Abundance of $C^{1}$ -robust homoclinic tangencies

Christian Bonatti and Lorenzo J. Díaz This paper was partially supported by CNPq, Faperj, and PRONEX (Brazil) and the Agreement in Mathematics Brazil-France. We acknowledge the warm hospitality of I.M.P.A, Institute de Mathématiques de Bourgogne, and PUC-Rio during the stays while preparing this paper

Abstract

A diffeomorphism $f$ has a $C^{1}$ -robust homoclinic tangency if there is a $C^{1}$ -neighbourhood ${\cal U}$ of $f$ such that every diffeomorphism in $g\in{\cal U}$ has a hyperbolic set $\Lambda_{g}$ , depending continuously on $g$ , such that the stable and unstable manifolds of $\Lambda_{g}$ have some non-transverse intersection. For every manifold of dimension greater than or equal to three, we exhibit a local mechanism (blender-horseshoes) generating diffeomorphisms with $C^{1}$ -robust homoclinic tangencies.

Using blender-horseshoes, we prove that homoclinic classes of $C^{1}$ -generic diffeomorphisms containing saddles with different indices and that do not admit dominated splittings (of appropriate dimensions) display $C^{1}$ -robust homoclinic tangencies.

keywords: chain recurrence set, dominated splitting, heterodimensional cycle, homoclinic class, homoclinic tangency, hyperbolic set.

MSC 2000: 37C05, 37C20, 37C25, 37C29, 37C70.

To Carlos Gutierrez (1944 -- 2008), in memoriam

1 Introduction

1.1 Framework and general setting

A homoclinic tangency is a dynamical mechanism which is at the heart of a great variety of non-hyperbolic phenomena: persistent coexistence of infinitely many sinks [22], Hénon-like strange attractors [5, 20], super-exponential growth of the number of periodic points [19], and non-existence of symbolic extensions [15], among others. Moreover, homoclinic bifurcations (homoclinic tangencies and heterodimensional cycles) are conjectured to be the main source of non-hyperbolic dynamics (Palis denseness conjecture, see [23]).

In this paper, we present a local mechanism generating $C^{1}$ -robust homoclinic tangencies. Using this construction, we show that the occurrence of robust tangencies is a quite general phenomenon in the non-hyperbolic setting, specially when the dynamics does not admit a suitable dominated splitting.

Let us now give some basic definitions (in Section 2, we will state precisely the definitions involved in this paper). A transitive hyperbolic set $\Lambda$ has a homoclinic tangency if there is a pair of points $x,y\in\Lambda$ such that the stable leaf $W^{\operatorname{s}}(x)$ of $x$ and the unstable leaf $W^{\operatorname{u}}(y)$ of $y$ have some non-transverse intersection

Given a hyperbolic set $\Lambda$ of a diffeomorphism $f$ , for $g$ close to $f$ , we denote by $\Lambda_{g}$ the hyperbolic set of $g$ which is the continuation of $\Lambda$ (i.e., $\Lambda_{g}$ is close to $\Lambda$ and the dynamics of $f$ on $\Lambda$ and $g$ on $\Lambda_{g}$ are conjugate).

Definition 1.1 (Robust cycles).

•

Robust homoclinic tangencies: A transitive hyperbolic set $\Lambda$ of a $C^{r}$ -diffeomorphism $f$ has a $C^{r}$ -robust homoclinic tangency if there is a $C^{r}$ -neighborhood ${\cal N}$ of $f$ such that for every $g\in{\cal N}$ the continuation $\Lambda_{g}$ of $\Lambda$ for $g$ has a homoclinic tangency.
•

Robust heterodimensional cycles: A diffeomorphism $f$ has a $C^{r}$ -robust heterodimensional cycle if there are transitive hyperbolic sets $\Lambda$ and $\Sigma$ of $f$ whose stable bundles have different dimensions and a $C^{r}$ -neighborhood ${\cal V}$ of $f$ such that $W^{\operatorname{s}}(\Lambda_{g})\cap W^{\operatorname{u}}(\Sigma_{g})\neq\emptyset$ and $W^{\operatorname{u}}(\Lambda_{g})\cap W^{\operatorname{s}}(\Sigma_{g})\neq\emptyset$ , for every diffeomorphism $g\in{\cal V}$ .

Note that, by Kupka-Smale theorem, $C^{r}$ -generically, invariant manifolds of periodic points are in general position. Hence, generically, the non-transverse intersections in a robust cycle (tangency or heterodimensional cycle) involve non-periodic points (i.e., at least a non-trivial hyperbolic set).

In [21], Newhouse constructed surface diffeomorphisms having hyperbolic sets (called thick horseshoes) exhibiting $C^{2}$ -robust homoclinic tangencies. Later, he proved that, in dimension two, homoclinic tangencies of $C^{2}$ -diffeomorphisms yield thick horseshoes with $C^{2}$ -robust homoclinic tangencies, [22] (see also [24] for a broad discussion of homoclinic bifurcations on surfaces). With the same $C^{2}$ -regularity assumption, theorems in [27, 25] extend Newhouse result, proving that homoclinic tangencies in any dimension lead to $C^{2}$ -robust homoclinic tangencies. In this paper, we study the occurrence of robust homoclinic tangencies in the $C^{1}$ -setting.

Newhouse construction (thick horseshoes with robust tangencies) involves distortion estimates which are typically $C^{2}$ . The results in [30] present some obstacles for carrying this construction to the $C^{1}$ -topology: $C^{1}$ -generic surface diffeomorphisms do not have thick horseshoes. Recent results by Moreira in [17] are a strong indication that there are no surface diffeomorphisms exhibiting $C^{1}$ -robust homoclinic tangencies¹¹1This question is closely related to the open problem of $C^{1}$ -density of hyperbolic diffeomorphisms on compact surfaces (Smale’s density conjecture). In fact, Moreira’s result imply that there are no $C^{1}$ -robust homoclinic tangencies associated to hyperbolic basic sets of surface diffeomorphisms. See [1] for a discussion of the current state of this conjecture..

Nevertheless, in higher dimensions, there are examples of diffeomorphisms having hyperbolic sets with $C^{1}$ -robust tangencies. For instance, the product of a non-trivial hyperbolic attractor by a normal expansion gives a hyperbolic set $\Lambda$ of saddle type, whose stable manifold has a topological dimension greater than the dimension of its stable bundle. Then the set $\Lambda$ can play the role of thick horseshoes in Newhouse construction. Geometrical constructions using these kind of “thick” hyperbolic sets provide examples of systems with $C^{1}$ -robust heterodimensional cycles²²2A heterodimensional cycle is a cycle associated to saddles having different indices. (see [3]) or $C^{1}$ -robust tangencies (see [28, 4]). But these constructions involve quite specific global dynamical configurations, thus they cannot translate to a general setting.

1.2 Robust homoclinic tangencies

The aim of this paper is to show that the existence of $C^{1}$ -robust homoclinic tangencies is a common phenomenon in the non-hyperbolic setting. For instance, next result is a consequence of the local mechanism for robust tangencies in Theorem 2.

Theorem 1.

Let $M$ be a compact manifold with $\dim(M)\geq 3$ . There is a residual subset ${\cal R}$ of $\mbox{{\rm Diff\,}}^{1}(M)$ such that, for every $f\in{\cal R}$ and every periodic saddle $P$ of $f$ such that

•

(index variability) the homoclinic class $H(P,f)$ of $P$ has a periodic saddle $Q$ with $\dim(E^{\operatorname{s}}(Q))\neq\dim(E^{\operatorname{s}}(P))$ ,
•

(non-domination) the stable/unstable splitting $E^{\operatorname{s}}(R)\oplus E^{\operatorname{u}}(R)$ over the set of saddles $R$ homoclinically related with $P$ is not dominated,

then the saddle $P$ belongs to a transitive hyperbolic set having a $C^{1}$ -robust homoclinic tangency.

For the precise definitions of homoclinic class and dominated splitting see Definitions 2.1 and 2.4. Let us reformulate Theorem 1 by focusing on the homoclinic class of a prescribed periodic orbit:

Corollary 1.

Let $M$ be a compact manifold with $\dim(M)\geq 3$ . Consider a diffeomorphism $f$ with a saddle $P_{f}$ whose continuation $P_{g}$ is defined for all $g$ in a neighborhood ${\cal U}$ of $f$ in $\mbox{{\rm Diff\,}}^{1}(M)$ . Assume that

•

(generic index variability) there is a residual subset ${\cal G}$ of ${\cal U}$ such that, for every $g\in{\cal G}$ , the homoclinic class of $P_{g}$ of $f$ contains a saddle $Q$ of different index,
•

(robust non-domination) for every $g\in{\cal U}$ , the stable/unstable splitting $E^{\operatorname{s}}(R)\oplus E^{\operatorname{u}}(R)$ over the set of saddles $R$ homoclinically related with $P_{g}$ is not dominated.

Then there is an open and dense subset ${\cal C}$ of ${\cal U}$ of diffeomorphisms $g$ such that the saddle $P_{g}$ belongs to a transitive hyperbolic set with a $C^{1}$ -robust homoclinic tangency.

Remark 1.2.

The diffeomorphisms $f$ in the residual subset ${\cal R}$ of $\mbox{{\rm Diff\,}}^{1}(M)$ in Theorem 1 satisfy the following properties (see [2, Section 2.1] and [13, Appendix B.1.1]):

•

Every homoclinic class $H(P_{f},f)$ of $f$ depends continuously on $f\in{\cal R}$ . Therefore, if $H(P_{f},f)$ has a dominated splitting then $H(P_{g},g)$ also has a dominated splitting whose bundles have constant dimension for all $g\in{\cal R}$ close to $f$ .
•

Assume that a homoclinic class $H(P_{f},f)$ of $f\in{\cal R}$ contains saddles of stable indices $j$ and $k$ , $j\neq k$ . Then the homoclinic class $H(P_{g},g)$ also contains saddles of stable indices $j$ and $k$ for every $g\in{\cal R}$ close to $f$ .

In other words, the conditions in Theorem 1 are $C^{1}$ -open in the residual set ${\cal R}$ of $\mbox{{\rm Diff\,}}^{1}(M)$ .

The index interval of a homoclinic class $H$ is the interval $[i,j]$ , where $i$ and $j$ are the minimum and the maximum of the ${\operatorname{s}}$ -indices (dimension of the stable bundle) of the periodic points in $H$ . The homoclinic class $H$ has index variation if $i<j$ . Given a transitive hyperbolic set $\Lambda$ its ${\operatorname{s}}$ -index is the dimension of its stable bundle.

Corollary 2.

For every diffeomorphism $f$ in the residual subset ${\cal R}$ of $\mbox{{\rm Diff\,}}^{1}(M)$ , any homoclinic class $H$ of $f$ with index variation, and every $k\in[i,j]$ , where $[i,j]$ is the index interval of $H$ , one has:

•

either there is a dominated splitting $E\oplus_{{}_{<}}F$ (i.e., $F$ dominates $E$ ) with $\dim(E)=k$ ,
•

or there is a hyperbolic transitive set $\Lambda\subset H$ with ${\operatorname{s}}$ -index $k$ having a $C^{1}$ -robust homoclinic tangency.

When we are interested only in the existence of robust homoclinic tangencies, without paying attention to the index of the hyperbolic set involved in their generation, there is the following reformulation:

Corollary 3.

There is a residual subset ${\cal G}$ of $\,\mbox{{\rm Diff\,}}^{1}(M)$ such that for every diffeomorphism $f\in{\cal G}$ and every homoclinic class $H(P,f)$ of $f$ with index interval $[i,j]$ , $j>i$ ,

•

either $H(P,f)$ has a dominated splitting

$T_{H(P,f)}M=E^{\operatorname{cs}}\oplus_{{}_{<}}E_{1}\oplus_{{}_{<}}\cdots\oplus_{{}_{<}}E_{j}\oplus_{{}_{<}}E^{\operatorname{cu}},$

where $\dim(E^{\operatorname{cs}})=i$ and $E_{1},\dots,E_{j}$ are one-dimensional,
•

or the homoclinic class $H(P,f)$ contains a transitive hyperbolic set with a robust homoclinic tangency.

In the first case of Corollary 3, we say that $H(P,f)$ has an indices adapted dominated splitting.

The previous results have an interesting formulation for tame diffeomorphisms, i.e., the $C^{1}$ -open set ${\cal T}(M)$ of $\mbox{{\rm Diff\,}}^{1}(M)$ of diffeomorphisms having finitely many chain recurrence classes (see Definition 2.3) in a robust way. We define ${\cal W}(M)\stackrel{{\scriptstyle\scriptscriptstyle\rm def}}{{=}}\mbox{{\rm Diff\,}}^{1}(M)\setminus\overline{{\cal T}(M)}$ as the set of wild diffeomorphisms. Let us observe that, for an open and dense subset of ${\cal T}(M)$ , a chain recurrence class is either hyperbolic or has index variation, see [2].

Given a chain recurrence class $C$ of $f$ we first consider the finest dominated splitting over $C$ (i.e., the bundles of this splitting can not be decomposed in a dominated way). Then we let $E^{\operatorname{s}}$ (resp. $E^{\operatorname{u}}$ ) be the sum of the uniformly contracting (resp. expanding) bundles of this splitting (these bundles may be trivial, see [14]). The bundles $E_{1},\dots,E_{k}$ are the remaining non-hyperbolic bundles of the finest dominated splitting of $C$ . In this way, we get a dominated splitting over $C$

T_{C}M=E^{\operatorname{s}}\oplus_{{}_{<}}E_{1}\oplus_{{}_{<}}\cdots\oplus_{{}_{<}}E_{k}\oplus_{{}_{<}}E^{\operatorname{u}},

where $E^{\operatorname{s}}$ and $E^{\operatorname{u}}$ are uniformly contracting and expanding, and $E_{1},\dots,E_{k}$ are indecomposable and non-hyperbolic. We call this splitting the finest central dominated splitting of the chain recurrence $C$ .

Remark 1.3.

Let $f$ be any tame diffeomorphisms and $H(P,f)$ any homoclinic class of $f$ which is far from robust homoclinic tangencies. Then the finest central dominated splitting of $H(P,f)$ is indices adapted. For tame diffeomorphisms, the corollary below gives a more precise description of the relation between the finest central dominated splitting and the robust homoclinic tangencies associated to a homoclinic class.

Corollary 4.

There is a $C^{1}$ -open and dense subset ${\cal O}$ of the set ${\cal T}(M)$ of tame diffeomorphisms such that, for every $f$ in ${\cal O}$ and every chain recurrence class $C$ of $f$ whose finest central dominated splitting is

T_{C}M=E^{\operatorname{s}}\oplus_{{}_{<}}E_{1}\oplus_{{}_{<}}\cdots\oplus_{{}_{<}}E_{k}\oplus_{{}_{<}}E^{\operatorname{u}},

then, for every $i=1,\dots,k$ ,

\dim(E_{i})>1\Longleftrightarrow\left\{\begin{array}[]{ll}&\mbox{for all $j\in\{1,\dots,\dim(E_{i})-1\}$,}\\ &\mbox{there is a transitive hyperbolic set $K$ of ${\operatorname{s}}$-index}\\ &{\mbox{\rm{ind}}^{\,{\operatorname{s}}}}(K)=\dim\left(E^{\operatorname{s}}\oplus E_{1}\oplus\cdots\oplus E_{i-1}\right)+j\\ &\mbox{having a $C^{1}$-robust homoclinic tangency.}\end{array}\right.

Remark 1.4.

1.

Under the hypotheses of Corollary 4, [9, Theorem 1.14] implies that (choosing appropriately the open and dense subset ${\cal O}$ of ${\cal T}(M)$ ) the hyperbolic set $K$ with a $C^{1}$ -robust homoclinic tangency is also involved in a $C^{1}$ -robust heterodimensional cycle.
2.

Corollary 4 can also be stated for isolated chain recurrence classes of $C^{1}$ -generic diffeomorphisms ³³3By $C^{1}$ -generic diffeomorphisms we mean diffeomorphisms in a residual subset of $\mbox{{\rm Diff\,}}^{1}(M)$ ..

This article proceed a program for studying the generation of robust cycles (homoclinic tangencies and heterodimensional cycles) in the $C^{1}$ -topology. In [9] we proved that homoclinic classes containing periodic points with different indices generate (by arbitrarily small $C^{1}$ -perturbations) $C^{1}$ -robust heterodimensional cycles. Here we show that these robust heterodimensional cycles generate blender-horseshoes, a sort of hyperbolic basic sets with geometrical properties resembling the thick horseshoes, see Section 3.2 and Theorem 6.4. We next see that, in the context of critical dynamics (some suitable non-domination property), blender-horseshoes yield $C^{1}$ -robust tangencies, see Theorem 2. In fact, the definition and construction of blender-horseshoes (a special class of ${\operatorname{cu}}$ -blenders defined in [8]) and Theorem 2 are the technical heart of our arguments and the main novelty of this paper.

The results in this paper and the ones in [9] support the following conjecture:

Conjecture 1 (Bonatti, [6]).

Every $C^{1}$ -diffeomorphism can be $C^{1}$ -approximated either by a hyperbolic diffeomorphism (Axiom A and no-cycle property) or by a diffeomorphism exhibiting a $C^{1}$ -robust cycle (homoclinic tangency or heterodimensional cycle).

This conjecture is a stronger version of the denseness conjecture by Palis in [23] (dichotomy hyperbolicity versus approximation by diffeomorphisms with homoclinic bifurcations). The novelty here is that the conjecture considers two disjoint open sets whose union is dense in the whole set of $C^{1}$ -diffeomorphisms: the hyperbolic ones and those with robust cycles. In the setting of tame diffeomorphisms, a strong version of Conjecture 1 was proved in [9, Theorem 1.14]: every tame diffeomorphism can be $C^{1}$ -approximated either by hyperbolic diffeomorphisms or by diffeomorphisms exhibiting robust heterodimensional cycles. Recall that Palis conjecture for surface $C^{1}$ -diffeomorphisms was proved in [26] (due to dimension deficiency, for surface diffeomorphisms the conjecture only involves homoclinic tangencies).

1.3 Newhouse domains

Following [19], we say that an open set ${\cal N}$ of $\mbox{{\rm Diff\,}}^{r}(M)$ is a $C^{r}$ -Newhouse domain if there is a dense subset ${\cal D}$ of ${\cal N}$ such that every $g\in{\cal D}$ has a homoclinic tangency (associated to some saddle). A preliminary step toward Conjecture 1 is the following question.

Question 1.

Let $M$ be a closed manifold and ${\cal N}$ be a $C^{1}$ -Newhouse domain of $\mbox{{\rm Diff\,}}^{1}(M)$ . Are the diffeomorphisms having $C^{1}$ -robust homoclinic tangencies dense in ${\cal N}$ ?

If it is not possible to answer positively this question in its full generality, it would be interesting to provide sufficient conditions for a $C^{1}$ -Newhouse domain to contain an open and dense subset of diffeomorphisms with $C^{1}$ -robust homoclinic tangencies. If the dimension of the ambient manifold is at least three, one may also ask about the interplay between robust homoclinic tangencies and robust heterodimensional cycles.

We now discuss briefly Question 1. Before going to our setting, lets us review the discussion in [1] about this question for $C^{1}$ -surface diffeomorphisms. Let $\textrm{Hyp}^{1}(M)$ denote the subset of $\mbox{{\rm Diff\,}}^{1}(M)$ consisting of Axiom A diffeomorphisms. By [26], for surface diffeomorphisms, the open set

{\cal N}^{1}(M^{2})\stackrel{{\scriptstyle\scriptscriptstyle\rm def}}{{=}}\textrm{Diff}^{1}(M^{2})\setminus\overline{\textrm{Hyp}^{1}(M^{2})}

is a Newhouse domain. The set ${\cal N}^{1}(M^{2})$ is the union of the closure of three pairwise disjoint open sets ${\cal O}_{1}(M^{2}),{\cal O}_{2}(M^{2})$ , and ${\cal O}_{3}(M^{2})$ defined as follows.

