multiple of $T$ with respect to ^s. We let be the time $T$ map of ^s and note that

•

is a strict, partially hyperbolic contactomorphism with stable and unstable bundle equal to those of ^s and central bundle given by the span of the Reeb vector-field of $\alpha_{s}$ .
•

is a closed Reeb orbit that is a normally hyperbolic fixed set of .
•

The stable bundle $E^{s}(\Phi)$ of _T is a smooth, integrable and uniformly contracted by on $U$ since agrees with _T on $U$ .

Note also the local stable manifolds of and for the set (or for any point $P\in\Gamma$ ) agree.

W^{s}_{\operatorname{loc}}(\Gamma,\Psi;U)=W^{s}_{\operatorname{loc}}(\Gamma,{}_{T};U)\qquad\text{and}\qquad W^{s}_{\operatorname{loc}}(P,\Psi;U)=W^{s}_{\operatorname{loc}}(P,{}_{T};U)

Since the points $P$ of are fixed, the stable manifold of $P$ is equal to the stable leaf of $P$ with respect to both and _T. It follows that

F^{s}(\Psi,P)=F^{s}(\Phi,P)\qquad\text{ on }W_{\operatorname{loc}}^{s}(\Gamma,\Psi;U)

This verifies the criteria in Theorem 4.2(a-c). To check Theorem 4.2(d) we apply the following lemma of Bonatti-Diaz.

Lemma 6.2.

[7, Lemma 4.3] Let be a partially hyprbolic map and , be closed orbits of different period, as constructed above. Then there are a pair of points $P,Q\in\Gamma$ and a heteroclinic orbit from $P$ to $Q$ .

The argument now proceeds identically to [7, p. 395] and we recall it here. Apply Theorem 4.2 to acquire a $1$ -parameter family of contactomorphisms _r and an $N>0$ such that

•

_r has hyperbolic fixed points $P$ and $Q$ on of index $n+1$ and $n$ , respectively.
•

There is a neighborhood $B_{r}(Q)$ of $Q$ such that $(B_{r}(Q),{}_{r}^{N})$ is a stable blender.
•

The intersection $W^{u}(P,{}_{r})\cap B_{r}(Q)$ contains a vertical disk $D_{Q}$ to the right of $W^{s}(Q,{}_{r})$ .

This implies that the tuple $((B_{r}(Q),{}_{r}^{N}),P)$ is a chain of blenders in the sense of Bonatti-Diaz (see [7, p. 369] or [7, §7.1]). Lemma 1.12 of [7] now states that there is a $C^{1}$ -neighborhood $\mathcal{U}$ of _r such that, for any ${}^{\prime}\in\mathcal{U}$ , there are fixed points $P^{\prime}$ and $Q^{\prime}$ (the continuations of $P$ and $Q$ , which are non-degenerate hyperbolic fixed points and thus persist in a $C^{1}$ -nieghborhood) such that

(6.1)

W^{s}(P^{\prime},{}^{\prime})\subset\operatorname{close}(W^{s}(Q^{\prime},{}^{\prime}))

Here $\operatorname{close}(-)$ denotes the topological closure.

Next, since ^s is Anosov and transitive, the time $T$ map is partially hyperbolic with well-defined center-stable and center-unstable foliations

F^{cs}(\Psi)\text{ tangent to }\operatorname{span}(R)\oplus E^{s}(\Psi)\qquad\text{and}\qquad F^{cu}(\Psi)\text{ tangent to }\operatorname{span}(R)\oplus E^{u}(\Psi)

Moreover, the center-stable and center-unstable foliations have dense leaves [32, Thm 6.2.10]. By Hirsch-Pugh-Robinson [39], these properties are $C^{1}$ -robust. Thus ^′ is partially hyperbolic with invariant foliations

F^{cs}({}^{\prime})\text{ tangent to }E^{c}({}^{\prime})\oplus E^{s}({}^{\prime})\qquad\text{and}\qquad F^{cu}({}^{\prime})\text{ tangent to }E^{c}({}^{\prime})\oplus E^{u}({}^{\prime})

with dense leaves. Now note that we have the following identifications.

F^{cs}({}^{\prime},P^{\prime})=W^{s}(P^{\prime},{}^{\prime})\qquad\text{and}\qquad F^{cu}({}^{\prime},Q^{\prime})=W^{u}(Q^{\prime},{}^{\prime})

Indeed, the uniqueness of local invariant manifolds near hyperbolic invariant sets [39, Thm 4.1(b)] implies equality near $P^{\prime}$ and $Q^{\prime}$ , and then global invariance implies global equality. In particular, $W^{s}(P^{\prime},{}^{\prime})$ and $W^{u}(Q^{\prime},{}^{\prime})$ are dense. Moreover, by (6.1) $W^{s}(Q^{\prime},{}^{\prime})$ is also dense.