•

The set ${\cal O}_{1}(M^{2})$ consists of diffeomorphisms having $C^{1}$ -robust homoclinic tangencies.
•

There is a residual subset ${\cal R}_{2}(M^{2})$ of ${\cal O}_{2}(M^{2})$ such that every $f\in{\cal R}_{2}(M^{2})$ has a homoclinic class $H(P,f)$ that robustly does not admit any dominated splitting. However, for every hyperbolic set $\Lambda$ contained in $H(P,f)$ the invariant manifolds of $\Lambda$ meet transversely. In this case, we say that the diffeomorphism $f$ has a persistently fragile homoclinic tangency associated to $P$ .
•

There is a residual subset ${\cal R}_{3}(M^{2})$ of ${\cal O}_{3}(M^{2})$ such that for every diffeomorphism $f\in{\cal R}_{3}(M^{2})$ and every (hyperbolic) periodic point $P$ of $f$ the homoclinic class $H(P,f)$ is hyperbolic. But there is a sequence of periodic points $(P_{n})_{n}$ of $f$ such that the hyperbolic homoclinic classes $H(P_{n},f)$ accumulate (Hausdorff limit) to an aperiodic class (i.e., a recurrence class without periodic points).

As mentions above, Moreira’s result in [17] provides strong evidences suggesting that ${\cal O}_{1}(M^{2})$ is empty. On the other hand, we do not know if the sets ${\cal O}_{2}(M^{2})$ and ${\cal O}_{3}(M^{2})$ are empty or not. In fact, Smale density conjecture (hyperbolic diffeomorphisms are dense in $\textrm{Diff}^{1}(M^{2})$ ) is equivalent to prove that these three sets are empty.

We now explain how the discussion above is translated to higher dimensions. As before, we first consider non-hyperbolic diffeomorphisms, that is, the set $\textrm{Diff}^{1}(M)\setminus\overline{\textrm{Hyp}^{1}(M)}$ . If $\dim(M)\geq 3$ this set is not a Newhouse domain: it contains open sets of diffeomorphisms without homoclinic tangencies. Thus we consider the sets $\textrm{Tang}^{1}(M)$ of diffeomorphisms having a homoclinic tangency associated to a saddle and ${\cal O}_{0}(M)$ of non-hyperbolic diffeomorphisms far from homoclinic tangencies,

{\cal O}_{0}(M)\stackrel{{\scriptstyle\scriptscriptstyle\rm def}}{{=}}\mbox{{\rm Diff\,}}^{1}(M)\setminus\overline{(\textrm{Tang}^{1}(M)\cup\textrm{Hyp}^{1}(M))}.

Note that this set is not-empty and it is an open question whether it is contained in the set of tame diffeomorphisms (in fact, the first author conjectured that ${\cal O}_{0}(M)$ consists of tame diffeomorphisms, [6]). The diffeomorphisms in ${\cal O}_{0}(M)$ were studied in several papers, let us just refer to [31, 32, 33].

From now on, we will focus on the set

{\cal N}^{1}(M)\stackrel{{\scriptstyle\scriptscriptstyle\rm def}}{{=}}\mbox{{\rm Diff\,}}^{1}(M)\setminus\overline{\left({\cal O}_{0}(M)\cup\textrm{Hyp}^{1}(M)\right)}.

By definition, this set is a Newhouse domain. As in the case of surface diffeomorphisms, we split the set ${\cal N}^{1}(M)$ into three closed sets with pairwise disjoint interiors. We first define the set ${\cal O}_{1}(M)$ similarly as the set ${\cal O}_{1}(M^{2})$ ,

{\cal O}_{1}(M)\stackrel{{\scriptstyle\scriptscriptstyle\rm def}}{{=}}\{\mbox{$f\in\mbox{{\rm Diff\,}}^{1}(M)$ with a transitive hyperbolic set with a robust homoclinic tangency}\}.

The results in this paper implies that ${\cal O}_{1}(M)$ is non-empty, see also [4, 28].

We define the set ${\cal O}_{2}(M)$ by

{\cal O}_{2}(M)\stackrel{{\scriptstyle\scriptscriptstyle\rm def}}{{=}}\{f\in(\mbox{{\rm Diff\,}}^{1}(M)\setminus\overline{{\cal O}_{1}(M)})\mbox{\,with a persistently fragile homoclinic tangency}\}.

Consider the residual set ${\cal G}$ of $\mbox{{\rm Diff\,}}^{1}(M)$ in Corollary 3. Then if $f$ is a diffeomorphism in ${\cal G}\cap{\cal O}_{2}(M)$ with a persistently fragile homoclinic tangency associated to $P$ then the homoclinic class $H(P,f)$ has no index variation (otherwise one gets robust homoclinic tangencies).

Finally, define ${\cal O}_{3}(M)$ by

{\cal O}_{3}(M)\stackrel{{\scriptstyle\scriptscriptstyle\rm def}}{{=}}\left(\mbox{{\rm Diff\,}}^{1}(M)\setminus\overline{\textrm{Hyp}^{1}(M)\cup{\cal O}_{0}(M)\cup{\cal O}_{1}(M)\cup{\cal O}_{2}(M)}\right).

Corollary 3 implies that if $f\in{\cal G}\cap{\cal O}_{3}(M)$ then every homoclinic class of $f$ has an indices adapted dominated splitting. The description of the accumulation of homoclinic classes of diffeomorphisms in ${\cal O}_{3}(M)$ is a subtle issue. For instance, by shrinking ${\cal G}$ , for diffeomorphisms $f\in{\cal G}\cap{\cal O}_{3}(M)$ , there are $k$ and a sequence of saddles $P_{n}$ of index $k$ such that every $H(P_{n},f)$ has a dominated splitting $E\oplus_{{{}_{<}}}F$ with $\dim(E)=k$ and the sequence of homoclinic classes $H(P_{n},f)$ accumulates to a set $\Lambda$ that does not admit a dominated splitting $E\oplus_{{{}_{<}}}F$ with $\dim(E)=k$ (the set $\Lambda$ is the Hausdorff limit of the sequence $(H(P_{n},f))$ ).

We observe that, as a consequence of Corollary 4, there is an open and dense subset of ${\cal O}_{2}(M)\cup{\cal O}_{3}(M)$ consisting of wild diffeomorphisms. Note that we do not known if the sets ${\cal O}_{2}(M)$ and ${\cal O}_{3}(M)$ are empty or not.

Summarizing, as in the case of surface diffeomorphisms, we have that the Newhouse domain ${\cal N}^{1}(M)$ is the closure of the union of the pairwise disjoint open sets ${\cal O}_{1}(M),{\cal O}_{2}(M)$ , and ${\cal O}_{3}(M)$ .

This paper is organized as follows. In Section 2, we recall some definitions and state some notations we will use throughout the paper. In Section 3, we review the notion of ${\operatorname{cu}}$ -blender in [8] and present the notion of blender-horseshoe, a key ingredient of our constructions. In Section 4, we introduce a class of sub-manifolds, called folding manifolds relative to a blender-horseshoe $\Lambda$ . The main result is that folding manifolds and the local stable manifold of the blender-horseshoe $\Lambda$ have $C^{1}$ -robust tangencies, see Theorem 2. Using this result, we state a sufficient condition for the generation of robust homoclinic tangencies by homoclinic tangencies associated to hyperbolic sets. In Section 5, we see that strong homoclinic intersections of non-hyperbolic periodic points (i.e., intersections between the strong stable and unstable manifolds) generate blender-horseshoes. We also see that such strong intersections naturally occur in the non-hyperbolic setting. Finally, in Section 6 we conclude the proof of Theorem 1. We also state a result about the occurrence of robust heterodimensional cycles inside non-hyperbolic chain recurrence classes, see Theorem 3, which is an extension of [9, Theorem 1.16].

2 Definitions and notations

In this section, we define precisely the notions involved in this paper and state some notations.

Given a closed manifold $M$ , we denote by $\mbox{{\rm Diff\,}}^{1}(M)$ the space of $C^{1}$ -diffeomorphisms endowed with the usual uniform topology.

A diffeomorphism $f$ has a homoclinic tangency associated to a (hyperbolic) saddle $R$ if the unstable manifold $W^{\operatorname{u}}(R,f)$ and the stable manifold $W^{\operatorname{s}}(R,f)$ of the orbit of $R$ have some non-transverse intersection.

The ${\operatorname{s}}$ -index (resp. ${\operatorname{u}}$ -index) of a hyperbolic periodic point $R$ , denoted by ${\mbox{\rm{ind}}^{\,{\operatorname{s}}}}(R)$ (resp. ${\mbox{\rm{ind}}^{\,{\operatorname{u}}}}(R)$ ), is the dimension of the stable bundle $E^{\operatorname{s}}$ (resp. dimension of $E^{\operatorname{u}}$ ) of $R$ . We similarly define the ${\operatorname{s}}$ -index and ${\operatorname{u}}$ -index of a transitive hyperbolic set $\Lambda$ , denoted by ${\mbox{\rm{ind}}^{\,{\operatorname{s}}}}(\Lambda)$ and ${\mbox{\rm{ind}}^{\,{\operatorname{u}}}}(\Lambda)$ , respectively.

A heterodimensional cycle of a diffeomorphism $f$ consists of two hyperbolic saddles $P$ and $Q$ of $f$ of different ${\operatorname{s}}$ -indices and two heteroclinic points $X\in W^{\operatorname{u}}(P,f)\cap W^{\operatorname{s}}(Q,f)$ and $Y\in W^{\operatorname{s}}(P,f)\cap W^{\operatorname{u}}(Q,f)$ . In this case, we say that the cycle is associated to $P$ and $Q$ . Note that (due to insufficient dimensions) at least one of these intersections is not transverse. The heterodimensional cycle has co-index $k$ if $|{\mbox{\rm{ind}}^{\,{\operatorname{s}}}}(Q)-{\mbox{\rm{ind}}^{\,{\operatorname{s}}}}(P)|=k$ (note that $k\geq 1$ ).

Definition 2.1 (Homoclinic class).

Consider a diffeomorphism $f$ and a saddle $P$ of $f$ . The homoclinic class of $P$ , denoted by $H(P,f)$ , is the closure of the transverse intersections of the stable and unstable manifolds of the orbit of $P$ .

Remark 2.2.

The homoclinic class $H(P,f)$ can be alternatively defined as the closure of the saddles $Q$ homoclinically related with $P$ : the stable manifold of the orbit of $Q$ transversely meets the unstable manifold of the orbit of $P$ and vice-versa. Although all saddles homoclinically related with $P$ have the same ${\operatorname{s}}$ -index as $P$ , the homoclinic class $H(P,f)$ may contain periodic orbits of different ${\operatorname{s}}$ -index as the one of $P$ (i.e., there are homoclinic classes having index variation). Finally, a homoclinic class is a transitive set with dense periodic points.

Definition 2.3 (Chain recurrence class).

A point $x$ is chain recurrent if for every $\varepsilon>0$ there are $\varepsilon$ -pseudo orbits starting and ending at $x$ . The chain recurrence class of $x$ for $f$ , denoted by $C(x,f)$ , is the set of points $y$ such that, for every $\varepsilon>0$ , there are $\varepsilon$ -pseudo orbits starting at $x$ , passing $\varepsilon$ -close to $y$ and ending at $x$ .

According to [7], for $C^{1}$ -generic diffeomorphisms, the chain recurrence class of any periodic point is its homoclinic class.

Definition 2.4 (Dominated splitting).

Consider a diffeomorphism $f$ and a compact $f$ -invariant set $\Lambda$ . A $Df$ -invariant splitting $T_{\Lambda}M=E\oplus F$ over $\Lambda$ is dominated if the fibers $E_{x}$ and $F_{x}$ of $E$ and $F$ have constant dimension and there exists $k\in{\mathbb{N}}$ such that

\frac{||D_{x}f^{k}(u)||}{||D_{x}f^{k}(w)||}<\frac{1}{2},

for every $x\in\Lambda$ and every pair of unitary vectors $u\in E_{x}$ and $w\in F_{x}$ .

This definition means that vectors in the bundle $F$ are uniformly more expanded than vectors in $E$ by the derivative $Df^{k}$ . If it occurs, we say that $F$ dominates $E$ and write $E\oplus_{{}_{<}}F$ .

Remark 2.5.

In some cases, one needs to consider splittings with more than two bundles. A $Df$ -invariant splitting $E_{1}\oplus E_{2}\oplus\cdots\oplus E_{k}$ over a set $\Lambda$ is dominated if for all $j\in\{1,\dots,k-1\}$ the splitting $E_{1}^{j}\oplus E_{j+1}^{k}$ is dominated, where $E_{i}^{r}=E_{i}\oplus\cdots\oplus E_{r}$ , $i<r$ .

We use the notation $E_{1}\oplus_{{}_{<}}E_{2}\oplus_{{}_{<}}\cdots\oplus_{{}_{<}}E_{k}$ , meaning that $E_{i+1}$ dominates $E_{i}$ , or equivalently that $E_{1}^{j}\oplus_{{}_{<}}E_{j+1}^{k}$ .

As mentioned before, the main goal of this paper is to construct hyperbolic sets exhibiting homoclinic tangencies in a robust way. We need the following definition.

Definition 2.6 (Robust tangency).

Given a diffeomorphism $f\colon M\to M$ , a hyperbolic set $\Gamma$ of $f$ with a hyperbolic splitting $E^{\operatorname{s}}\oplus E^{\operatorname{u}}$ , and a submanifold $N\subset M$ with dimension $\dim(N)=\dim(E^{\operatorname{u}})$ , we say that the stable manifold $W^{\operatorname{s}}(\Gamma)$ of $\Gamma$ and the submanifold $N$ have a $C^{1}$ -robust tangency if for every diffeomorphism $g$ $C^{1}$ close to $f$ and every submanifold $N_{g}$ $C^{1}$ -close to $N$ , the stable manifold $W^{\operatorname{s}}(\Gamma_{g})$ of $\Gamma_{g}$ has some non-transverse intersection with $N_{g}$ .

We are specially interested in the case where $N$ is the unstable manifold $W^{\operatorname{u}}(P)$ of a periodic point $P$ of a non-trivial hyperbolic set $\Gamma$ and $N_{g}=W^{\operatorname{u}}(P_{g})$ . In that case, one gets $C^{1}$ -robust homoclinic tangencies (associated to $\Gamma$ ), recall Definition 1.1.

Standing notation: Throughout this paper we use the following notation:

•

Given a diffeomorphism $f$ and a hyperbolic set $\Lambda_{f}$ of $f$ there is a $C^{1}$ -neighborhood ${\cal U}$ of $f$ such that every $g\in{\cal U}$ has a hyperbolic set $\Lambda_{g}$ called the continuation of $\Lambda_{f}$ . The set $\Lambda_{g}$ is close to $\Lambda_{f}$ and the restrictions of $f$ to $\Lambda_{f}$ and of $g$ to $\Lambda_{g}$ are conjugate. If $P_{f}$ a hyperbolic periodic point, we denote by $P_{g}$ the continuation of $P_{f}$ for $g$ close to $f$ .
•

Given a periodic point $P$ of $f$ we denote by $\pi(P)$ its period.
•

The perturbations we consider are always arbitrarily small. Thus the sentence there is a $C^{r}$ -perturbation $g$ of $f$ means there is $g$ arbitrarily $C^{r}$ -close to $f$ .

3 Blender-horseshoes

In this section, we introduce precisely the definition of a blender-horseshoe, a particular case of the blenders in [8]. In fact, blender-horseshoes are the main ingredient of this paper and the key tool for getting robust homoclinic tangencies. We beging by reviewing the notion of a blender.

3.1 Blenders

The notion of a ${\operatorname{cu}}$ -blender was introduced in [8] as a class of examples, without a precise and formal definition. Blenders were used to get $C^{1}$ -robust transitivity, [8], and robust heterodimensional cycles, [9]. The relevance of blenders comes from their internal geometry and not from their dynamics: a ${\operatorname{cu}}$ -blender is a (uniformly) hyperbolic transitive set whose stable set robustly has Hausdorff dimension greater than its stable bundle. In some sense, this property resembles and plays a similar role as the thick horseshoes introduced by Newhouse, [21]. Following [13, Definition 6.11], we now give a tentative formal definition of a ${\operatorname{cu}}$ -blender:

Definition 3.1 ( ${\operatorname{cu}}$ -blender).

Let $f\colon M\to M$ be a diffeomorphism. A transitive hyperbolic set $\Gamma$ of $f$ with ${\mbox{\rm{ind}}^{\,{\operatorname{u}}}}(\Gamma)=k\geq 2$ is a ${\operatorname{cu}}$ -blender if there are a $C^{1}$ -neighborhood ${\cal U}$ of $f$ and a $C^{1}$ -open set ${\cal D}$ of embeddings of $(k-1)$ -dimensional disks $D$ into $M$ such that, for every diffeomorphism $g\in{\cal U}$ , every disk $D\in{\cal D}$ intersects the local stable manifold $W^{s}_{{\operatorname{loc}}}(\Gamma_{g})$ of the continuation $\Gamma_{g}$ of $\Gamma$ for $g$ . The set ${\cal D}$ is called the superposition region of the blender.

By definition, the property of a diffeomorphism having a ${\operatorname{cu}}$ -blender is a $C^{1}$ -robust property.

We do not know whether ${\operatorname{cu}}$ -blenders yield robust tangencies in the sense of Definition 2.6. This leads to the following questions:

Question 2.

Let $f\colon M\to M$ be a diffeomorphism having a ${\operatorname{cu}}$ -blender $\Gamma$ with $k={\mbox{\rm{ind}}^{\,{\operatorname{u}}}}(\Gamma)$ .

•

Does it exist a submanifold $N\subset M$ with $\dim(N)=k$ such that that $W^{s}(\Gamma)$ and $N$ have a robust tangency?
•

Suppose that a submanifold $L$ of dimension $k$ and $W^{s}(\Gamma)$ have a tangency. Does this tangency yield robust tangencies? More precisely, does there exist an open set ${\cal U}$ of $\mbox{{\rm Diff\,}}^{1}(M)$ , $f$ in the closure of ${\cal U}$ , of diffeomorphisms $g$ with robust tangencies associated to $\Gamma_{g}$ and “continuations” of $L$ ?

We note that, even for the first ${\operatorname{cu}}$ -blenders constructed in [8, Section 1], these questions remain open. We will give a partial answer to this question in Theorem 2. For that we will introduce a special class of ${\operatorname{cu}}$ -blenders, conjugate to the usual Smale horseshoe, that we call blender-horseshoes.

3.2 Blender-horseshoes

In this section, we give the precise definition of a blender-horseshoe. This definition involves several concepts as invariant cone-fields, hyperbolicity, partial hyperbolicity, and Markov partitions, which we will present separately. Our presentation follows closely [8, Section 1], thus some details of our construction are just sketched.

3.2.1 Cone-fields

Consider ${\mathbb{R}}^{n}={\mathbb{R}}^{s}\oplus{\mathbb{R}}\oplus{\mathbb{R}}^{u}$ , where $s>0,u>0$ , and $n=s+u+1$ . For $\alpha\in(0,1)$ , denote by ${\cal C}^{{\operatorname{s}}}_{\alpha}$ , ${\cal C}^{{\operatorname{u}}}_{\alpha}$ , and ${\cal C}^{\operatorname{uu}}_{\alpha}$ the following cone-fields:

\begin{array}[]{ll}{\cal C}^{{\operatorname{s}}}_{\alpha}(x)&=\{v=(v^{s},v^{c},v^{u})\in{\mathbb{R}}^{s}\oplus{\mathbb{R}}\oplus{\mathbb{R}}^{u}=T_{x}M\quad\colon\quad\|v^{c}+v^{u}\|\leq\alpha\,\|v^{s}\|\},\\ {\cal C}^{{\operatorname{u}}}_{\alpha}(x)&=\{v=(v^{s},v^{c},v^{u})\in{\mathbb{R}}^{s}\oplus{\mathbb{R}}\oplus{\mathbb{R}}^{u}=T_{x}M\quad\colon\quad\|v^{s}\|\leq\alpha\,\|v^{c}+v^{u}\|\},\\ {\cal C}^{{\operatorname{uu}}}_{\alpha}(x)&=\{v=(v^{s},v^{c},v^{u})\in{\mathbb{R}}^{s}\oplus{\mathbb{R}}\oplus{\mathbb{R}}^{u}=T_{x}M\quad\colon\quad\|v^{s}+v^{c}\|\leq\alpha\,\|v^{u}\|\}.\end{array}

As $\alpha\in(0,1)$ , one gets that ${\cal C}^{{\operatorname{s}}}_{\alpha}$ is transverse to ${\cal C}_{\alpha}^{{\operatorname{u}}}$ , that is, ${\cal C}_{\alpha}^{{\operatorname{s}}}(x)\cap{\cal C}_{\alpha}^{{\operatorname{u}}}(x)=0_{x}\in T_{x}M$ . Moreover, ${\cal C}_{\alpha}^{{\operatorname{uu}}}(x)\subset{\cal C}_{\alpha}^{{\operatorname{uu}}}(x)$ for all $x$ .

Consider the cube

{\mathbb{C}}=[-1,1]^{n}=[-1,1]^{s}\times[-1,1]\times[-1,1]^{u}.

We split the boundary of ${\mathbb{C}}$ into three parts:

\begin{array}[]{ll}\partial^{\operatorname{s}}{\mathbb{C}}&=\partial\big{(}[-1,1]^{s}\big{)}\times[-1,1]\times[-1,1]^{u},\\ \partial^{\operatorname{c}}{\mathbb{C}}&=[-1,1]^{s}\times\{-1,1\}\times[-1,1]^{u},\quad\mbox{and}\\ \partial^{\operatorname{uu}}{\mathbb{C}}&=[-1,1]^{s}\times[-1,1]\times\partial\big{(}[-1,1]^{u}\big{)}.\end{array}

We also consider

\partial^{\operatorname{u}}{\mathbb{C}}=[-1,1]^{s}\times\partial\big{(}[-1,1]\times[-1,1]^{u}\big{)}=\partial^{c}{\mathbb{C}}\cup\partial^{\operatorname{uu}}{\mathbb{C}}.

We now consider a local diffeomorphism $f\colon{\mathbb{C}}\to{\mathbb{R}}^{n}$ and formulate conditions BH1)–BH6) for the maximal invariant set $\Lambda$ of $f$ in the cube ${\mathbb{C}}$ ,

\Lambda=\bigcap_{i\in{\mathbb{Z}}}f^{i}({\mathbb{C}}),

to be a blender-horseshoe, see Definition 3.8.

BH1) The intersection $f({\mathbb{C}})\cap({\mathbb{R}}^{s}\times{\mathbb{R}}\times[-1,1]^{u})$ consists of two connected components, denoted $f({\cal A})$ and $f({\cal B})$ . Furthermore,

f({\cal A})\cup f({\cal B})\subset(-1,1)^{s}\times{\mathbb{R}}\times[-1,1]^{u},\mbox{ and }

({\cal A}\cup{\cal B})\cap\partial^{\operatorname{uu}}({\mathbb{C}})=\emptyset.

We denote $f_{{}_{\cal A}}\colon{\cal A}\to f({\cal A})$ and $f_{{}_{\cal B}}\colon{\cal B}\to f({\cal B})$ the restrictions of $f$ to ${\cal A}$ and ${\cal B}$ , respectively. See Figure 1.

\psfrag{C}{${\mathbb{C}}$}\psfrag{Ru}{${\mathbb{R}}^{u}$}\psfrag{B}{${\cal B}$}\psfrag{A}{${\cal A}$}\psfrag{Rc}{${\mathbb{R}}$}\psfrag{dC}{$\partial^{\operatorname{uu}}{\mathbb{C}}$}\includegraphics[height=115.63243pt]{bh1.eps}

Figure 1: Projection in

{\mathbb{R}}\oplus{\mathbb{R}}^{u}

of a blender horseshoe. Condition BH1).

BH2) Cone-fields: The cone-field ${\cal C}_{\alpha}^{\operatorname{s}}$ is strictly $Df^{-1}$ -invariant and the cone-fields ${\cal C}^{{\operatorname{u}}}_{\alpha}$ and ${\cal C}^{\operatorname{uu}}_{\alpha}$ are strictly $Df$ -invariant. More precisely, there is $0<\alpha^{\prime}<\alpha$ such that, for every $x\in f({\cal A})\cup f({\cal B})$ one has

Df^{-1}({\cal C}^{\operatorname{s}}_{\alpha}(x))\subset{\cal C}^{{\operatorname{s}}}_{\alpha^{\prime}}(f^{-1}x).

In the same way, for every $x\in{\cal A}\cup{\cal B}$ , one has

Df({\cal C}^{\operatorname{uu}}_{\alpha}(x))\subset{\cal C}^{{\operatorname{uu}}}_{\alpha^{\prime}}(f(x))\quad\mbox{and}\quad Df({\cal C}^{\operatorname{u}}_{\alpha}(x))\subset{\cal C}^{{\operatorname{u}}}_{\alpha^{\prime}}(f(x)).

Moreover, the cone-fields ${\cal C}^{\operatorname{u}}_{\alpha}$ and ${\cal C}^{\operatorname{s}}_{\alpha}$ are uniformly expanding and contracting, respectively.

Note that that property BH2) is open: by increasing slightly $\alpha^{\prime}<\alpha$ , it holds for every diffeomorphism $g$ in a $C^{1}$ neighborhood of $f$ .

Since $f(\partial^{\operatorname{uu}}{\mathbb{C}})$ is disjoint from ${\mathbb{R}}^{s+1}\times[-1,1]^{u}$ and $f({\cal A})\cup f({\cal B})\subset(-1,1)^{s}\times{\mathbb{R}}\times[-1,1]^{u}$ , from condition BH1) one has that

f(\partial({\mathbb{C}}))\cap\partial([-1,1]^{s}\times{\mathbb{R}}\times[-1,1]^{u})\subset f(\partial^{\operatorname{s}}{\mathbb{C}}\cup\partial^{\operatorname{c}}{\mathbb{C}}).

Furthermore, this is a $C^{1}$ -robust property.

By BH2), the components of $f(\partial({\mathbb{C}}))\cap\partial([-1,1]^{s}\times{\mathbb{R}}\times[-1,1]^{u})$ are foliated by disks $\Delta$ tangent to ${\cal C}_{\alpha}^{\operatorname{uu}}$ , i.e., $T_{x}\Delta\subset{\cal C}^{\operatorname{uu}}_{\alpha}(x)$ . Hence these disks are transverse to $\partial([-1,1]^{s}\times{\mathbb{R}}\times[-1,1]^{u})$ . As a consequence, one gets the following:

Remark 3.2.

Under the ( $C^{1}$ -robust) hypothesis BH2), hypothesis BH1) is also a $C^{1}$ -robust property.

Remark 3.3 (Hyperbolicity).

Consider the maximal invariant set $\Lambda$ of $f$ in ${\mathbb{C}}$

\Lambda=\bigcap_{i\in{\mathbb{Z}}}f^{i}({\mathbb{C}}).

By BH1) and BH2) the set $\Lambda$ is compact and satisfies

\Lambda\subset\mbox{\rm int}({\cal A}\cup{\cal B})\subset\mbox{\rm int}({\mathbb{C}}).

Moreover, the set $\Lambda$ has a dominated splitting $T_{\Lambda}M=E\oplus_{{}_{<}}F\oplus_{{}_{<}}F$ , where $E\subset{\cal C}^{\operatorname{s}}$ , $F\oplus G\subset{\cal C}^{\operatorname{u}}$ , and $G\subset{\cal C}^{{\operatorname{uu}}}$ , and $F$ is one-dimensional.

By BH2), the set $\Lambda$ has a hyperbolic splitting $E^{\operatorname{s}}\oplus E^{\operatorname{u}}$ , where $E^{\operatorname{s}}=E$ and $E^{\operatorname{u}}=F\oplus G$ and $\dim(E^{\operatorname{s}})=s$ and $\dim(E^{\operatorname{u}})=u+1$ . Furthermore, the set $\Lambda$ also has a partially hyperbolic splitting

T_{\Lambda}M=E^{\operatorname{s}}\oplus_{{}_{<}}E^{{\operatorname{cu}}}\oplus_{{}_{<}}E^{{\operatorname{uu}}}

with three non-trivial directions, where $E^{{\operatorname{cu}}}=F$ and $E^{{\operatorname{uu}}}=G$ and $\dim(E^{\operatorname{uu}})=u$ . Note that $E^{\operatorname{u}}=E^{\operatorname{cu}}\oplus E^{\operatorname{uu}}$ . We say that $E^{\operatorname{uu}}$ is the strong unstable bundle of $\Lambda$ .

3.2.2 Markov partitions

Write

{\mathbb{A}}=f^{-1}\left(f({\cal A})\cap{\mathbb{C}}\right)\quad\mbox{and}\quad{\mathbb{B}}=f^{-1}\left(f({\cal B})\cap{\mathbb{C}}\right).

BH3) Associated Markov partition:

•

The sets ${\mathbb{A}}$ and ${\mathbb{B}}$ are both non-empty and connected. That is, the sets ${\mathbb{A}}$ and ${\mathbb{B}}$ are the connected components of $f^{-1}({\mathbb{C}})\cap{\mathbb{C}}$ .
•

The sets ${\mathbb{A}}$ and ${\mathbb{B}}$ are horizontal sub-cubes of ${\mathbb{C}}$ and their images $f({\mathbb{A}})$ and $f({\mathbb{B}})$ are vertical sub-cubes of ${\mathbb{C}}$ . More precisely,

$f({\mathbb{A}})\cup f({\mathbb{B}})\subset(-1,1)^{s}\times[-1,1]\times[-1,1]^{u},\mbox{ and}$

${\mathbb{A}}\cup{\mathbb{B}}\subset[-1,1]^{s}\times(-1,1)\times(-1,1)^{u}.$

In other words, $f({\mathbb{A}})\cup f({\mathbb{B}})$ is disjoint from $\partial^{\operatorname{s}}{\mathbb{C}}$ and ${\mathbb{A}}\cup{\mathbb{B}}$ is disjoint from $\partial^{\operatorname{u}}{\mathbb{C}}$ .

As a consequence of BH2) and BH3), one gets that $\{{\mathbb{A}},{\mathbb{B}}\}$ is a Markov partition generating $\Lambda$ . Therefore the dynamics of $f$ in $\Lambda$ is conjugate to the full shift of two symbols. In particular, the hyperbolic set $\Lambda$ contains exactly two fixed points of $f$ , $P\in{\mathbb{A}}$ and $Q\in{\mathbb{B}}$ . See Figure 2.

\psfrag{C}{${\mathbb{C}}$}\psfrag{b}{${\mathbb{B}}$}\psfrag{B}{${\cal B}$}\psfrag{A}{${\cal A}$}\psfrag{a}{${\mathbb{A}}$}\includegraphics[height=115.63243pt]{bh4.eps}

Figure 2: Projection in

{\mathbb{R}}\times{\mathbb{R}}^{u}

. BH3) Markov partition of a blender-horseshoe.

3.2.3 ${\operatorname{uu}}$ -disks and their iterates

Definition 3.4 ( ${\operatorname{s}}$ - and ${\operatorname{uu}}$ -disks).

A disk $\Delta$ of dimension $s$ contained in ${\mathbb{C}}$ is an ${\operatorname{s}}$ -disk if

•

it is tangent to ${\cal C}_{\alpha}^{\operatorname{s}}$ , i.e., $T_{x}\Delta\subset{\cal C}_{\alpha}^{\operatorname{s}}(x)$ for all $x\in\Delta$ , and
•

its boundary $\partial\Delta$ is contained in $\partial^{\operatorname{s}}({\mathbb{C}})$ .

A disk $\Upsilon\subset{\mathbb{R}}^{s}\times{\mathbb{R}}\times[-1,1]^{u}$ of dimension $u$ is a ${\operatorname{uu}}$ -disk if

•

it is tangent to ${\cal C}_{\alpha}^{\operatorname{uu}}$ , i.e. $T_{x}\Upsilon\subset{\cal C}_{\alpha}^{\operatorname{uu}}(x)$ for all $x\in\Upsilon$ , and
•

$\partial\Upsilon\subset{\mathbb{R}}^{s}\times{\mathbb{R}}\times\partial([-1,1]^{u})$ .

Given a point $x\in\Lambda$ , there is a unique $f$ -invariant manifold of dimension $u$ tangent at $x$ to the strong unstable bundle $E^{\operatorname{uu}}(x)$ , the strong unstable manifold $W^{{\operatorname{uu}}}(x)$ of $x$ . For points $x\in\Lambda$ , the local invariant manifolds $W^{\operatorname{s}}_{\operatorname{loc}}(x)$ , $W^{{\operatorname{u}}}_{\operatorname{loc}}(x)$ , and $W^{{\operatorname{uu}}}_{\operatorname{loc}}(x)$ are the connected components of the intersections $W^{\operatorname{s}}(x)\cap{\mathbb{C}}$ , $W^{{\operatorname{u}}}(x)\cap{\mathbb{C}}$ , and $W^{{\operatorname{uu}}}(x)\cap{\mathbb{C}}$ containing $x$ , respectively.

As a consequence of BH1)–BH3) one gets

Remark 3.5.

For every $x\in\Lambda$ , $W^{\operatorname{s}}_{\operatorname{loc}}(x)$ is an ${\operatorname{s}}$ -disk and $W^{\operatorname{uu}}_{\operatorname{loc}}(x)$ and is a ${\operatorname{uu}}$ -disk.

BH4) ${\operatorname{uu}}$ -disks through the local stable manifolds of $P$ and $Q$ : Let $D$ and $D^{\prime}$ be ${\operatorname{uu}}$ -disks such that $D\cap W^{\operatorname{s}}_{\operatorname{loc}}(P)\neq\emptyset$ and $D^{\prime}\cap W^{\operatorname{s}}_{\operatorname{loc}}(Q)\neq\emptyset$ . Then

D\cap\partial^{\operatorname{c}}({\mathbb{C}})=D^{\prime}\cap\partial^{\operatorname{c}}({\mathbb{C}})=D\cap D^{\prime}=\emptyset,

see Figure 3.

\psfrag{D}{$D$}\psfrag{Dp}{$D^{\prime}$}\psfrag{b}{${\mathbb{B}}$}\psfrag{a}{${\mathbb{A}}$}\psfrag{c}{${\mathbb{C}}$}\psfrag{p}{$P$}\psfrag{q}{$Q$}\psfrag{wuuP}{$W^{\operatorname{uu}}_{\operatorname{loc}}(P)$}\psfrag{wuuQ}{$W^{\operatorname{uu}}_{\operatorname{loc}}(Q)$}\psfrag{Cuu}{${\cal C}^{{\operatorname{uu}}}_{\alpha}$}\includegraphics[height=144.54pt]{bh5.eps}

Figure 3: Projection in

{\mathbb{R}}\oplus{\mathbb{R}}^{u}

{\operatorname{uu}}

-disks. Condition BH4).

Given any ${\operatorname{s}}$ -disk $\Delta$ , there are two different homotopy classes of ${\operatorname{uu}}$ -disks contained in $[-1,1]^{s}\times{\mathbb{R}}\times[-1,1]^{u}$ and disjoint from $\Delta$ . We call these classes ${\operatorname{uu}}$ -disks at the right and at the left of $\Delta$ . We use the following criterion: the ${\operatorname{uu}}$ -disks disjoint from $W^{\operatorname{s}}_{\operatorname{loc}}(P)$ in the homotopy class of $W^{\operatorname{uu}}_{\operatorname{loc}}(Q)$ are at the right of $W^{\operatorname{s}}_{\operatorname{loc}}(P)$ . The ${\operatorname{uu}}$ -disks disjoint from $W^{\operatorname{s}}_{\operatorname{loc}}(P)$ in the other homotopy class are at the left of $W^{\operatorname{s}}_{\operatorname{loc}}(P)$ . We define similarly ${\operatorname{uu}}$ -disks at the left and at the right of $W^{\operatorname{s}}_{\operatorname{loc}}(Q)$ , where ${\operatorname{uu}}$ -disks at the left of $W^{\operatorname{s}}_{\operatorname{loc}}(Q)$ are those in the class of $W^{{\operatorname{uu}}}_{\operatorname{loc}}(P)$ .

According to BH4), ${\operatorname{uu}}$ -disks at the left of $W^{\operatorname{s}}_{\operatorname{loc}}(P)$ are also at the left of $W^{\operatorname{s}}_{\operatorname{loc}}(Q)$ . Analogously, ${\operatorname{uu}}$ -disks at the right of $W^{\operatorname{s}}_{\operatorname{loc}}(Q)$ are also at the right of $W^{\operatorname{s}}_{\operatorname{loc}}(P)$ .

Summarizing, there are five possibilities for a ${\operatorname{uu}}$ -disk $D$ in $[-1,1]^{s}\times{\mathbb{R}}\times[-1,1]^{u}$ :

•

either $D$ is at the left of $W^{\operatorname{s}}_{\operatorname{loc}}(P)$ ,
•

or $D\cap W^{\operatorname{s}}_{\operatorname{loc}}(P)\neq\emptyset$ ,
•

or $D$ is at the right of $W^{\operatorname{s}}_{\operatorname{loc}}(Q)$ ,
•

or $D\cap W^{\operatorname{s}}_{\operatorname{loc}}(Q)\neq\emptyset$ ,
•

or else $D$ is at the right of $W^{{\operatorname{s}}}_{\operatorname{loc}}(P)$ and at the left of $W^{{\operatorname{s}}}_{\operatorname{loc}}(Q)$ . In this case, we say that the ${\operatorname{uu}}$ -disk $D$ is in between $W^{{\operatorname{s}}}_{\operatorname{loc}}(P)$ and $W^{{\operatorname{s}}}_{\operatorname{loc}}(Q)$ .

As a consequence of BH4) one gets the following.

Remark 3.6 ( ${\operatorname{uu}}$ -disks in between $W^{{\operatorname{s}}}_{\operatorname{loc}}(P)$ and $W^{{\operatorname{s}}}_{\operatorname{loc}}(Q)$ ).

1.

There is a non-empty open subset $U$ of ${\mathbb{C}}$ such that any ${\operatorname{uu}}$ -disk through a point $x\in U$ is in between $W^{{\operatorname{s}}}_{\operatorname{loc}}(P)$ and $W^{{\operatorname{s}}}_{\operatorname{loc}}(Q)$ . In particular, there exist ${\operatorname{uu}}$ -disks in between $W^{{\operatorname{s}}}_{\operatorname{loc}}(P)$ and $W^{{\operatorname{s}}}_{\operatorname{loc}}(Q)$ .
2.

Every ${\operatorname{uu}}$ -disk $\Delta\subset[-1,1]^{s}\times{\mathbb{R}}\times[-1,1]^{u}$ in between $W^{{\operatorname{s}}}_{\operatorname{loc}}(P)$ and $W^{{\operatorname{s}}}_{\operatorname{loc}}(Q)$ is contained in ${\mathbb{C}}$ and is disjoint from $\partial^{\operatorname{c}}({\mathbb{C}})$ .

Consider a ${\operatorname{uu}}$ -disk $\Delta\subset{\mathbb{C}}$ and write

f_{{}_{\cal A}}(\Delta)=f(\Delta\cap{\cal A})\quad\mbox{and}\quad f_{{}_{\cal B}}(\Delta)=f(\Delta\cap{\cal B}).

According to BH1) and BH2) one gets,

Remark 3.7.

For every ${\operatorname{uu}}$ -disk $\Delta\subset{\mathbb{C}}$ , $f_{{}_{\cal A}}(\Delta)$ and $f_{{}_{\cal B}}(\Delta)$ are ${\operatorname{uu}}$ -disks in $[-1,1]^{s}\times{\mathbb{R}}\times[-1,1]^{u}$ .