Now [10, Lem 7.3] states that ^′ is robustly transitive (and in fact mixing). We recall the argument. Let $U$ and $V$ be neighborhoods in $Y$ . Then

U\cap W^{s}(Q^{\prime},{}^{\prime})\neq\emptyset\qquad\text{and}\qquad V\cap W^{u}(Q^{\prime},{}^{\prime})\neq\emptyset

since these invariant manifolds are dense. This implies that $({}^{\prime})^{j}(U)\cap({}^{\prime})^{-k}(V)$ is non-empty for all sufficiently large $j$ and $k$ . In other words, ^′ is topologically mixing. ∎

References

[1] Dmitry Victorovich Anosov. Ergodic properties of geodesic flows on closed riemannian manifolds of negative curvature. In Doklady Akademii Nauk, volume 151, pages 1250–1252. Russian Academy of Sciences, 1963.
[2] Russell Avdek. An algebraic generalization of giroux’s criterion. arXiv preprint arXiv:2307.09068, 2023.
[3] Russell Avdek and Zhengyi Zhou. Bourgeois’ contact manifolds are tight. arXiv preprint arXiv:2404.16311, 2024.
[4] John A Baldwin and Steven Sivek. A contact invariant in sutured monopole homology. In Forum of Mathematics, Sigma, volume 4, page e12. Cambridge University Press, 2016.
[5] Mélanie Bertelson and Gael Meigniez. Conformal symplectic structures, foliations and contact structures. arXiv preprint arXiv:2107.08839, 2021.
[6] Christian Bonatti, Sylvain Crovisier, Lorenzo Diaz, and Amie Wilkinson. What is… a blender? arXiv preprint arXiv:1608.02848, 2016.
[7] Christian Bonatti and Lorenzo J. Diaz. Persistent nonhyperbolic transitive diffeomorphisms. Annals of Mathematics, 143(2):357–396, 1996.
[8] Christian Bonatti and Lorenzo J Díaz. Robust heterodimensional cycles and-generic dynamics. Journal of the Institute of Mathematics of Jussieu, 7(3):469–525, 2008.
[9] Christian Bonatti, Lorenzo J Díaz, and Enrique R Pujals. A c1-generic dichotomy for diffeomorphisms: weak forms of hyperbolicity or infinitely many sinks or sources. Annals of Mathematics, pages 355–418, 2003.
[10] Christian Bonatti, Lorenzo J Díaz, and Marcelo Viana. Dynamics beyond uniform hyperbolicity: A global geometric and probabilistic perspective, volume 3. Springer Science & Business Media, 2004.
[11] Armand Borel. Compact clifford-klein forms of symmetric spaces. Topology, 2(1-2):111–122, 1963.
[12] Joseph Breen. Morse-smale characteristic foliations and convexity in contact manifolds. Proceedings of the American Mathematical Society, 149(9):3977–3989, 2021.
[13] Joseph Breen. Folded symplectic forms in contact topology. Journal of Geometry and Physics, 201:105213, 2024.
[14] Joseph Breen, Ko Honda, and Yang Huang. The giroux correspondence in arbitrary dimensions. arXiv preprint arXiv:2307.02317, 2023.
[15] Dmitri Burago and Sergei Ivanov. Partially hyperbolic diffeomorphisms of 3-manifolds with abelian fundamental groups. J. Mod. Dyn, 2(4):541–580, 2008.
[16] A. Candel and L. Conlon. Foliations. Number v. 2 in Foliations. Mathematical Society of Japan, 1985.
[17] Robert Cardona. Stability is not open or generic in symplectic four-manifolds. arXiv preprint arXiv:2305.13158, 2023.
[18] Julian Chaidez and Shira Tanny. Elementary sft spectral gaps and the strong closing property. arXiv preprint arXiv:2312.17211, 2023.
[19] Rima Chatterjee, John B Etnyre, Hyunki Min, and Anubhav Mukherjee. Existence and construction of non-loose knots. arXiv preprint arXiv:2310.04908, 2023.
[20] Kai Cieliebak and Evgeny Volkov. First steps in stable hamiltonian topology. Journal of the European Mathematical Society, 17(2):321–404, 2015.
[21] Vincent Colin, Paolo Ghiggini, Ko Honda, and Michael Hutchings. Sutures and contact homology i. Geometry & Topology, 15(3):1749–1842, 2011.
[22] Vincent Colin, Emmanuel Giroux, and Ko Honda. On the coarse classification of tight contact structures. arXiv preprint math/0305186, 2003.
[23] Vincent Colin, Emmanuel Giroux, and Ko Honda. Finitude homotopique et isotopique des structures de contact tendues. Publications mathématiques, 109:245–293, 2009.
[24] James Conway and Hyunki Min. Classification of tight contact structures on surgeries on the figure-eight knot. Geometry & Topology, 24(3):1457–1517, 2020.
[25] Sylvain Crovisier and Rafael Potrie. Introduction to partially hyperbolic dynamics. School on Dynamical Systems, ICTP, Trieste, 3(1), 2015.
[26] Yakov Eliashberg and Dishant Pancholi. Honda-huang’s work on contact convexity revisited. arXiv preprint arXiv:2207.07185, 2022.