BH5) Positions of images of ${\operatorname{uu}}$ -disks (I): Given any ${\operatorname{uu}}$ -disk $\Delta\subset{\mathbb{C}}$ , the following holds:

1.

if $\Delta$ is at the right of $W^{\operatorname{s}}_{\operatorname{loc}}(P)$ then $f_{{}_{\cal A}}(\Delta)$ is a ${\operatorname{uu}}$ -disk at the the right of $W^{\operatorname{s}}_{\operatorname{loc}}(P)$ ,
2.

if $\Delta$ is at the left of $W^{\operatorname{s}}_{\operatorname{loc}}(P)$ then $f_{{}_{\cal A}}(\Delta)$ is a ${\operatorname{uu}}$ -disk at the the left of $W^{\operatorname{s}}_{\operatorname{loc}}(P)$ ,
3.

if $\Delta$ is at the right of $W^{\operatorname{s}}_{\operatorname{loc}}(Q)$ then $f_{{}_{\cal B}}(\Delta)$ is a ${\operatorname{uu}}$ -disk at the the right of $W^{\operatorname{s}}_{\operatorname{loc}}(Q)$ ,
4.

if $\Delta$ is at the left of $W^{\operatorname{s}}_{\operatorname{loc}}(Q)$ then $f_{{}_{\cal B}}(\Delta)$ is a ${\operatorname{uu}}$ -disk at the the left of $W^{\operatorname{s}}_{\operatorname{loc}}(Q)$ ,
5.

if $\Delta$ is at the left of $W^{\operatorname{s}}_{\operatorname{loc}}(P)$ or $\Delta\cap W^{\operatorname{s}}_{\operatorname{loc}}(P)\neq\emptyset$ then $f_{{}_{\cal B}}(\Delta)$ is a ${\operatorname{uu}}$ -disk at the the left of $W^{\operatorname{s}}_{\operatorname{loc}}(P)$ , and
6.

if $\Delta$ is at the right of $W^{\operatorname{s}}_{\operatorname{loc}}(Q)$ or $\Delta\cap W^{s}_{\operatorname{loc}}(Q)\neq\emptyset$ then $f_{{}_{\cal A}}(\Delta)$ is a ${\operatorname{uu}}$ -disk at the the right of $W^{\operatorname{s}}_{\operatorname{loc}}(Q)$ .

Finally, we state the last condition (which will play a key role) in the definition of blender-horseshoe:

: BH6) Positions of images of ${\operatorname{uu}}$ -disks (II): Let $\Delta$ be a ${\operatorname{uu}}$ -disk in between $W^{{\operatorname{s}}}_{\operatorname{loc}}(P)$ and $W^{{\operatorname{s}}}_{\operatorname{loc}}(Q)$ . Then either $f_{{}_{\cal A}}(\Delta)$ or $f_{{}_{\cal B}}(\Delta)$ is a ${\operatorname{uu}}$ -disk in between $W^{{\operatorname{s}}}_{\operatorname{loc}}(P)$ and $W^{{\operatorname{s}}}_{\operatorname{loc}}(Q)$ .

Conditions BH5)–BH6) are depicted in Figure 4.

\psfrag{D}{$D$}\psfrag{E}{$E$}\psfrag{F}{$F$}\psfrag{a}{${\mathbb{A}}$}\psfrag{b}{${\mathbb{B}}$}\psfrag{wuuP}{$W^{\operatorname{uu}}_{\operatorname{loc}}(P)$}\psfrag{wuuQ}{$W^{\operatorname{uu}}_{\operatorname{loc}}(Q)$}\psfrag{fad}{$f_{{}_{\cal A}}(D)$}\psfrag{fae}{$f_{{}_{\cal A}}(E)$}\psfrag{fbe}{$f_{{}_{\cal B}}(E)$}\psfrag{fbf}{$f_{{}_{\cal B}}(F)$}\psfrag{fbd}{$f_{{}_{\cal B}}(D)$}\psfrag{faf}{$f_{{}_{\cal A}}(F)$}\psfrag{p}{$P$}\psfrag{q}{$Q$}\psfrag{text}{$f_{{}_{\cal B}}(D)$ and $f_{{}_{\cal A}}(F)$ are not in between $W^{\operatorname{s}}_{\operatorname{loc}}(P)$ and $W^{\operatorname{s}}_{\operatorname{loc}}(Q)$}\includegraphics[height=151.76744pt]{bh6.eps}

Figure 4: Projection in

{\mathbb{R}}\oplus{\mathbb{R}}^{u}

. Disks in between

W^{\operatorname{s}}_{\operatorname{loc}}(P)

and

W^{\operatorname{s}}_{\operatorname{loc}}(Q)

and their images.

3.2.4 Definition of blender-horseshoe

We are now ready for defining blender-horseshoes⁴⁴4In some cases, we will use the terminology ${\operatorname{cu}}$ -blender-horseshoe for emphasizing that the central one-dimensional direction is expanding. Using this terminology, a ${\operatorname{cs}}$ -blender-horseshoe is a blender-horseshoe for $f^{-1}$ .:

Definition 3.8 (Blender-horseshoe).

Consider a manifold $M$ of dimension $n\geq 3$ and a diffeomorphism $f\colon M\to M$ . A hyperbolic set $\Lambda$ of $f$ is a blender-horseshoe if (in some coordinate system) there are a cube ${\mathbb{C}}$ and families of cone-fields ${\cal C}^{\operatorname{s}}$ , ${\cal C}^{\operatorname{u}}$ , and ${\cal C}^{{\operatorname{uu}}}$ verifying conditions BH1)–BH6 ) above.

We say that ${\mathbb{C}}$ is the reference cube of the blender-horseshoe $\Lambda$ and that the saddles $P$ and $Q$ are the reference saddles of $\Lambda$ , where $P$ is the left saddle and $Q$ is the right saddle.

Recall that given a hyperbolic set $\Lambda$ of a diffeomorphism $f$ there is a $C^{1}$ -neighborhood ${\cal U}_{f}$ of $f$ such that every diffeomorphism $g\in{\cal U}_{f}$ has a hyperbolic set $\Lambda_{g}$ which is close and conjugate to $\Lambda$ , called the continuation of $\Lambda$ for $g$ . Following [8, Lemma 1.11], one can prove the following:

Lemma 3.9.

Let $\Lambda$ be a blender-horseshoe of a diffeomorphism $f$ with reference cube ${\mathbb{C}}$ and reference saddles $P$ and $Q$ . Then there is a neighborhood ${\cal U}_{f}$ of $f$ in $\mbox{{\rm Diff\,}}^{1}(M)$ such that for all $g\in{\cal U}_{g}$ the continuation $\Lambda_{g}$ of $\Lambda$ for $g$ is a blender-horseshoe with reference cube ${\mathbb{C}}$ and reference saddles $P_{g}$ and $Q_{g}$ .

Arguing as in [8] (see also [12] for a simple toy model and the example in Section 5.1) one gets the following:

Remark 3.10.

Every ${\operatorname{uu}}$ -disk in between in between $W^{{\operatorname{s}}}_{\operatorname{loc}}(P)$ and $W^{{\operatorname{s}}}_{\operatorname{loc}}(Q)$ intersects $W^{\operatorname{s}}_{\operatorname{loc}}(\Lambda)$ . Therefore the blender-horseshoe $\Lambda$ is a ${\operatorname{cu}}$ -blender in the sense of Definition 3.1, where the ${\operatorname{uu}}$ -disks in between $W^{{\operatorname{s}}}_{\operatorname{loc}}(P)$ and $W^{{\operatorname{s}}}_{\operatorname{loc}}(Q)$ define its superposition region.

4 Robust tangencies

In this section, we introduce a class of sub-manifolds called folding manifolds relative to a blender-horseshoe $\Lambda$ . The main technical step is Proposition 4.4, which claims that folding manifolds are tangent to the local stable manifold $W^{\operatorname{s}}_{\operatorname{loc}}(\Lambda)$ of the blender-horseshoe $\Lambda$ . Moreover, these tangencies are $C^{1}$ -robust, see Theorem 2. Finally, using blender-horseshoes, in Theorem 4.8 we give sufficient conditions for the generation of robust tangencies by a homoclinic tangency.

4.1 Folding manifolds and tangencies associated to blender-horseshoes

Let $\Lambda$ be a blender-horseshoe with reference cube ${\mathbb{C}}$ as in Section 3.2. Recall that the dimension of the unstable bundle of $\Lambda$ , $E^{\operatorname{u}}=E^{\operatorname{cu}}\oplus E^{\operatorname{uu}}$ , is $(u+1)$ . We say that the ${\operatorname{u}}$ -index of the blender-horseshoe $\Lambda$ is $(u+1)$ (i.e., the ${\operatorname{u}}$ -index of $\Lambda$ as a hyperbolic set). Define the local stable manifold of $\Lambda$ by

W^{\operatorname{s}}_{{\operatorname{loc}}}(\Lambda)=\bigcup_{x\in\Lambda}W^{\operatorname{s}}_{{\operatorname{loc}}}(x),

where $W^{\operatorname{s}}_{{\operatorname{loc}}}(x)$ is the connected component of $W^{\operatorname{s}}(x)\cap{\mathbb{C}}$ containing $x$ . .

Remark 4.1.

The local stable manifold $W^{\operatorname{s}}_{{\operatorname{loc}}}(\Lambda)$ of the blender-horseshoe $\Lambda$ is the set of points $x\in{\mathbb{C}}$ whose forward orbit remains in the reference cube ${\mathbb{C}}$ .

A new ingredient of this section is the notion of folding manifold defined as follows:

Definition 4.2 (Folding manifold).

Consider a blender-horseshoe $\Lambda$ of ${\operatorname{u}}$ -index $(u+1)$ with reference cube ${\mathbb{C}}$ and reference saddles $P$ and $Q$ . A sub-manifold ${\cal S}\subset{\mathbb{C}}$ of dimension $(u+1)$ is a folding manifold of $\Lambda$ (relative to the saddle $P$ ) if there is a family $({\cal S}_{t})_{t\in[0,1]}$ of ${\operatorname{uu}}$ -disks depending continuously on $t$ such that:

•

${\cal S}=\bigcup_{t\in[0,1]}{\cal S}_{t},$
•

${\cal S}_{0}$ and ${\cal S}_{1}$ intersects $W^{\operatorname{s}}_{\operatorname{loc}}(P)$ , and
•

for every $t\in(0,1)$ , the ${\operatorname{uu}}$ -disk ${\cal S}_{t}$ is in between $W^{\operatorname{s}}_{\operatorname{loc}}(P)$ and $W^{\operatorname{s}}_{\operatorname{loc}}(Q)$ .

We similarly define a folding manifold (relative to $Q$ ). A folding manifold of the blender-horseshoe $\Lambda$ is a folding manifold relative either to $P$ or to $Q$ .

Remark 4.3.

Let ${\cal S}$ be a folding manifold of the blender-horseshoe $\Lambda$ . Then there are a point $x\in{\cal S}$ and a non-zero vector $v\in T_{x}{\cal S}$ such that $v\in{\cal C}^{\operatorname{s}}(x)$ .

A key property of folding manifolds of blender-horseshoes is the following:

Proposition 4.4.

Let ${\cal S}$ be a folding manifold of a blender-horseshoe $\Lambda$ . Then ${\cal S}$ and $W^{\operatorname{s}}_{\operatorname{loc}}(\Lambda)$ are tangent at some point $z$ .

To prove this proposition we need the following lemma:

Lemma 4.5.

Consider a diffeomorphism $f$ having a blender-horseshoe $\Lambda$ as above. The image by $f$ of a folding manifold ${\cal S}$ of $\Lambda$ contains a folding manifold of $\Lambda$ .

Proof: Let us assume, for instance, that the folding manifold ${\cal S}$ is relative to $P$ . We will prove that either $f_{{}_{\cal A}}({\cal S})$ is a folding manifold relative to $P$ or $f_{{}_{\cal B}}({\cal S})$ contains a folding manifold relative to $P$ .

If $f_{{}_{\cal A}}({\cal S})$ is a folding manifold relative to $P$ we are done. So we can assume that $f_{{}_{\cal A}}({\cal S})$ is not a folding manifold. As the ${\operatorname{uu}}$ -disks ${\cal S}_{0}$ and ${\cal S}_{1}$ meet $W^{\operatorname{s}}_{\operatorname{loc}}(P)$ , by Remark 3.7, their images $f_{{}_{\cal A}}({\cal S}_{0})$ and $f_{{}_{\cal A}}({\cal S}_{1})$ are ${\operatorname{uu}}$ -disks intersecting $W^{\operatorname{s}}_{\operatorname{loc}}(P)$ . Furthermore, by item 1 in BH6), the ${\operatorname{uu}}$ -disk $f_{{}_{\cal A}}({\cal S}_{t})$ is at the right of $W^{\operatorname{s}}_{\operatorname{loc}}(P)$ , for every $t\in(0,1)$ .

Since we are assuming that $f_{{}_{\cal A}}({\cal S})$ is not a folding manifold relative to $P$ , by definition of a folding manifold relative to $P$ , there is some $t_{0}\in(0,1)$ such that $f_{{}_{\cal A}}({\cal S}_{t_{0}})$ is not at the left of $W^{\operatorname{s}}_{\operatorname{loc}}(Q)$ (i.e., it is either at the left of $W^{\operatorname{s}}_{\operatorname{loc}}(Q)$ or it meets $W^{\operatorname{s}}_{\operatorname{loc}}(Q)$ ). Thus, by continuity of the disks $f_{{}_{\cal A}}({\cal S}_{t})$ , there is $t_{1}\in(0,t_{0})$ such that $f_{{}_{\cal A}}({\cal S}_{t_{1}})\cap W^{\operatorname{s}}_{\operatorname{loc}}(Q)\neq\emptyset$ . As ${\cal S}_{t_{1}}$ is in between $W^{\operatorname{s}}_{\operatorname{loc}}(P)$ and $W^{\operatorname{s}}_{\operatorname{loc}}(Q)$ , by BH6), the image $f_{{}_{\cal B}}({\cal S}_{t_{1}})$ is in between $W^{\operatorname{s}}_{\operatorname{loc}}(P)$ and $W^{\operatorname{s}}_{\operatorname{loc}}(Q)$ .

By the definition of folding manifold relative to $P$ , every ${\cal S}_{t}$ is at the left of $W^{\operatorname{s}}_{\operatorname{loc}}(Q)$ . Therefore, by item 4 in BH5), $f_{{}_{\cal B}}({\cal S}_{t})$ is a ${\operatorname{uu}}$ -disk at the left of $W^{\operatorname{s}}_{\operatorname{loc}}(Q)$ , for every $t\in(0,1)$ . Moreover, by item 5 in BH5), the images $f_{{}_{\cal B}}({\cal S}_{0})$ and $f_{{}_{\cal B}}({\cal S}_{1})$ are ${\operatorname{uu}}$ -disks at the left of $W^{\operatorname{s}}_{{\operatorname{loc}}}(P)$ .

Now by continuity of the disks $f_{{}_{\cal B}}({\cal S}_{t})$ and since $f_{{}_{\cal B}}({\cal S}_{t_{1}})$ is at the right of $W^{\operatorname{s}}_{\operatorname{loc}}(P)$ ), there are parameters $t_{2}$ and $t_{3}$ , with $t_{2}<t_{1}<t_{3}$ , such that $f_{{}_{\cal B}}({\cal S}_{t_{2}})$ and $f_{{}_{\cal B}}({\cal S}_{t_{3}})$ are ${\operatorname{uu}}$ -disks intersecting $W^{\operatorname{s}}_{{\operatorname{loc}}}(P)$ and $f_{{}_{\cal B}}({\cal S}_{t})$ is a ${\operatorname{uu}}$ -disk at the right of $W^{\operatorname{s}}_{\operatorname{loc}}(P)$ , for all $t\in(t_{2},t_{3})$ . Since, by item 4 in BH5), these disks are at the left of $W^{\operatorname{s}}_{\operatorname{loc}}(Q)$ , they are in between $W^{\operatorname{s}}_{\operatorname{loc}}(P)$ and $W^{\operatorname{s}}_{\operatorname{loc}}(Q)$ . This implies that

\bigcup_{t\in[t_{1},t_{2}]}f_{{}_{\cal B}}({\cal S}_{t})\subset f_{{}_{\cal B}}({\cal S})

is a folding manifold relative to $P$ , ending the proof of the lemma. $\Box$

We are now ready to conclude the proof of Proposition 4.4.

Proof of Proposition 4.4: Write ${\cal S}_{0}={\cal S}$ . By Lemma 4.5, there is a folding manifold ${\cal S}_{1}$ contained in $f({\cal S}_{0})$ . Using Lemma 4.5 and arguing inductively, we define a sequence of folding manifolds $({\cal S}_{i})_{i}$ of the blender-horseshoe $\Lambda$ such that, for every $i\geq 0$ , ${\cal S}_{i+1}$ is contained in $f(S_{i})$ . Let

\tilde{\cal S}_{i}=f^{-i}(S_{i}).

In this way we get a nested sequence $(\tilde{\cal S}_{i})_{i}$ , $\tilde{\cal S}_{i+1}\subset\tilde{\cal S}_{i}\subset{\cal S}$ , of connected and compact sets. Thus, by construction, the intersection set

{\cal S}_{\infty}=\bigcap_{i=0}^{\infty}\tilde{\cal S}_{i}\neq\emptyset

is connected and compact. Moreover, ${\cal S}_{\infty}\subset{\cal S}_{0}$ .

By construction, the whole forward orbit of the set ${\cal S}_{\infty}$ is contained in the reference cube ${\mathbb{C}}$ of the blender. By Remark 4.1, the set ${\cal S}_{\infty}$ is contained in $W^{\operatorname{s}}_{\operatorname{loc}}(\Lambda)$ . Note that, as $\Lambda$ is totally disconnected (a Cantor set) and ${\cal S}_{\infty}$ is connected, hence there is some $z\in\Lambda$ such that

{\cal S}_{\infty}\subset W^{\operatorname{s}}_{\operatorname{loc}}(z).

Since ${\cal S}_{i}$ is a folding manifold for every $i$ , Remark 4.3 implies that there are a point $x_{i}\in{\cal S}_{i}$ and a non-zero vector $v_{i}\in T_{x_{i}}{\cal S}_{i}$ such that $v_{i}\in{\cal C}^{\operatorname{s}}_{\alpha}(x_{i})$ .

Consider the point $\tilde{x}_{i}=f^{-i}(x_{i})\in\tilde{\cal S}_{i}\subset{\cal S}$ and an unitary vector $\tilde{v}_{i}$ parallel to $D_{x_{i}}f^{-i}(v_{i})$ . Note that $\tilde{v}_{i}\in T_{\tilde{x}_{i}}\tilde{\cal S}_{i}$ , thus $\tilde{v}_{i}\in T_{\tilde{x}_{i}}{\cal S}$ . By the $(Df^{-1})$ -invariance of the cone-field ${\cal C}^{\operatorname{s}}_{\alpha}$ , condition BH2), we have that $\tilde{v}_{i}\in{\cal C}^{\operatorname{s}}_{\alpha}(\tilde{x}_{i})$ . We can assume (taking a subsequence if necessary) that

\tilde{x}_{i}\to x_{\infty}\in{\cal S}_{\infty}\subset{\cal S}\cap W^{\operatorname{s}}_{\operatorname{loc}}(\Lambda)\qquad\mbox{and}\qquad\tilde{v}_{i}\to v_{\infty}.

Hence $x_{\infty}\in{\cal S}_{\infty}$ and $x_{\infty}\in W^{\operatorname{s}}_{\operatorname{loc}}(z)$ . Our construction also implies that $v_{\infty}\in T_{x_{\infty}}{\cal S}$ . Finally, also by construction, the vector $v_{\infty}$ belongs to the intersection

\bigcap_{i\geq 0}Df^{-i}({\cal C}^{\operatorname{s}}_{\alpha}(f^{i}(x_{\infty})))=T_{x_{\infty}}W^{\operatorname{s}}_{\operatorname{loc}}(z).

This completes the proof of the proposition. $\Box$

4.2 Robust tangencies

We now return to the problem of robust tangencies in Question 2. We need the following definition.

Definition 4.6 (Folded submanifolds with respect to a blender-horseshoe).

Let $f\colon M\to M$ be a diffeomorphism having a blender-horseshoe $\Lambda$ with reference saddles $P$ and $Q$ and $N\subset M$ be a submanifold of dimension ${\mbox{\rm{ind}}^{\,{\operatorname{u}}}}(\Lambda)$ .

We say that $N$ is folded with respect to $\Lambda$ if the interior of $N$ contains a folding manifold ${\cal S}=({\cal S}_{t})_{t\in[0,1]}$ relative to some reference saddle $A\in\{P,Q\}$ of the blender, here $({\cal S}_{t})_{t\in[0,1]}$ is the family of ${\operatorname{uu}}$ -disks in Definition 4.2, such that:

•

${\cal S}_{0}\cap W^{\operatorname{s}}_{\operatorname{loc}}(A)$ and ${\cal S}_{1}\cap W^{\operatorname{s}}_{\operatorname{loc}}(A)$ are transverse intersection points of $N$ with $W^{\operatorname{s}}_{\operatorname{loc}}(A)$ ;
•

There is $0<\alpha^{\prime}<\alpha$ such that the ${\operatorname{uu}}$ -disks ${\cal S}_{t}$ , $t\in[0,1]$ , are tangent to the cone-field ${\cal C}^{\operatorname{uu}}_{\alpha^{\prime}}$ .

To emphasize the reference saddle $A$ of the blender we consider, we say that the submanifold $N$ is folded with respect to $(\Lambda,A)$ .

Remark 4.7.

A submanifold to be folded with respect to a blender-horseshoe is a $C^{1}$ -open property.

As a direct consequence of Proposition 4.4 and Remark 4.7 one gets:

Theorem 2.

Let $N\subset M$ be a folded submanifold with respect to a blender-horseshoe $\Lambda$ . Then $N$ and $W^{\operatorname{s}}_{\operatorname{loc}}(\Lambda)$ have a $C^{r}$ -robust tangency.

4.3 Robust homoclinic tangencies

In this section we prove that homoclinic tangencies associated to blender-horseshoes yield $C^{r}$ -robust homoclinic tangencies.

Theorem 4.8.

Consider a transitive hyperbolic set $\Sigma$ of a $C^{r}$ -diffeomorphism $f$ containing a ${\operatorname{cu}}$ -blender-horseshoe and a saddle with a homoclinic tangency. Then there is a diffeomorphism $g$ arbitrarily $C^{r}$ -close to $f$ such that the continuation $\Sigma_{g}$ of $\,\Sigma$ has a $C^{r}$ -robust homoclinic tangency.

We need the following lemma.

Lemma 4.9.

Consider a $C^{r}$ -diffeomorphism $f$ with a ${\operatorname{cu}}$ -blender-horseshoe $\Lambda$ . Assume that there is a saddle $R$ with ${\mbox{\rm{ind}}^{\,{\operatorname{u}}}}(R)={\mbox{\rm{ind}}^{\,{\operatorname{u}}}}(\Lambda)$ and such that $W^{\operatorname{u}}(R)$ has a tangency with $W^{\operatorname{s}}(A)$ , where $A$ is a reference saddle of the blender $\Lambda$ . Then there is a diffeomorphism $g$ arbitrarily $C^{r}$ -close to $f$ such that $W^{\operatorname{u}}(R_{g})$ is a folded manifold with respect to the continuation $\Lambda_{g}$ of the blender-horseshoe $\Lambda$ .

By Theorem 2 one gets:

Corollary 5.

In Lemma 4.9, the stable manifold $W^{\operatorname{s}}_{\operatorname{loc}}(\Lambda_{g})$ of the blender-horseshoe and $W^{\operatorname{u}}(R_{g})$ have a $C^{r}$ -robust tangency.

Proof of Lemma 4.9: We suppose that $A=P$ is the left reference saddle of the blender. The proof involves a string of $C^{r}$ -perturbations of the diffeomorphism $f$ . For simplicity, we also denote these perturbations by $f$ .

We begin by noting that the center stable bundle $E^{{\operatorname{cs}}}$ is well defined for every point $x$ in the local stable manifold $W^{\operatorname{s}}_{\operatorname{loc}}(\Lambda)$ . Recall that $W^{\operatorname{s}}_{\operatorname{loc}}(\Lambda)$ is the set of points whose forward orbit remains in the reference cube ${\mathbb{C}}$ of the blender-horseshoe, Remark 4.1. Given a point $x\in W^{\operatorname{s}}_{\operatorname{loc}}(\Lambda)$ , the subspace $E^{{\operatorname{cs}}}(x)$ is the set of vectors $v\in T_{x}M$ such that $Df^{n}(v)\notin({\cal C}^{\operatorname{uu}}(f^{n}(x))\setminus\{0\})$ , for every $n\geq 0$ . The space $E^{\operatorname{cs}}(x)$ has dimension ${\mbox{\rm{ind}}^{\,{\operatorname{s}}}}(\Lambda)+1$ and depends continuously on the point $x\in W^{\operatorname{s}}_{\operatorname{loc}}(\Lambda)$ and on the diffeomorphism $f$ .

First, after considering forward iterations, we can assume that the tangency intersection point $B$ between $W^{\operatorname{u}}(R)$ and $W^{\operatorname{s}}(P)$ is in $W^{\operatorname{s}}_{\operatorname{loc}}(P)$ . Therefore the whole forward orbit of $B$ is the reference cube ${\mathbb{C}}$ . Recall that $P$ is the left reference saddle of the blender-horseshoe, thus $P$ is in the “rectangle” ${\mathbb{A}}$ of the Markov partition. Thus, for any $n\geq 0$ , $f_{{}_{\cal A}}^{n}(B)$ is defined and belongs to ${\mathbb{C}}$ . Hence $E^{\operatorname{cs}}(f^{i}(B))$ is well defined for all $i\geq 0$ (recall the comment above). Note that

\dim(T_{B}W^{\operatorname{u}}(R))+\dim(E^{\operatorname{cs}}(B))=\dim(T_{B}W^{\operatorname{u}}(R))+\dim(T_{B}W^{\operatorname{s}}(P))+1=\dim(M)+1.

Thus after a perturbation, we can assume that $T_{B}W^{\operatorname{u}}(R)$ is transverse to $E^{\operatorname{cs}}(B)$ . Hence there is a subspace ${\mathbb{V}}\subset T_{B}W^{\operatorname{u}}(R)$ of dimension $u$ with ${\mathbb{V}}\oplus E^{\operatorname{cs}}(B)=T_{B}M$ , where $(u+1)={\mbox{\rm{ind}}^{\,{\operatorname{u}}}}(\Lambda)$ .

Consider any $\alpha^{\prime}\in(0,\alpha)$ ( $\alpha$ is the constant in the definition of the cone-fields of the blender). Then, for every $n>0$ large enough, one has that $Df^{n}({\mathbb{V}})$ is contained in the cone ${\cal C}^{\operatorname{uu}}_{\alpha^{\prime}}(f^{n}(B))$ . For simplicity, let us assume that $n=0$ , that is ${\mathbb{V}}\subset{\cal C}^{\operatorname{uu}}_{\alpha^{\prime}}(B)$ . This implies that (up to increase slightly the constant $\alpha^{\prime}<\alpha$ ) there is a small submanifold $\tilde{\cal S}\subset W^{\operatorname{u}}(R)$ such that the point $B$ is in the interior of $\tilde{\cal S}$ , and $\tilde{\cal S}$ is foliated by disks $(\tilde{\cal S}_{t})_{t\in[-1,1]}$ of dimension $u$ tangent to the cone-field ${\cal C}^{\operatorname{uu}}_{\alpha^{\prime}}$ .

Using the expansion by $Df$ in the cone-field ${\cal C}^{\operatorname{uu}}_{\alpha^{\prime}}$ and considering forward iterations of $\tilde{\cal S}$ by $f$ , we get $k\geq 0$ and a submanifold ${\cal S}\subset f^{k}(\tilde{\cal S})$ such that:

•

${\cal S}$ contains $f^{k}(B)$ in its interior and is tangent to $W^{\operatorname{s}}(P)$ at $f^{k}(B)$ , and
•

${\cal S}$ is foliated by ${\operatorname{uu}}$ -disks ${\cal S}_{t}\subset f^{k}(\tilde{\cal S}_{t})$ , $t\in[-1,1]$ , where the disks ${\cal S}_{t}$ are tangent to ${\cal C}^{\operatorname{uu}}_{\alpha^{\prime}}$ .

Again for simplicity, we assume that $k=0$ and that $B\in{\cal S}_{0}$ .

After a new perturbation, we can assume that the contact between ${\cal S}$ and $W^{\operatorname{s}}_{\operatorname{loc}}(P)$ at the point $B$ is quadratic. In particular, there is small $\epsilon>0$ , such that either all the ${\operatorname{uu}}$ -disks ${\cal S}_{t}$ , $t\neq 0$ and $t\in[-\epsilon,\epsilon]$ , are at the left of $W^{\operatorname{s}}_{\operatorname{loc}}(P)$ (case (a)), or all the ${\operatorname{uu}}$ -disks ${\cal S}_{t}$ , $t\neq 0$ and $t\in[-\epsilon,\epsilon]$ , are at the right of $W^{\operatorname{s}}_{\operatorname{loc}}(P)$ (case (b)). So after discarding some disks and reparametrizing the family ${\cal S}_{t}$ , we can assume that $\epsilon=1$ .

\psfrag{p}{$P$}\psfrag{s}{${\cal S}$}\psfrag{text}{perturbation}\psfrag{proj}{projection in ${\mathbb{R}}^{s}\oplus{\mathbb{R}}^{u}$}\includegraphics[height=144.54pt]{casea.eps}

Figure 5: Folded manifold: case (a).

Case (a): for $t\neq 0$ , every ${\cal S}_{t}$ is at the left of $W^{\operatorname{s}}_{\operatorname{loc}}(P)$ . After a new perturbation, we can assume that ${\cal S}$ is a folding manifold relative to $P$ , see Figure 5. Since, by construction, ${\cal S}$ is contained in $W^{\operatorname{u}}(R)$ this concludes the proof in the first case.

Case (b): for $t\neq 0$ every ${\cal S}_{t}$ is at the right of $W^{\operatorname{s}}_{\operatorname{loc}}(P)$ . By considering positive iterations of ${\cal S}$ by $f_{{}_{\cal A}}$ , one gets a large $i>0$ such that $f_{{}_{\cal A}}^{i}({\cal S})$ meets transversely $W^{\operatorname{s}}_{\operatorname{loc}}(Q)$ at some points in $f_{{}_{\cal A}}^{i}({\cal S}_{t_{1}})$ and $f_{{}_{\cal A}}^{i}({\cal S}_{t_{2}})$ , where $t_{1}<0<t_{2}$ . Once again, let us assume that $i=0$ .

\psfrag{p}{$P$}\psfrag{q}{$Q$}\psfrag{s}{${\cal S}$}\psfrag{q}{$Q$}\psfrag{fi}{$f^{i}$}\psfrag{fs}{$f_{{}_{\cal A}}^{i}({\cal S})$}\psfrag{text}{perturbation}\psfrag{proj}{projection in ${\mathbb{R}}^{s}\oplus{\mathbb{R}}^{u}$}\includegraphics[height=173.44756pt]{caseb.eps}

Figure 6: Folded manifold: case (b).

We can choose $t_{1}$ and $t_{2}$ such that the disks ${\cal S}_{t}$ are at the left of $W^{s}_{\operatorname{loc}}(Q)$ for every $t\in(t_{1},t_{2})$ . Recall that, by hypothesis, for all $t\in[t_{1},t_{2}]\setminus\{0\}$ the disks ${\cal S}_{t}$ are at the right of $W^{\operatorname{s}}_{\operatorname{loc}}(P)$ . Therefore, we can perform a final perturbation so that $\hat{\cal S}=\bigcup_{t\in[t_{1},t_{2}]}{\cal S}_{t}$ is a folding manifold relative to $Q$ , see Figure 6. Since $\hat{\cal S}$ is contained in $W^{\operatorname{u}}(R)$ this ends the proof of the lemma. $\Box$

We are now ready to prove Theorem 4.8.

Proof of Theorem 4.8: It is enough to observe that after a $C^{r}$ -perturbation, one can assume that the homoclinic tangency of $\Sigma$ occurs between the unstable manifold of a periodic point $R\in\Sigma$ and the left reference saddle of the blender-horseshoe. $\Box$

5 Generation of blender-horseshoes

In this section we see how blender-horseshoes arise naturally in our non-hyperbolic setting. First, in Section 5.1, we review constructions in [12] providing simple examples of blender-horseshoes. In Section 5.2, using these constructions, we see that partially hyperbolic saddles (saddle-node and flip points) with strong homoclinic intersections (intersections between the strong stable and strong unstable manifolds of a non-hyperbolic saddle) yield blender-horseshoes. Finally, following [9], in Sections 5.3 and 5.4 we prove that co-index one heterodimensional cycles generate blender-horseshoes. We will see in Section 6.2 that co-index one cycles occur naturally in the non-hyperbolic setting.

5.1 Prototypical blender-horseshoes

In this section, we consider a local diffeomorphism $f$ having an affine horseshoe $\Lambda$ with a dominated splitting with three non-trivial bundles, $E^{{\operatorname{s}}}\oplus_{{}_{<}}E^{\operatorname{cu}}\oplus_{{}_{<}}E^{\operatorname{uu}}$ , where $E^{\operatorname{cu}}$ is one dimensional and $E^{\operatorname{u}}=E^{\operatorname{cu}}\oplus E^{\operatorname{uu}}$ is the unstable bundle of $\Lambda$ . We suppose that $\Lambda$ is contained in a hyperplane $\Pi$ tangent to $E^{\operatorname{s}}\oplus E^{{\operatorname{uu}}}$ and that the expansion along the direction $E^{\operatorname{cu}}$ is close to one. Under these assumptions we prove that there are perturbations $g$ of $f$ such that the continuations $\Lambda_{g}$ of $\Lambda$ for $g$ are blender-horseshoes, see Proposition 5.1. We now go to the details of this construction, we borrow from [12].

Let ${\mathbb{D}}=[-1,1]^{n}$ and $n=s+u,\,s,u\geq 1$ . Consider a diffeomorphism $F\colon{\mathbb{R}}^{n}\to{\mathbb{R}}^{n}$ having a horseshoe $\Sigma=\cap_{k\in{\mathbb{Z}}}F^{k}({\mathbb{D}})$ such that:

•

$F^{-1}({\mathbb{D}})\cap{\mathbb{D}}$ consists of two connected components ${\mathbb{D}}_{1}=[-1,1]^{s}\times{\mathbb{U}}_{1}$ and ${\mathbb{D}}_{2}=[-1,1]^{s}\times{\mathbb{U}}_{2}$ , where ${\mathbb{U}}_{1}$ and ${\mathbb{U}}_{2}$ are disjoint topological compact disks of dimension $u$ .
•

The map $F$ is affine on each rectangle ${\mathbb{D}}_{i}$ : there are linear maps $S_{i}\colon{\mathbb{R}}^{s}\to{\mathbb{R}}^{s}$ and $U_{i}\colon{\mathbb{R}}^{u}\to{\mathbb{R}}^{u}$ , $i=1,2$ , such that

$DF|_{{\mathbb{D}}_{i}}=\left(\begin{matrix}S_{i}&0\\ 0&U_{i}\end{matrix}\right),\qquad||S_{i}||,||U_{i}^{-1}||<1/2,\quad i=1,2,$

where $||A||$ is the norm of the linear map $A$ .

We suppose that, in the usual coordinates $(x^{s},x^{u})$ in ${\mathbb{R}}^{n}={\mathbb{R}}^{s}\times{\mathbb{R}}^{u}$ , the fixed saddles of the horseshoe $\Sigma$ are $p=(0^{s},0^{u})\in{\mathbb{D}}_{1}$ and $q=(a^{s},a^{u})\in{\mathbb{D}}_{2}$ .

\psfrag{Ru}{${\mathbb{R}}^{u}$}\psfrag{Rs}{${\mathbb{R}}^{s}$}\psfrag{R}{${\mathbb{R}}$}\psfrag{D1}{${\mathbb{D}}_{1}$}\psfrag{D2}{${\mathbb{D}}_{2}$}\psfrag{I}{$[-1,1]^{n}$}\includegraphics[height=144.54pt]{horseshoe.eps}

Figure 7: An affine horseshoe. The map

f_{\lambda,0}

Consider $\lambda\in(1,2)$ and the family of local diffeomorphisms $(f_{\lambda,\mu})_{\mu\in[-\epsilon,\epsilon]}$ of ${\mathbb{R}}^{n+1}$ given by:

f_{\lambda,\mu}(x^{s},x^{u},x)=\left\{\begin{array}[]{lll}&(F(x^{s},x^{u}),\lambda\,x),&\quad\mbox{if $x^{u}\in{\mathbb{U}}_{1}$},\\ &(F(x^{s},x^{u}),\lambda\,x-\mu),&\quad\mbox{if $x^{u}\in{\mathbb{U}}_{2}$}.\end{array}\right.

By definition, for small $\mu$ , the diffeomorphism $f_{\lambda,\mu}$ has two fixed saddles: $P=(0^{s},0^{u},0)$ (independent of $\lambda$ and $\mu$ ) and $Q_{\lambda,\mu}=(a^{s},a^{u},\mu/(\lambda-1))$ .

Let $\Lambda_{\lambda,0}$ be the maximal invariant set of $f_{\lambda,0}$ in $({\mathbb{D}}_{1}\cup{\mathbb{D}}_{2})\times[-1,1]$ . Note that $\Lambda_{\lambda,0}=\Sigma\times\{0\}$ is a hyperbolic set of $f_{\lambda,0}$ . We say that $\Lambda_{\lambda,0}$ is an affine horseshoe of $f_{\lambda,0}$ with central expansion $\lambda$ . Observe that that the hyperplane ${\mathbb{R}}^{n}\times\{0\}$ is not normally hyperbolic for $f_{\lambda,0}$ .

We denote by $\Lambda_{\lambda,\mu}$ the maximal invariant set of $f_{\lambda,\mu}$ in $({\mathbb{D}}_{1}\cup{\mathbb{D}}_{2})\times[-1,1]$ . For small $\mu$ , the set $\Lambda_{\lambda,\mu}$ is the continuation of $\Lambda_{\lambda,0}$ . More precisely, fixed small $\delta>0$ and the cube ${\mathbb{C}}_{\delta}=[-1,1]^{s}\times[-1,1]^{u}\times[-\delta,\delta]$ , for $|\mu|<(\lambda-1)\,\delta$ , the set $\Lambda_{\lambda,\mu}$ is the maximal invariant set of $f_{\lambda,\mu}$ in ${\mathbb{C}}_{\delta}$ . Clearly, $P,Q_{\lambda,\mu}\in\Lambda_{\lambda,\mu}$ .

Proposition 5.1.

For every $\lambda>1$ close to $1$ and $\mu>0$ , the set $\Lambda_{\lambda,\mu}$ is a blender-horseshoe with reference cube ${\mathbb{C}}_{\delta}$ and reference saddles $P$ and $Q_{\lambda,\mu}$ ( $P$ is the left saddle and $Q_{\lambda,\mu}$ the right one).

In this section, for notational convenience, we write the central coordinates in the third position.

Proof: We fix $\lambda>1$ and $\mu>0$ and we simply write $\Lambda$ , $f$ , $P$ , and $Q$ , omitting the dependence on the parameters. The hyperbolicity of $\Lambda$ follows from the hyperbolicity of $F$ and from the normal expansion by $\lambda>1$ . Consider the constant bundles

E^{\operatorname{s}}=\left({\mathbb{R}}^{s}\times\{(0^{u},0)\}\right),\quad E^{\operatorname{c}}=\left(\{0^{s},0^{u}\}\times{\mathbb{R}}\right),\quad E^{\operatorname{uu}}=\left(\{0^{s}\}\times{\mathbb{R}}^{u}\times\{0\}\right).

Since $\lambda$ is less than $2$ , then

T_{x}{\mathbb{R}}^{n+1}=E^{\operatorname{s}}\oplus_{{}_{<}}E^{\operatorname{c}}\oplus_{{}_{<}}E^{\operatorname{uu}},\quad x\in\Lambda,

is a dominated splitting over $\Lambda$ . Furthermore, as the bundles above are constant, the cone-fields ${\cal C}^{\operatorname{cu}}_{\alpha}$ and ${\cal C}^{\operatorname{uu}}_{\alpha}$ are $Df$ -invariant and ${\cal C}^{\operatorname{s}}_{\alpha}$ is $(Df^{-1})$ -invariant, for every $\alpha\in(0,1)$ . Finally, for small $\alpha$ , ${\cal C}^{\operatorname{s}}_{\alpha}$ is uniformly contracting and ${\cal C}^{\operatorname{u}}_{\alpha}$ is uniformly expanding. This gives condition BH2).