[27] John B Etnyre and Ko Honda. On the nonexistence of tight contact structures. Annals of Mathematics, pages 749–766, 2001.
[28] John B Etnyre and Ko Honda. Tight contact structures with no symplectic fillings. Inventiones Mathematicae, 148(3):609, 2002.
[29] John B Etnyre, Hyunki Min, and Anubhav Mukherjee. Non-loose torus knots. arXiv preprint arXiv:2206.14848, 2022.
[30] François-Simon Fauteux-Chapleau and Joseph Helfer. Exotic tight contact structures on ⁿ. arXiv preprint arXiv:2111.07590, 2021.
[31] Eduardo Fernández and Hyunki Min. Spaces of legendrian cables and seifert fibered links. arXiv preprint arXiv:2310.12385, 2023.
[32] Todd Fisher and Boris Hasselblatt. Hyperbolic flows. European Mathematical Society, 2019.
[33] Hansjörg Geiges. An introduction to contact topology, volume 109. Cambridge University Press, 2008.
[34] Paolo Ghiggini, Paolo Lisca, András I Stipsicz, et al. Classification of tight contact structures on small seifert 3-manifolds with $e_{0}\geq 0$ . In Proc. Amer. Math. Soc, volume 134, pages 909–916, 2006.
[35] Emmanuel Giroux. Convexity in contact topology. Comment. Math. Helv, 66(4):637–677, 1991.
[36] Michael Gromov and Ilya Piatetski-Shapiro. Non-arithmetic groups in lobachevsky spaces. Publications Mathématiques de l’IHÉS, 66:93–103, 1987.
[37] Victor Guillemin, Ana Cannas da Silva, and Christopher Woodward. On the unfolding of folded symplectic structures. Mathematical Research Letters, 7(1):35–53, 2000.
[38] F Rodriguez Hertz, J Rodriguez Hertz, and Raúl Ures. Partially hyperbolic dynamics. Publicaçoes Matemáticas do IMPA, 2011.
[39] M.W. Hirsch, C.C. Pugh, and M. Shub. Invariant Manifolds. Lecture Notes in Mathematics. Springer Berlin Heidelberg, 2006.
[40] Ko Honda. On the classification of tight contact structures i. arXiv preprint math/9910127, 1999.
[41] Ko Honda. On the classification of tight contact structures ii. Journal of Differential Geometry, 55(1):83–143, 2000.
[42] Ko Honda and Yang Huang. Bypass attachments in higher-dimensional contact topology. arXiv preprint arXiv:1803.09142, 2018.
[43] Ko Honda and Yang Huang. Convex hypersurface theory in contact topology. arXiv preprint arXiv:1907.06025, 2019.
[44] Ko Honda, William H Kazez, and Gordana Matić. Convex decomposition theory. International Mathematics Research Notices, 2002(2):55–88, 2002.
[45] Ko Honda, William H Kazez, and Gordana Matić. Tight contact structures on fibred. Journal of Differential Geometry, 64(2):305–358, 2003.
[46] Ko Honda, William H Kazez, and Gordana Matic. Contact structures, sutured floer homology and tqft. arXiv preprint arXiv:0807.2431, 2008.
[47] Dongchen Li. Blender-producing mechanisms and a dichotomy for local dynamics for heterodimensional cycles. arXiv preprint arXiv:2403.15818, 2024.
[48] John J Millson. On the first betti number of a constant negatively curved manifold. Annals of Mathematics, 104(2):235–247, 1976.
[49] Hyunki Min and Isacco Nonino. Tight contact structures on a family of hyperbolic l-spaces. arXiv preprint arXiv:2311.14103, 2023.
[50] Atsuhide Mori. Reeb foliations on s^ 5 and contact 5-manifolds violating the thurston-bennequin inequality. arXiv preprint arXiv:0906.3237, 2009.
[51] Atsuhide Mori. On the violation of thurston-bennequin inequality for a certain non-convex hypersurface. arXiv preprint arXiv:1111.0383, 2011.
[52] Emmy Murphy. Loose Legendrian embeddings in high dimensional contact manifolds. Stanford University, 2012.
[53] Meysam Nassiri and Enrique R Pujals. Robust transitivity in hamiltonian dynamics. In Annales scientifiques de l’École normale supérieure, volume 45, pages 191–239, 2012.
[54] Workshop Participants. Aim problem list in higher dimensional contact topology, 2024.
[55] Charles Pugh, Michael Shub, and Amie Wilkinson. Hölder foliations. Duke Math. J., 90(1):517–546, 1997.
[56] Dennis Sullivan. Hyperbolic geometry and homeomorphisms. In Geometric topology, pages 543–555. Elsevier, 1979.
[57] Ichiro Torisu. Convex contact structures and fibered links in 3-manifolds. International Mathematics Research Notices, 2000(9):441–454, 2000.
[58] Alan Weinstein. Symplectic manifolds and their lagrangian submanifolds. Advances in Mathematics, 6(3):329–346, 1971.
[59] Chris Wendl. Lectures on symplectic field theory. arXiv preprint arXiv:1612.01009, 2016.
[60] Kanda Yutaka. The classification of tight contact structures on the 3-torus. Communications in Analysis and Geometry, 5(3):413–438, 1997.