To get conditions BH1) and BH3) just let

{\cal A}=[-1,1]^{s}\times U^{-1}_{1}([-1,1]^{u})\times[-\delta,\delta],\qquad{\cal B}=[-1,1]^{s}\times U^{-1}_{2}([-1,1]^{u})\times[-\delta,\delta],

{\mathbb{A}}=[-1,1]^{s}\times U^{-1}_{1}([-1,1]^{u})\times[-\lambda^{-1}\,\delta,\lambda^{-1}\,\delta],\qquad{\mathbb{B}}=[-1,1]^{s}\times U^{-1}_{2}([-1,1]^{u})\times\left[\frac{-\delta+\mu}{\lambda},\frac{\delta+\mu}{\lambda}\right],

and observe that

f({\mathbb{A}})=S_{1}([-1,1]^{s})\times[-1,1]^{u}\times[-\delta,\delta]\quad\mbox{ and }\quad f({\mathbb{B}})=S_{2}([-1,1]^{s})\times[-1,1]^{u}\times[-\delta,\delta].

\psfrag{Cd}{${\mathbb{C}}_{\delta}$}\psfrag{cB}{${\cal B}$}\psfrag{cA}{${\cal A}$}\psfrag{BB}{${\mathbb{B}}$}\psfrag{AA}{${\mathbb{A}}$}\psfrag{flm}{$f_{\lambda,\mu}$}\includegraphics[height=130.08621pt]{prototypal.eps}

Figure 8: Prototypical blender-horseshoe.

Observe that the local invariant manifolds of $P$ and $Q$ are:

\begin{array}[]{lll}&W^{{\operatorname{s}}}_{{\operatorname{loc}}}(P,f)=[-1,1]^{s}\times\{(0^{u},0)\},&W^{{\operatorname{s}}}_{{\operatorname{loc}}}(Q,f)=[-1,1]^{s}\times\{(a^{u},\frac{\mu}{\lambda-1})\},\\ \\ &W^{{\operatorname{uu}}}_{{\operatorname{loc}}}(P,f)=\{0^{s}\}\times[-1,1]^{u}\times\{0\},&W^{{\operatorname{uu}}}_{{\operatorname{loc}}}(Q,f)=\{a^{s}\}\times[-1,1]^{u}\times\{\frac{\mu}{\lambda-1}\}.\end{array}

It is immediate to check that vertical disks of the form $\{x^{s}\}\times[-1,1]^{u}\times\{x^{c}\}$ satisfy condition BH4). To get BH4) for ${\operatorname{uu}}$ -disks it is enough to take $\alpha\in(0,1)$ small enough in the definition of the cone-fields.

Condition BH5) follows from the fact that $f_{{}_{\cal A}}$ and $f_{{}_{\cal B}}$ are affine maps preserving the dominated splitting and whose center eigenvalue $\lambda$ is positive.

\psfrag{wuuP}{$W^{\operatorname{uu}}_{\operatorname{loc}}(P,f)$}\psfrag{wuuQ}{$W^{\operatorname{uu}}_{\operatorname{loc}}(Q,f)$}\psfrag{de}{$\Delta$}\psfrag{I1}{$I_{1}$}\psfrag{I2}{$I_{2}$}\includegraphics[height=130.08621pt]{onedim.eps}

Figure 9:

{\operatorname{uu}}

-disks and one-dimensional reduction.

It remains to check condition BH6). We first consider vertical disks $\Delta$ parallel to $E^{\operatorname{uu}}$

\Delta=\{x^{s}\}\times[-1,1]^{u}\times\{x^{c}\}

which are in between $W^{{\operatorname{s}}}_{{\operatorname{loc}}}(P,f)$ and $W^{{\operatorname{s}}}_{{\operatorname{loc}}}(Q,f)$ . This means that $x^{c}\in(0,\frac{\mu}{\lambda-1})$ . We consider two cases:

Case 1: $x^{c}\in I_{1}=(0,\frac{\mu}{\lambda\,(\lambda-1)})$ . In this case, one has that

f_{{}_{\cal A}}(\Delta)=\{\bar{x}^{s}\}\times[-1,1]^{u}\times\{\lambda\,x^{c}\}\quad\mbox{where}\quad\lambda\,x^{c}\in\left(0,\frac{\mu}{\lambda-1}\right).

Thus $f_{{}_{\cal A}}(\Delta)$ is in between $W^{{\operatorname{s}}}_{{\operatorname{loc}}}(P,f)$ and $W^{{\operatorname{s}}}_{{\operatorname{loc}}}(Q,f)$ .

Case 2: $x^{c}\in I_{2}=(\frac{\mu}{\lambda},\frac{\mu}{\lambda-1})$ . Note that in this case one gets

f_{{}_{\cal B}}(\Delta)=\{\bar{x}^{s}\}\times[-1,1]^{u}\times\{\lambda\,x^{c}-\mu\}\quad\mbox{where}\quad\lambda\,x^{c}-\mu\in\left(0,\frac{\mu}{\lambda-1}\right).

Hence $f_{{}_{\cal B}}(\Delta)$ is in between $W^{{\operatorname{s}}}_{{\operatorname{loc}}}(P,f)$ and $W^{{\operatorname{s}}}_{{\operatorname{loc}}}(Q,f)$ .

This completes the proof of BH6) for disks parallel to $E^{\operatorname{uu}}$ .

We next consider general ${\operatorname{uu}}$ -disks. We begin with two claims.

Claim 5.2.

Consider $\tau_{1}<\frac{\mu}{\lambda\,(\lambda-1)}$ and any ${\operatorname{uu}}$ -disk $\Delta$ (i.e. tangent to the cone-field ${\cal C}^{\operatorname{uu}}_{\alpha}$ ) through a point $(x^{s},x^{u},x^{c})$ with $x^{c}\leq\tau_{1}$ . Then, for every $\alpha\in(0,1)$ small enough,

•

$f_{{}_{\cal A}}(\Delta)$ is at the left of $W^{\operatorname{s}}_{\operatorname{loc}}(Q)$ ,
•

assume that $\Delta$ is at the right of $W^{\operatorname{s}}_{\operatorname{loc}}(P)$ , then $f_{{}_{\cal A}}(\Delta)$ is in between $W^{{\operatorname{s}}}_{{\operatorname{loc}}}(P,f)$ and $W^{{\operatorname{s}}}_{{\operatorname{loc}}}(Q,f)$ .

Proof: The first statement follows from the compactness of the set of vertical disks with $x^{c}\leq\tau_{1}$ and the uniform convergence in $\alpha$ of the ${\operatorname{uu}}$ -disks to the vertical disks (parallel to $E^{\operatorname{uu}}$ ) as $\alpha\to 0$ .

From the first part of the claim we know that $f_{{}_{\cal A}}(\Delta)$ is at the left of $W^{{\operatorname{s}}}_{{\operatorname{loc}}}(Q,f)$ . It remains to check that $f_{{}_{\cal A}}(\Delta)$ is at the right of $W^{{\operatorname{s}}}_{{\operatorname{loc}}}(P,f)$ . The disk $\Delta$ contains a point of the form $(x^{s},0^{u},x^{c})$ with $x^{c}>0$ . Thus $f_{{\cal A}}(\Delta)$ contains the point $(S_{1}(x^{s}),0^{u},\lambda\,x^{c})$ . This implies that $f_{{\cal A}}(\Delta)$ is at the right of $W^{{\operatorname{s}}}_{{\operatorname{loc}}}(P,f)$ . $\Box$

Arguing as above and using Case 2, one also deduces the following:

Claim 5.3.

Consider $\tau_{2}>\frac{\mu}{\lambda}$ and any ${\operatorname{uu}}$ -disk $\Delta$ through a point $(x^{s},x^{u},x^{c})$ with $x^{c}\geq\tau_{2}$ . Then, for every $\alpha\in(0,1)$ small enough,

•

$f_{{}_{\cal B}}(\Delta)$ is at the right of $W^{\operatorname{s}}_{\operatorname{loc}}(P)$ ,
•

assume that $\Delta$ is at the left of $W^{\operatorname{s}}_{\operatorname{loc}}(Q)$ , then $f_{{}_{\cal B}}(\Delta)$ is in between $W^{{\operatorname{s}}}_{{\operatorname{loc}}}(P,f)$ and $W^{{\operatorname{s}}}_{{\operatorname{loc}}}(Q,f)$ .

To get condition BH6), note that, since that $(\lambda-1)\in(0,1)$ , one has $\frac{\mu}{\lambda}<\frac{\mu}{\lambda\,(\lambda-1)}$ . Now it is enough to note that, for $\alpha\in(0,1)$ small enough, for every point $(x^{s},x^{u},x^{c})$ in a ${\operatorname{uu}}$ -disk $\Delta$ in between $W^{{\operatorname{s}}}_{{\operatorname{loc}}}(P,f)$ and $W^{{\operatorname{s}}}_{{\operatorname{loc}}}(Q,f)$ one has:

•

either $x^{c}<\frac{\mu}{\lambda\,(\lambda-1)}$ and then, by Claim 5.2, $f_{{}_{\cal A}}(\Delta)$ is in between $W^{{\operatorname{s}}}_{{\operatorname{loc}}}(P,f)$ and $W^{{\operatorname{s}}}_{{\operatorname{loc}}}(Q,f)$ ,
•

or $x^{c}>\frac{\mu}{\lambda}$ and then, by Claim 5.3, $f_{{}_{\cal B}}(\Delta)$ is in between $W^{{\operatorname{s}}}_{{\operatorname{loc}}}(P,f)$ and $W^{{\operatorname{s}}}_{{\operatorname{loc}}}(Q,f)$ .

We have checked that the set $\Lambda$ satisfies conditions BH1)–BH6). Therefore the set $\Lambda$ is a blender-horseshoe and the proof of Proposition 5.1 is complete. $\Box$

5.2 Strong homoclinic intersections and generation of blender-horseshoes

In this section, we state the generation of blender-horseshoes by saddle-node and flip periodic points with strong homoclinic intersections.

Let $f$ be a diffeomorphism with a periodic point $S$ such that the tangent bundle of $M$ at $S$ has a $Df^{\pi(S)}$ -invariant dominated splitting $T_{S}M=E^{{\operatorname{ss}}}\oplus_{{}_{<}}E^{\operatorname{c}}\oplus_{{}_{<}}E^{\operatorname{uu}}$ , where $E^{{\operatorname{ss}}}$ is uniformly contracting, $E^{\operatorname{uu}}$ is uniformly expanding, and $E^{\operatorname{c}}$ is one-dimensional. We say that $S$ is a saddle-node (resp. flip) if the eigenvalue of $Df^{\pi(S)}$ corresponding to the (one-dimensional) central direction $E^{c}$ is $1$ (resp. $-1$ ).

Consider the strong stable and unstable manifolds of the orbit of $S$ (denoted by $W^{\operatorname{ss}}(S)$ and $W^{\operatorname{uu}}(S)$ ). We say that the $S$ has a strong homoclinic intersection if there is a point $X\in W^{\operatorname{ss}}(S)\cap W^{\operatorname{uu}}(S)$ such that $X\neq f^{i}(S)$ for all $i$ . The point $X$ is a strong homoclinic point of $f$ associated to $S$ .

Let $f$ be a diffeomorphism with a strong homoclinic intersection associated to a saddle-node $S$ . In [9, Section 4.1] it is shown that there are $k\geq 1$ and a $C^{1}$ -perturbation $g$ of $f$ such that $g^{k}$ has an a affine horseshoe $\Lambda$ associated to $S$ whose central expansion is arbitrarily close to $1$ (recall Section 5.1). Considering perturbations similar to the ones in Proposition 5.1, [9, Sections 4.1.1-2] gives the following:

Proposition 5.4.

Consider a diffeomorphism $f$ having a strong homoclinic intersection associated to a saddle-node $S$ . There is a diffeomorphism $g$ arbitrarily $C^{1}$ -close to $f$ with a ${\operatorname{cu}}$ -blender-horseshoe having $S$ as a reference saddle.

The same statement holds for ${\operatorname{cs}}$ -blender-horseshoes, i.e. ${\operatorname{cu}}$ -blender-horseshoes for $f^{-1}$ .

We observe that in [9] the terminology blender-horseshoe it is not used. However, the constructions in [9] provide prototypical blender-horseshoes exactly as the ones in Section 5.1. In fact, these constructions are the main motivation (and model) for our definition of blender-horseshoe. Thus Proposition 5.4 just reformulates some results in [9] using this new terminology of blender-horseshoes.

For diffeomorphisms having flip points we need the following lemma (see [9, Remark 4.6]):

Lemma 5.5.

Consider a diffeomorphism $f$ having a strong homoclinic intersection associated to a flip point $S$ . There is a diffeomorphism $g$ arbitrarily $C^{1}$ -close to $f$ having a saddle node $S^{\prime}$ with a strong homoclinic intersection and such that the orbit of $S^{\prime}$ remains in an arbitrarily small neighborhood of the orbit of the initial flip point $S$ .

Proof: Consider a $2$ -parameter family of deformations $f_{s,t}$ of $f=f_{0,0}$ such that

•

the parameter $s$ corresponds to a (non-generic) unfolding of the flip, generating a saddle-node $S^{\prime}$ close to the flip $S$ of period twice the period of $S$ , and
•

the parameter $t$ corresponds to the unfolding of the strong homoclinic intersection of the flip $S$ , the local strong stable manifold of $S$ “passing from the left to the right” of the local unstable manifold of $S$ .

Then for every $s\neq 0$ small enough, there is a small parameter $t=t(s)$ , $t(s)\to 0$ as $s\to 0$ , such that the saddle-node $S^{\prime}$ has a strong homoclinic intersection. $\Box$

5.3 Co-index one cycles and blender horseshoes

In this section, we borrow some arguments and results from [2, 9] in order to prove that diffeomorphisms with co-index one heterodimensional cycles yield blender-horseshoes.

Proposition 5.6.

Let $f$ be a diffeomorphism with a heterodimensional cycle associated to saddles $P$ and $Q$ with ${\mbox{\rm{ind}}^{\,{\operatorname{s}}}}(P)={\mbox{\rm{ind}}^{\,{\operatorname{s}}}}(Q)+1$ . Then there is $g$ arbitrarily $C^{1}$ -close to $f$ with a saddle $R$ such that:

1.

${\mbox{\rm{ind}}^{\,{\operatorname{s}}}}(R)={\mbox{\rm{ind}}^{\,{\operatorname{s}}}}(Q)$ , the orbit of $R$ has a dominated splitting $E^{\operatorname{ss}}\oplus_{{}_{<}}E^{\operatorname{c}}\oplus_{{}_{<}}E^{\operatorname{uu}}$ with three non-trivial bundles such that $E^{\operatorname{ss}}$ and $E^{\operatorname{uu}}$ are uniformly contracting and expanding, respectively, $\dim(E^{\operatorname{ss}})={\mbox{\rm{ind}}^{\,{\operatorname{s}}}}(Q)$ , and $\dim(E^{\operatorname{c}})$ has dimension one and is expanding,
2.

there is ${\operatorname{cu}}$ -blender-horseshoe of $g$ having $R$ as a reference saddle,
3.

$W^{\operatorname{s}}(R)$ intersects transversely $W^{\operatorname{u}}(Q)$ , and
4.

$W^{\operatorname{uu}}(R)$ meets transversely $W^{\operatorname{s}}(P)$ .

The arguments for proving this proposition can be found scattered along several constructions in [9]. But, unfortunately, this result is not stated explicitly there and its prove involves some adaptations of the constructions in [9]. As the proof of Proposition 5.6 is somewhat technical, we next explain the sequence of arguments we borrow from [9] and their adaptations in order to prove this proposition. The proof of Proposition 5.6 consists of several reductions to simpler cases we proceed to explain. Let us begin with a definition.

Definition 5.7 (Strong-intermediate saddles).

Let $f$ be a diffeomorphism having two periodic saddles $P$ and $Q$ of indices ${\mbox{\rm{ind}}^{\,{\operatorname{s}}}}(P)={\mbox{\rm{ind}}^{\,{\operatorname{s}}}}(Q)+1$ . A periodic point $R$ is strong-intermediate with respect to $P$ and $Q$ , denoted by $Q\prec_{{\operatorname{u}},{\operatorname{ss}}}R\prec_{{\operatorname{uu}},{\operatorname{s}}}P$ , if:

•

the orbit of $R$ is partially hyperbolic and has a dominated splitting $E^{\operatorname{ss}}\oplus_{{}_{<}}E^{\operatorname{c}}\oplus_{{}_{<}}E^{\operatorname{uu}}$ with three non-trivial bundles such that $E^{\operatorname{ss}}$ and $E^{\operatorname{uu}}$ are uniformly contracting and expanding, $\dim(E^{\operatorname{ss}})={\mbox{\rm{ind}}^{\,{\operatorname{s}}}}(Q)$ , and $\dim(E^{\operatorname{c}})=1$ ,

•

the strong stable manifold of $R$ meets transversely the unstable manifold of $Q$ and the strong unstable manifold of $R$ meets transversely the stable manifold of $P$ , in a formula,

W^{\operatorname{ss}}(R)\pitchfork W^{\operatorname{u}}(Q)\neq\emptyset\quad\mbox{and}\quad W^{\operatorname{uu}}(R)\pitchfork W^{\operatorname{s}}(P)\neq\emptyset.

Note that if a (hyperbolic) saddle $R$ with ${\mbox{\rm{ind}}^{\,{\operatorname{s}}}}(R)={\mbox{\rm{ind}}^{\,{\operatorname{s}}}}(Q)$ is strong-intermediate with respect to $P$ and $Q$ then it satisfies items 3 and 4 in Proposition 5.6.

We need the following lemma:

Lemma 5.8.

Consider two saddles $P$ and $Q$ in the same chain recurrence class $C$ and a periodic point $R$ which is strong-intermediate with respect to $P$ and $Q$ . Then $R\in C$ .

Proof: We construct a pseudo-orbit going from $R$ to $P$ (the other pseudo-orbits are obtained similarly). Take a point $X\in W^{\operatorname{ss}}(R)\cap W^{\operatorname{u}}(Q)$ . Note that there are arbitrarily large $n$ and $m$ such that $\{f^{-n}(X),\dots,f^{m}(X)\}$ is a segment of orbit starting (arbitrarily) close to $R$ and ending close to $Q$ . Since $Q$ and $P$ are in the same chain recurrent class, there is a finite pseudo-orbit going from $Q$ to $P$ . A pseudo-orbit going from $R$ to $P$ is obtained concatenating these two pseudo-orbits. This concludes the sketch of the proof of the lemma. $\Box$

We now explain the generation of strong-intermediate saddles.

5.3.1 Reduction to the case of cycles associated to saddles with real central eigenvalues

Given a periodic point $R$ of a diffeomorphism $f$ , write $\lambda_{1}(R),\dots,\lambda_{n}(R)$ the eigenvalues of $Df^{\pi(R)}(R)$ counted with multiplicity and ordered in increasing modulus ( $|\lambda_{i}(R)|\leq|\lambda_{i+1}(R)|$ ). We say that $\lambda_{i}(R)$ is the $i$ -th multiplier of $R$ .

Consider a diffeomorphism $f$ having a co-index one cycle associated to period points $A$ and $B$ with ${\mbox{\rm{ind}}^{\,{\operatorname{s}}}}(P)={\mbox{\rm{ind}}^{\,{\operatorname{s}}}}(Q)+1=s+1$ . The cycle has real central eigenvalues if $\lambda_{s+1}(A)$ and $\lambda_{s+1}(B)$ are both real and

|\lambda_{s}(A)|<|\lambda_{s+1}(A)|<1<|\lambda_{s+2}(A)|\quad\mbox{and}\quad|\lambda_{s}(B)|<1<|\lambda_{s+1}(B)|<|\lambda_{s+2}(B)|.

Before proving Proposition 5.6, we recall the following two facts:

: Fact 1: [9, Theorem 2.1] claims that, if $f$ has a co-index one cycle associated to $A$ and $B$ then there is $g$ arbitrarily $C^{1}$ -close to $f$ with a co-index one cycle with real central eigenvalues. Moreover, this cycle can be chosen associated to saddles $A^{\prime}_{g}$ and $B^{\prime}_{g}$ homoclinically related to the continuations $A_{g}$ and $B_{g}$ of $A$ and $B$ , respectively.
: Fact 2: Assume that there is a diffeomorphism $h$ arbitrarily $C^{1}$ -close to $g$ having a ${\operatorname{cu}}$ -blender-horseshoe $\Lambda$ with a reference saddle $R_{h}$ which is strong-intermediate to $A^{\prime}_{h}$ and $B^{\prime}_{h}$ . Since two saddles being homoclinically related is a $C^{1}$ -robust relation, one has $R_{h}$ is strong-intermediate with respect to $A_{h}$ and $B_{h}$ . In this case, the proof of Proposition 5.6 is complete.

In view of these two facts, to prove Proposition 5.6 it is enough to consider the case where the saddles $P$ and $Q$ in the cycle have real central eigenvalues and to check that these cycles generate strong-intermediate saddles as in Fact 2, see Proposition 5.9. We now go to the details of this construction.

5.3.2 Reduction to the generation of saddle-node or flip points

In this section, we show that Proposition 5.6 is a consequence of the following result.

Proposition 5.9.

Let $f$ be a diffeomorphism with a co-index one cycle associated to saddles $P$ and $Q$ with real central eigenvalues. Then there is $g$ arbitrarily $C^{1}$ -close to $f$ having a saddle-node or flip periodic point $R_{g}$ such that:

•

$R_{g}$ has a strong homoclinic intersection and
•

$R_{g}$ is strong-intermediate to $P_{g}$ and $Q_{g}$ .

This proposition is a stronger version of [9, Theorem 2.3], adding the strong-intermediate property of $R$ with respect to $P$ and $Q$ .

Proposition 5.9 implies Proposition 5.6: First, observe that the strong unstable and strong stable manifolds of $R_{g}$ depend continuously on the diffeomorphism (while there is defined a continuation of $R_{g}$ ).

If $R=R_{g}$ is a saddle-node then Proposition 5.4 gives a diffeomorphism $h$ arbitrarily $C^{1}$ -close to $g$ (thus arbitrarily close to $f$ ) with a blender-horseshoe having $R$ as a reference saddle. Therefore, for $h$ close enough to $g$ , one gets the announced intersections between the strong invariant manifolds (intermediate intersections).

If $R$ is a flip then Lemma 5.5 gives a perturbation $h$ of $g$ with a saddle-node with a strong homoclinic intersection and with the strong-intermediate property. Thus we are in the first case. This completes the proof of our claim. $\Box$

Therefore it is enough to prove Proposition 5.9. The proof of the proposition is similar to the one of [9, Theorem 2.3] and consists of several steps. We next explain and adapt these steps.

5.3.3 Reduction to the creation of weak hyperbolic saddles

We now see that Proposition 5.9 (hence of Proposition 5.6) follows from:

Proposition 5.10.

Let $f$ be a diffeomorphism having a co-index one cycle associated to saddles $P$ and $Q$ , ${\mbox{\rm{ind}}^{\,{\operatorname{s}}}}(P)={\mbox{\rm{ind}}^{\,{\operatorname{s}}}}(Q)+1$ , with real central eigenvalues. Then there are a constant $C>1$ and a sequence of diffeomorphisms $(f_{n})$ , $f_{n}\stackrel{{\scriptstyle C^{1}}}{{\longrightarrow}}f$ , such that every $f_{n}$ has a periodic point $R_{n}$ such that:

•

$R_{n}$ has a one-dimensional center-unstable direction whose corresponding multiplier $\lambda_{n}^{c}$ satisfies $|\lambda_{n}^{c}|\in[\frac{1}{C},C]$ ,
•

$W^{\operatorname{uu}}(R_{n})$ and $W^{\operatorname{ss}}(R_{n})$ have a quasi-transverse intersection, therefore $R_{n}$ has strong homoclinic intersections
•

the periods of $R_{n}$ go to infinity as $n\to\infty$ , and
•

$R_{n}$ is strong-intermediate with respect to $P_{n}$ and $Q_{n}$ (the continuations of $P$ and $Q$ for $f_{n}$ ).

This proposition is a stronger version [9, Proposition 3.3], adding the intersection property of the strong invariant manifolds.

Proposition 5.10 implies Proposition 5.9: We proceed exactly as in [9, page 484] (proof of Theorem 2.3 using Proposition 3.3). We just perform a local $C^{1}$ -perturbation of $f_{n}$ supported in a small neighborhood of $R_{n}$ , turning the the central eigenvalue of $R_{n}$ equal to $\pm 1$ while keeping the strong homoclinic point of $R_{n}$ and the transverse intersections $W^{\operatorname{ss}}(R_{n})\pitchfork W^{\operatorname{u}}(Q_{n})\neq\emptyset$ and $W^{\operatorname{uu}}(R_{n})\pitchfork W^{\operatorname{s}}(P_{n})\neq\emptyset$ .

In this way, we get diffeomorphisms $g$ (arbitrarily close to $f$ ) with saddle-node or flip points $R_{g}$ (depending if $\lambda_{n}^{c}$ is positive or negative) with strong homoclinic intersections and being strong-intermediate with respect to $P_{g}$ and $Q_{g}$ . This completes the proof of Proposition 5.9. $\Box$

5.4 Proof of Proposition 5.10

The following steps of the proof of [9, Proposition 3.3] are described in [9, page 484]:

: Step 1: One first puts the cycle in a kind of normal form called simple cycle. In fact, [9, Proposition 3.5] implies that, after a $C^{1}$ -perturbation, one can assume that the cycle is simple.
: Step 2: One shows that the dynamics in a simple cycle is given (up to a renormalization) by a model family, denoted by $F^{\pm,\pm}_{\lambda,\beta,t}$ . Moreover, perturbations of this model family correspond to perturbations of the initial dynamics.

Therefore, to prove Proposition 5.10, it is enough to consider model families $F^{\pm,\pm}_{\lambda,\beta,t}$ and their perturbations. Hence it is enough to adapt the perturbations of these normal families in order to get the intersection properties between the strong invariant manifolds.

Proposition 3.8 in [9] claims that the unfolding of co-index one cycles generates sequences of saddles $A_{n,m}$ whose orbits are contained in a neighborhood of the cycle. We now see that these saddles can be taken with the strong-intermediate property (relative to saddles in the initial cycle).

Lemma 5.11.

In [9, Proposition 3.8], for every integer $n,m$ large enough, (in fact larger that the integer $N$ in the statement), all periodic points $A_{n,m}$ in the proposition are strong-intermediate with respect to $P$ and $Q$ .

Note that we have the following string of implications:

\mbox{Lemma~\ref{l.intermediate}}\Rightarrow\mbox{Proposition~\ref{p.weak}}\Rightarrow\mbox{Proposition~\ref{p.saddleblenders}}\Rightarrow\mbox{Proposition~\ref{p.coindexblenders}}.

Therefore to prove Proposition 5.6 it remains to prove Lemma 5.11.

Proof: The model maps $F^{\pm,\pm}_{\lambda,\beta,t}$ are defined on some cubes and their restrictions to each of these cubes are affine maps ${\cal A}_{\lambda}$ , ${\cal B}_{\beta}$ , ${\cal T}^{\pm}_{1,t}$ , and ${\cal T}^{\pm}_{2}$ which preserve a constant dominated splitting: the strong stable bundle is the horizontal space ${\mathbb{R}}^{s}\times\{(0,0^{u})\}$ , the strong unstable bundles is the vertical one $\{(0^{s},0)\}\times{\mathbb{R}}^{u}$ , and the center bundle is one dimensional $\{0^{s}\}\times{\mathbb{R}}\times\{0^{u}\}$ . More precisely (see Figure 10):

\psfrag{Yp}{$Y$}\psfrag{Xp}{$f^{N_{2}}(X)$}\psfrag{Yq}{$f^{N_{1}}(Y)$}\psfrag{Xq}{$X$}\psfrag{Yq}{$f^{N_{1}}(Y)$}\psfrag{Xq}{$X$}\psfrag{A}{${\cal A}_{\lambda}$}\psfrag{B}{${\cal B}_{\beta}$}\psfrag{Es}{$E^{ss}$}\psfrag{Eu}{$E^{uu}$}\psfrag{Ec}{$E^{c}$}\psfrag{P}{$P$}\psfrag{Q}{$Q$}\psfrag{T1}{${\mathfrak{T}}_{1,t}$}\psfrag{T2}{${\mathfrak{T}}_{2}$}\includegraphics[height=130.08621pt]{transitions.eps}

Figure 10: The model maps

F^{\pm,\pm}_{\lambda,\beta,t}

•

The maps ${\cal A}_{\lambda}$ and ${\cal B}_{\beta}$ are $Df^{\pi(P)}(P)$ and $Df^{\pi(Q)}(Q)$ and correspond to iterates of $f$ close to $P$ and $Q$ , respectively.
•

The sub-scripts $\lambda$ and $\beta$ correspond to the central multipliers of $Df^{\pi(P)}(P)$ and $Df^{\pi(Q)}(Q)$ .
•

The maps ${\cal T}^{\pm}_{1,t}$ and ${\cal T}^{\pm}_{2}$ are the transitions of the cycle. The map ${\cal T}^{\pm}_{2}$ corresponds to a fixed number $N_{2}$ of iterates from a neighborhood of $Q$ to a neighborhood of $P$ following a segment of orbit of a transverse heteroclinic point $X$ in $W^{\operatorname{s}}(P)\pitchfork W^{\operatorname{u}}(Q)$ .

Similarly, the map ${\cal T}^{\pm}_{1,t}$ corresponds to a fixed number $N_{1}$ of iterates from a neighborhood of $P$ to a neighborhood of $Q$ following a segment of orbit of a fixed quasi-transverse heteroclinic point $Y$ in $W^{\operatorname{u}}(P)\pitchfork W^{\operatorname{s}}(Q)$ . The parameter $t$ of ${\cal T}^{\pm}_{1,t}$ corresponds to the unfolding of the cycle.
•

The super-script $\pm$ is positive if the transition map preserves the orientation in the central bundle and negative if otherwise.

For details see [9, page 488].

\psfrag{Yp}{$Y$}\psfrag{Xp}{$f^{N_{2}}(X)$}\psfrag{Yq}{$f^{N_{1}}(Y)$}\psfrag{Xq}{$X$}\psfrag{Yq}{$f^{N_{1}}(Y)$}\psfrag{Xq}{$X$}\psfrag{A}{${\cal A}_{\lambda}$}\psfrag{amn}{$A_{m,n}$}\psfrag{B}{${\cal B}_{\beta}$}\psfrag{Es}{$E^{ss}$}\psfrag{Eu}{$E^{uu}$}\psfrag{Ec}{$E^{c}$}\psfrag{wsp}{$W^{s}(P)$}\psfrag{wuq}{$W^{u}(Q)$}\psfrag{P}{$P$}\psfrag{Q}{$Q$}\psfrag{T1}{${\mathfrak{T}}_{1,t}$}\psfrag{T2}{${\mathfrak{T}}_{2}$}\includegraphics[height=144.54pt]{amn.eps}

Figure 11: The intermediate saddle

A_{m,n}

By definition, the points $A_{m,n}=(a^{s},a,a^{u})$ are fixed points of the composition ${\cal B}_{\beta}^{n}\circ{\cal T}^{\pm}_{1,t}\circ{\cal A}_{\lambda}^{m}\circ{\cal T}^{\pm}_{2}$ . In particular, the point $A_{m,n}$ belongs to domain of definition $\Sigma_{Q}$ of ${\cal T}^{\pm}_{2}$ , see Figure 11. This domain (defined in [9, page 488]) is the cube

\Sigma_{Q}=[-1,1]^{s}\times[b_{Q}-\delta,b_{Q}+\delta]\times[-1,1]^{u},

where

[-1,1]^{s}\times[b_{Q}-\delta,b_{Q}+\delta]\times\{0^{u}\}\subset W^{\operatorname{s}}(P)\quad\mbox{and}\quad\{0^{s}\}\times[b_{Q}-\delta,b_{Q}+\delta]\times[-1,1]^{u}\subset W^{\operatorname{u}}(Q).

Moreover, the local strong stable and strong unstable manifolds of the point $A_{m,n}=(a^{s},a,a^{u})$ are (see [9, page 495, first paragraph]):

W^{\operatorname{ss}}_{\operatorname{loc}}(A_{m,n})=[-1,1]^{s}\times\{(a,a^{u})\}\quad\mbox{and}\quad W^{\operatorname{uu}}_{\operatorname{loc}}(A_{m,n})=\{(a^{s},a)\}\times[-1,1]^{u}.

This means that

W^{\operatorname{ss}}(A_{m,n})\pitchfork W^{\operatorname{u}}(Q)\neq\emptyset\quad\mbox{and}\quad W^{\operatorname{uu}}_{\operatorname{loc}}(A_{m,n})\pitchfork W^{\operatorname{s}}(P)\neq\emptyset.

Therefore the saddle $A_{m,n}$ is strong-intermediate with respect to $P$ and $Q$ , ending the proof of the lemma. $\Box$

6 Robust tangencies and heterodimensional cycles for $C^{1}$ -generic diffeomorphisms

In this section, we prove Theorem 1. First, in Section 6.1, we state some properties about $C^{1}$ -generic diffeomorphisms. In Section 6.2, we state the $C^{1}$ -generic occurrence of blender-horseshoes in homoclinic classes containing saddles of different indices (Theorem 6.4). Finally, in Section 6.3, we state the existence of robust homoclinic tangencies inside homoclinic classes with index variation and lack of domination (Proposition 6.11), completing the proof of Theorem 1. We close this paper presenting and extension of [9, Theorem 1.16] about the occorrence of robust heterodimensional cycles inside chain recurrence classes (see Theorem 3 in Section 6.4).

6.1 $C^{1}$ -generic properties of $C^{1}$ -diffeomorphisms

We now collect some properties of $C^{1}$ -generic diffeomorphisms. According to [7, Remarque 1.10] and [2, Theorem 1], there is a residual subset set ${\cal G}$ of $\mbox{{\rm Diff\,}}^{1}(M)$ such that, for every $f\in{\cal G}$ ,

•

every periodic point of $f$ is hyperbolic,
•

for every periodic point $P$ of $f$ , its homoclinic class $H(P,f)$ and its chain recurrence class $C(P,f)$ are equal,
•

any homoclinic class $H(P,f)$ containing periodic points of ${\operatorname{u}}$ -indices $\alpha$ and $\beta$ also contains saddles of ${\operatorname{u}}$ -index $\tau$ , for every $\tau\in[\alpha,\beta]\cap{\mathbb{N}}$ .

Lemma 6.1.

([2, Lemma 2.1]) There is a residual subset ${\cal G}_{0}\subset{\cal G}$ of $\mbox{{\rm Diff\,}}^{1}(M)$ such that, for every $f\in{\cal G}_{0}$ and every pair of periodic points $P_{f}$ and $Q_{f}$ of $f$ , there is a neighborhood ${\cal U}_{f}$ of $f$ in ${\cal G}_{0}$ such that:

•

either $H(P_{g},g)=H(Q_{g},g)$ for all $g\in{\cal U}_{f}\cap{\cal G}_{0}$ ,
•

or $H(P_{g},g)\cap H(Q_{g},g)=\emptyset$ for all $g\in{\cal U}_{f}\cap{\cal G}_{0}$ .

Remark 6.2 (Proof of Claim 2.2 in [2]).

Using a filtration given by Conley theory, one has that the property of two hyperbolic saddles to be in different chain recurrence classes is $C^{1}$ -robust.

Next lemma claims that, for $C^{1}$ -generic diffeomorphisms, the property of the chain recurrence classes of two periodic points to be equal is also a $C^{1}$ -robust property.

Lemma 6.3.

Let ${\cal G}_{0}$ be the residual set of of $\mbox{{\rm Diff\,}}^{1}(M)$ in Lemma 6.1. For every $f\in{\cal G}_{0}$ and every pair of periodic points $P_{f}$ and $Q_{f}$ of $f$ , the property of $Q_{f}$ belonging to the chain recurrence class $C(P_{f},f)$ of $P_{f}$ is $C^{1}$ -robust: if $Q_{f}\in C(P_{f},f)$ then $Q_{g}\in C(P_{g},g)$ for all $g$ $C^{1}$ -close to $f$ .

Proof: Let $f\in{\cal G}_{0}$ and suppose that $Q_{f}\in C(P_{f},f)$ . Since $f\in{\cal G}$ one has that $C(P_{f},f)=H(P_{f},f)$ . As $f\in{\cal G}_{0}$ , by Lemma 6.1, there is a $C^{1}$ -open neighborhood ${\cal U}_{f}$ of $f$ such that $H(P_{g},g)=H(Q_{g},g)$ for every $g\in{\cal U}_{f}\cap{\cal G}_{0}$ . In particular, $Q_{g}\in H(P_{g},g)$ for all $g\in{\cal U}_{f}\cap{\cal G}_{0}$ .

Assume that there is $g\in{\cal U}_{f}$ such that $Q_{g}\notin C(P_{g},g)$ . By Remark 6.2, one has that $Q_{h}\notin C(P_{h},h)$ for every $h$ in a small neighborhood ${\cal V}_{g}$ of $g$ contained in ${\cal U}_{f}$ . Choosing $h\in{\cal G}_{0}\cap{\cal V}_{g}$ one gets that $Q_{h}\not\in C(P_{h},h)=H(P_{h},h)$ , a contradiction. $\Box$

6.2 Generation of blender-horseshoes

In this section, we prove that blender-horseshoes occur $C^{1}$ -generically for homoclinic classes with index variability (i.e., containing saddles with different indices).

Theorem 6.4.

There is a residual subset ${\cal R}$ of $\mbox{{\rm Diff\,}}^{1}(M)$ of diffeomorphisms $f$ such that for every homoclinic class $H(P,f)$ containing a hyperbolic saddle $Q$ with ${\mbox{\rm{ind}}^{\,{\operatorname{s}}}}(Q)>{\mbox{\rm{ind}}^{\,{\operatorname{s}}}}(P)$ there is a transitive hyperbolic set $\Sigma$ containing $P$ and a ${\operatorname{cu}}$ -blender-horseshoe $\Lambda$ .

Applying Theorem 6.4 to $f^{-1}$ one gets the following.

Remark 6.5.

Under the assumptions of Theorem 6.4, every $C^{1}$ -generic diffeomorphism $f$ has a ${\operatorname{cs}}$ -blender-horseshoe containing $Q$ and contained in a transitive hyperbolic set.

We first prove a version of Theorem 6.4 for a given fixed saddle $P_{f}$ :

Proposition 6.6.

Let ${\cal U}$ be an open subset of $\mbox{{\rm Diff\,}}^{1}(M)$ and $f\mapsto P_{f}$ be a continuous map defined on ${\cal U}$ associating to each $f\in{\cal U}$ a hyperbolic periodic point $P_{f}$ of $f$ .

There is a residual subset ${\cal R}={\cal R}_{P}$ of $\mbox{{\rm Diff\,}}^{1}(M)$ with the following property. For every diffeomorphism $f\in{\cal R}\cap{\cal U}$ such that $H(P_{f},f)$ contains a saddle $B_{f}$ with ${\mbox{\rm{ind}}^{\,{\operatorname{s}}}}(B_{f})>{\mbox{\rm{ind}}^{\,{\operatorname{s}}}}(P_{f})$ there is a transitive hyperbolic set $\Sigma_{f}$ containing $P_{f}$ and a ${\operatorname{cu}}$ -blender-horseshoe $\Lambda_{f}$ .

Theorem 6.4 will follow from this proposition using standard genericity arguments (the details can be found in the end of this subsection).