Robustly Non-Convex Hypersurfaces In Contact Manifolds

Abstract.

1. Introduction

Theorem 1 (Giroux).

Theorem 2 (Honda-Huang).

Question 3.

Remark 4.

Theorem 5 (Main Theorem).

1.1. Characteristic Foliations

Definition 6 (Dividing Set).

Definition 7 (Topological Mixing).

Lemma 8.

Proof.

Definition 9.

1.2. Suspension

Theorem 10 (Proposition 2.17).

1.3. Robustly Mixing Contactomorphisms And Blenders

Theorem 11 (Theorem 6.1).

Example 12.

Remark 1.1.

1.4. Non-Convex Hypersurfaces

Theorem 13.

Theorem 14.

Theorem 15.

Lemma 16 (Lemma 2.20).

Theorem 17.

Proof.

1.5. Open Problems

Definition 18 (Robust Non-Convexity).

Question 19.

Theorem 20.

Question 21.

Question 22.

Remark 23.

Outline

Acknowledgements

2. Contact Hamiltonian Manifolds

2.1. Fundamentals

Definition 2.1.

Example 2.2 (Product).

Definition 2.3.

Lemma 2.4.

Proof.

Definition 2.5 (Hamiltonian Vector-fields).

2.2. Contact Hamiltonian Hypersurfaces

Definition 2.6.

Lemma 2.7.

Proof.

Definition 2.8.

Lemma 2.9.

Proof.

Definition 2.10 (Deformation).

Lemma 2.11 (Collar Neighborhood).

Proof.

2.3. Suspensions

Definition 2.12 (Contact Suspension).

Remark 2.13.

Lemma 2.14 (Hamiltonian Perturbation).

Proof.

Example 2.15 (Mapping Torus Of Identity).

Lemma 2.16 (Surface Perturbation).

Proof.

Proposition 2.17 (Theorem 10).

Proof.

2.4. Constructions Of Contact-Hamiltonian Hypersurfaces

Lemma 2.18 (Disk Neighborhood).

Proof.

Lemma 2.19.

Proof.

Lemma 2.20 (Lemma 16).

Proof.

3. Partial Hyperbolicity

Remark 3.1.

3.1. Fundamentals

Definition 3.2 (Expansion/Contraction).

Definition 3.3 (Domination).

Theorem 3.4 (C1C^{1}-Persistence).

Theorem 3.5 (Foliations).

Theorem 3.6 (Invariant Manifolds).

Notation 3.7 (Local Invariant Manifolds).

Theorem 3.4 ( $C^{1}$ -Persistence).

Construction 4.6 (Hamiltonian $H$ ).

Construction 4.7 (Hamiltonian $G$ ).

Definition 5.2 (Blender Set $A$ ).

Definition 5.5 (Blender Set $A^{\prime}$ ).

Lemma 5.8 (Bounds On $\tilde{A}$ ).