Proof of Proposition 6.4: Consider the residual subset ${\cal G}_{0}$ of $\mbox{{\rm Diff\,}}^{1}(M)$ in Lemma 6.1. Given any $f\in{\cal G}_{0}\cap{\cal U}$ , if $H(P,f)$ contains a saddle $B_{f}$ with ${\mbox{\rm{ind}}^{\,{\operatorname{s}}}}(B_{f})>{\mbox{\rm{ind}}^{\,{\operatorname{s}}}}(P_{f})$ , then there is a saddle $Q_{f}\in H(P_{f},f)$ with ${\mbox{\rm{ind}}^{\,{\operatorname{s}}}}(Q_{f})={\mbox{\rm{ind}}^{\,{\operatorname{s}}}}(P_{f})+1$ . Furthermore, by Lemma 6.3, if $f\in{\cal G}_{0}$ the point $Q_{f}$ belongs robustly to the chain recurrence class $C(P_{f},f)$ . Moreover, $C(P_{f},f)=H(P_{f},f)$ .

Let ${\cal W}$ be the set of diffeomorphisms:

{\cal W}=\left\{f\in{\cal U}\colon\mbox{there is a saddle $Q_{f}\in\mbox{{\rm Per}}(f)$ with}\quad\begin{array}[]{ll}&{\mbox{\rm{ind}}^{\,{\operatorname{s}}}}(Q_{f})={\mbox{\rm{ind}}^{\,{\operatorname{s}}}}(P_{f})+1,\\ &\mbox{and}\\ &\mbox{$Q_{f}$ is $C^{1}$-robustly in $C(P_{f},f)$}\end{array}\right\}

By definition, the set ${\cal W}$ is an open subset of ${\cal U}$ . Let

{\cal U}_{1}={\cal U}\setminus\overline{{\cal W}}.

By construction the set ${\cal U}_{1}$ is open and ${\cal U}_{1}\cup{\cal W}$ is dense in ${\cal U}$ .

Claim 6.7.

Let $f\in{\cal G}_{0}\cap{\cal U}_{1}$ . Then $C(P_{f},f)$ does not contain any hyperbolic periodic point $\tilde{Q}$ with ${\mbox{\rm{ind}}^{\,{\operatorname{s}}}}(\tilde{Q})>{\mbox{\rm{ind}}^{\,{\operatorname{s}}}}(P_{f})$ .

Proof: The proof is by contradiction. Suppose that there is a saddle $\tilde{Q}\in C(P_{f},f)$ with ${\mbox{\rm{ind}}^{\,{\operatorname{s}}}}(\tilde{Q})>{\mbox{\rm{ind}}^{\,{\operatorname{s}}}}(P_{f})$ . As $f\in{\cal G}_{0}$ , there is a saddle $Q_{f}\in C(P_{f},f)=H(P_{f},f)$ with ${\mbox{\rm{ind}}^{\,{\operatorname{s}}}}(Q_{f})={\mbox{\rm{ind}}^{\,{\operatorname{s}}}}(P_{f})+1$ . As $f\in{\cal G}_{0}$ , the saddle $Q_{f}$ belongs $C^{1}$ -robustly to $C(P_{f},f)$ . Therefore $f\in{\cal W}$ , contradicting the definition of ${\cal U}_{1}$ . $\Box$

Thus, by Lemma 6.3, it is enough to prove Proposition 6.6 for diffeomorphisms in ${\cal W}$ . In other words, next lemma implies Proposition 6.6.

Lemma 6.8.

The open set of diffeomorphisms $f$ having a transitive hyperbolic set $\Sigma_{f}$ containing $P_{f}$ and a ${\operatorname{cu}}$ -blender-horseshoe $\Lambda_{f}$ is dense in ${\cal W}$ .

Proof: Consider $f\in{\cal W}$ and a saddle $Q_{f}$ with ${\mbox{\rm{ind}}^{\,{\operatorname{s}}}}(Q_{f})={\mbox{\rm{ind}}^{\,{\operatorname{s}}}}(P_{f})+1$ which belongs robustly to $C(P_{f},f)$ . Next lemma is an immediate consequence of Hayashi connecting lemma in [18]:

Lemma 6.9 ([18]).

Let $f$ be a diffeomorphism having a pair of (hyperbolic) saddles $A_{f}$ and $B_{f}$ whose orbits are different and such that $B_{f}\in H(A_{f},f)$ . Then there is $g$ arbitrarily $C^{1}$ -close to $f$ such that $W^{\operatorname{u}}(A_{g},g)\cap W^{\operatorname{s}}(B_{g},g)\neq\emptyset$ .

Note that as $f\in{\cal W}$ we have $Q_{f}\in C(P_{f},f)$ in a robust way. After a first perturbation, we can assume that $f\in{\cal G}_{0}$ so that $Q_{f}\in H(P_{f},f)$ . We apply Lemma 6.9 to the saddles $P_{f}$ and $Q_{f}$ to get a diffeomorphism $g$ close to $f$ with a transverse intersection between $W^{\operatorname{u}}(P_{g},g)$ and $W^{\operatorname{s}}(Q_{g},g)$ . Note that this transverse intersection persists after perturbation, so that we can assume that $g\in{\cal W}\cap{\cal G}_{0}$ . After a new application of Lemma 6.9 we can suppose that $W^{\operatorname{s}}(P_{g},g)\cap W^{\operatorname{u}}(Q_{g},g)\neq\emptyset$ . Therefore there is $g$ arbitrarily $C^{1}$ -close to $f$ , having a co-index one cycle associated to $P_{g}$ and $Q_{g}$ .

Applying Proposition 5.6 to the diffeomorphism $g$ with a co-index one cycle (associated to $P_{g}$ and $Q_{g}$ ), we get $h$ close to $g$ , thus close to $f$ , with a periodic point $R_{h}$ such that

•

$R_{h}$ is strong-intermediate with respect to $P_{h}$ and $Q_{h}$ ,
•

$R_{h}$ has the same index as $P_{h}$ , and
•

$R_{h}$ is a reference saddle of a ${\operatorname{cu}}$ -blender-horseshoe $\Lambda_{h}$ .

Recall that, by Lemma 3.9, the continuation of this blender-horseshoe is defined in a neighborhood of $h$ . Thus we can assume that $h\in{\cal G}_{0}$ .

As the saddle $R_{h}$ is (robustly) intermediate with respect to $P_{h}$ and $Q_{h}$ and $Q_{h}\in C(P_{h},h)$ , Lemma 5.8 implies that $R_{h}\in C(P_{h},h)$ . Thus, from Lemma 6.3, we have that $R_{h}$ belongs robustly to $C(P_{h},h)$ . As $R_{h}$ and $P_{h}$ have the same index, Lemma 6.9 gives a perturbation $\varphi$ of $h$ with transverse intersections between the invariant manifolds of these saddles. That is, the saddles $R_{\varphi}$ and $P_{\varphi}$ are homoclinically related, thus their homoclinic classes are equal. Therefore there is a transitive hyperbolic set $\Sigma_{\varphi}$ containing $P_{\varphi}$ and the ${\operatorname{cu}}$ -blender-horseshoe $\Lambda_{\varphi}$ (the continuation of $\Lambda_{h}$ ). This ends the proof of the lemma. $\Box$

The proof of Proposition 6.6 is now complete. $\Box$

Proof of Theorem 6.4: Given a diffeomorphism $f$ , denote by $\mbox{{\rm Per}}_{n}(f)$ the set of periodic points $P$ of $f$ of period $\pi(P)\leq n$ . To prove Theorem 6.4 it is enough to see that, for every $n\in{\mathbb{N}}$ , there is a residual set ${\cal R}_{\leq n}$ of diffeomorphisms $f$ such that the conclusion of the theorem holds for every periodic orbit $P\in\mbox{{\rm Per}}_{n}(f)$ . Then it is enough to take ${\cal R}=\cap_{n}{\cal R}_{\leq n}$ .

Note that there is a $C^{1}$ -open an dense subset ${\cal O}_{n}\subset\mbox{{\rm Diff\,}}^{1}(M)$ of diffeomorphisms $f$ such that every periodic point $P\in\mbox{{\rm Per}}_{n}(f)$ is hyperbolic. In particular, the cardinal of $\mbox{{\rm Per}}_{n}(f)$ is finite an locally constant in ${\cal O}_{n}$ . Moreover, each periodic point $P\in\mbox{{\rm Per}}_{n}(f)$ has a hyperbolic continuation in each (open) connected component ${\cal U}$ of the open set ${\cal O}_{n}$ . That is, there are a constant $k=k({\cal U})$ and continuous maps $f\mapsto P_{i,f}$ , $i=1,\dots,k$ , such that $\mbox{{\rm Per}}_{n}(f)=\{P_{1,f},\dots,P_{k,f}\}$ , for every $f\in{\cal U}$ .

Note that the set $\mbox{cc}({\cal O}_{n})$ of connected components of ${\cal O}_{n}$ is countable. Therefore to prove Theorem 6.4 for periods $\pi\leq n$ it is enough to see that this result holds in each connected component ${\cal U}$ of ${\cal O}_{n}$ . More precisely, for each connected component ${\cal U}$ of ${\cal O}_{n}$ , we first build a residual subset $\widetilde{{\cal R}_{\cal U}}$ of $\mbox{{\rm Diff\,}}^{1}(M)$ such that the conclusion holds in the set $\widetilde{{\cal R}_{\cal U}}\cap{\cal U}$ . We now consider the set

{\cal R}_{\cal U}=\widetilde{{\cal R}_{\cal U}}\cup({\cal O}_{n}\setminus{\cal U}).

Note that the set ${\cal O}_{n}\setminus{\cal U}$ is the union of the open connected components of ${\cal O}_{n}$ different from ${\cal U}$ . Thus the set ${\cal O}_{n}\setminus{\cal U}$ is open (and closed) in ${\cal O}_{n}$ and therefore the set ${\cal R}_{\cal U}$ is residual in $\mbox{{\rm Diff\,}}^{1}(M)$ . Finally, the announced residual subset ${\cal R}_{\leq n}$ of $\mbox{{\rm Diff\,}}^{1}(M)$ is the countable intersection of the residual subsets ${\cal R}_{\cal U}$ of $\mbox{{\rm Diff\,}}^{1}(M)$ :

{\cal R}_{\leq n}=\bigcap_{{\cal U}\in\mbox{cc}({\cal O}_{n})}{\cal R}_{\cal U}.

To complete the proof of the theorem it remains to define $\widetilde{{\cal R}_{\cal U}}$ for each component ${\cal U}$ of ${\cal O}_{n}$ . Given $f\in{\cal U}$ consider $\mbox{{\rm Per}}_{n}(f)=\{P_{1,f},\dots,P_{k,f}\}$ . For each $i=1,\dots,k$ , Proposition 6.6 gives a residual subset ${\cal R}_{P_{i}}$ of $\mbox{{\rm Diff\,}}^{1}(M)$ where the conclusion holds. The residual set $\widetilde{{\cal R}_{\cal U}}$ is the finite intersection of the residual sets ${\cal R}_{P_{i}}$ . The proof of Theorem 6.4 is now complete. $\Box$

6.3 Robust homoclinic tangencies under lack of domination

In this section, we conclude the proof of Theorem 1 about $C^{1}$ -generic existence of robust homoclinic tangencies inside homoclinic classes with index variation and lack of domination. We first recall a key result stating the relation between lack of domination and homoclinic tangencies.

Theorem 6.10 (Theorem 1.1 in [16]).

Let $P_{f}$ be a saddle of a diffeomorphism $f$ such that the stable/unstable splitting defined over the set of periodic points homoclinically related with $P_{f}$ is not dominated. Then there is a diffeomorphism $h$ arbitrarily $C^{1}$ -close to $f$ with a homoclinic tangency associated to $P_{h}$ .

As in Section 6.2, we begin with a version of Theorem 1 for a given fixed saddle.

Proposition 6.11.

Consider a diffeomorphism $g$ and a hyperbolic saddle $P_{g}$ of $g$ . Let ${\cal U}$ be an open subset of $\mbox{{\rm Diff\,}}^{1}(M)$ such that the map $f\mapsto P_{f}$ ( $P_{f}$ a hyperbolic saddle) is continuous and well defined. Then there is a residual subset ${\cal G}_{\cal U}$ of ${\cal U}$ with the following property. Let $f\in{\cal G}_{\cal U}$ be any diffeomorphism such that:

•

the chain recurrence class $C(P_{f},f)$ has a periodic point $Q_{f}$ with ${\mbox{\rm{ind}}^{\,{\operatorname{s}}}}(Q_{f})>{\mbox{\rm{ind}}^{\,{\operatorname{s}}}}(P_{f})$ and
•

$C(P_{f},f)$ does not admit a dominated splitting $E\oplus_{{}_{<}}F$ with $\dim(E)={\mbox{\rm{ind}}^{\,{\operatorname{s}}}}(P_{f})$ .

The $C(P_{f},f)$ has a transitive hyperbolic set containing $P_{f}$ with a $C^{1}$ -robust homoclinic tangency.

Proof: Let ${\cal W}_{0}\subset{\cal U}$ be the set of diffeomorphisms $f$ such that the chain recurrence class $C(P_{f},f)$ of $P_{f}$ contains robustly a hyperbolic periodic point $Q_{f}$ with ${\mbox{\rm{ind}}^{\,{\operatorname{s}}}}(Q_{f})>{\mbox{\rm{ind}}^{\,{\operatorname{s}}}}(P_{f})$ (i.e., $Q_{g}\in C(P_{g},g)$ for all $g$ $C^{1}$ -close to $f$ ). By definition and Remark 6.2, the set ${\cal W}_{0}$ is open and non-empty. Let

{\cal U}_{0}={\cal U}\setminus\overline{{\cal W}_{0}}.

Note that ${\cal U}_{0}\cup{\cal W}_{0}$ is an open and dense subset of ${\cal U}$ .

Let ${\cal U}_{1}\subset{\cal U}$ be the set of diffeomorphisms $f$ such that $C(P_{f},f)$ has a dominated splitting $E\oplus_{{}_{<}}F$ with $\dim(E)={\mbox{\rm{ind}}^{\,{\operatorname{s}}}}(P_{f})$ . Since the map $f\mapsto C(P_{f},f)$ is upper-semi-continuous and a dominated splitting persists in a neighborhood of $C(P_{f},f)$ by perturbations (for instance, see [13, Chapter B.1]), one gets that the set ${\cal U}_{1}$ is open. Let

{\cal W}_{1}={\cal U}\setminus\overline{{\cal U}_{1}}.

Then the set ${\cal U}_{1}\cup{\cal W}_{1}$ is open and dense in ${\cal U}$ . As a consequence, the open sets ${\cal U}_{0}\cup{\cal U}_{1}$ and ${\cal W}_{0}\cap{\cal W}_{1}$ are disjoint and their union is dense in ${\cal U}$ .

Note that we are interested in the subset of ${\cal U}$ of diffeomorphisms $f$ whose chain recurrence class $C(P_{f},f)$ contains points of different indices and has not an appropriate dominated splitting, that is, the set set ${\cal W}_{0}\cap{\cal W}_{1}$ . Thus next lemma implies the proposition.

Lemma 6.12.

The set ${\cal T}$ of diffeomorphisms $g$ having a hyperbolic set $\Sigma_{g}$ containing $P_{g}$ and with a $C^{1}$ -robust homoclinic tangency is open and dense in ${\cal W}_{0}\cap{\cal W}_{1}$ .

Proof: It is enough to prove the density of the set ${\cal T}$ . Let $g\in{\cal W}_{0}\cap{\cal W}_{1}$ . As $g\in{\cal W}_{0}$ there is a saddle $Q_{g}$ with ${\mbox{\rm{ind}}^{\,{\operatorname{s}}}}(Q_{g})>{\mbox{\rm{ind}}^{\,{\operatorname{s}}}}(P_{g})$ such that $Q_{g}$ belongs robustly to $C(P_{g},g)$ . After a $C^{1}$ -perturbation, we can assume that the diffeomorphism $g$ simultaneously belongs to the residual set ${\cal G}$ where $C(P_{g},g)=H(P_{g},g)$ and to the residual set ${\cal R}$ in Theorem 6.4. Note that:

•

By Theorem 6.4, the set of diffeomorphisms $g$ having a ${\operatorname{cu}}$ -blender horseshoe $\Lambda_{g}$ which is contained in a transitive hyperbolic set containing $P_{g}$ is open and dense in ${\cal W}_{0}$ .
•

As $g\in{\cal W}_{1}$ and $H(P_{g},g)=C(P_{g},g)$ , one has that the stable/unstable splitting defined over the set of periodic points homoclinically related with $P_{g}$ is not dominated. Otherwise, this dominated splitting could be extended to the closure of these points (the whole $H(P_{g},g)$ ) in a dominated way (see [13, Chapter B.1]), which is a contradiction.
•

Since the stable/unstable splitting defined over the set of periodic points homoclinically related with $P_{g}$ is not dominated, Theorem 6.10 implies that there is a diffeomorphism $h$ arbitrarily $C^{1}$ -close to $g$ with a homoclinic tangency associated to $P_{h}$ .

Theorem 4.8 now implies that there is a diffeomorphism $\varphi$ arbitrarily close to $g$ with a transitive hyperbolic set containing $P_{\varphi}$ and having a robust homoclinic tangency. This ends the proof of the lemma. $\Box$

The proof of Proposition 6.11 is now complete. $\Box$

6.3.1 End of the proof of Theorem 1

The proof of Theorem 1 using Proposition 6.11 is almost identical to the proof of Theorem 6.4 using Proposition 6.6. It is enough to see that, for every $n\in{\mathbb{N}}$ , there is a residual set ${\cal G}_{\leq n}$ of diffeomorphisms $f$ for which the conclusion of the theorem holds for the points in $\mbox{{\rm Per}}_{n}(f)$ .

This proof is similar to the one of Theorem 6.4. So we will omit some details. As in Theorem 6.4, we consider the $C^{1}$ -open an dense subset ${\cal O}_{n}\subset\mbox{{\rm Diff\,}}^{1}(M)$ of diffeomorphisms $f$ such that every periodic point in $\mbox{{\rm Per}}_{n}(f)$ is hyperbolic. Recall that the number of elements of $\mbox{{\rm Per}}_{n}(f)$ is finite and locally constant, and that each periodic point in $\mbox{{\rm Per}}_{n}(f)$ has a hyperbolic continuation on each (open) connected component of ${\cal O}_{n}$ .

To state Theorem 1 for periodic points in $\mbox{{\rm Per}}_{n}(f)$ , it is enough to prove it in each connected component ${\cal U}\in\mbox{cc}({\cal O}_{n})$ (recall that $\mbox{cc}({\cal O}_{n})$ is countable): for each connected component ${\cal U}$ , we construct a residual subset $\widetilde{{\cal G}_{\cal U}}$ such that the conclusion holds in the set $\widetilde{{\cal G}_{\cal U}}\cap{\cal U}$ . Then we let

{\cal G}_{\cal U}=\widetilde{{\cal G}_{\cal U}}\cup({\cal O}_{n}\setminus{\cal U})

and define

{\cal G}_{\leq n}=\bigcap_{{\cal U}\in\mbox{cc}({\cal O}_{n})}{\cal G}_{\cal U}.

To define $\widetilde{{\cal G}_{\cal U}}$ for a component ${\cal U}$ of ${\cal O}_{n}$ , given $f\in{\cal U}$ write $\mbox{{\rm Per}}_{n}(f)=\{P_{1,f},\dots,P_{k,f}\}$ ( $k=k({\cal U})$ ) and consider the continuous maps $f\mapsto P_{i,f}$ , $i\in\{1,\dots,k\}$ , defined on ${\cal U}$ . For each $i$ , Proposition 6.11 provides a residual subset where the conclusion of the theorem holds for $P_{i,f}$ . Now it is enough to define $\widetilde{{\cal G}_{\cal U}}$ as the intersection of these residual sets. The proof of Theorem 1 is now complete. $\Box$

6.4 Robust cycles in non-hyperbolic chain recurrence classes

We close this paper by stating an extension of [9, Theorem 1.16]. The novelty of this version is that the hyperbolic sets involved in the robust cycle are contained in a prescribed chain recurrence class. We note that [9] does not give information about the relation between the hyperbolic set involved in the robust cycle and the saddles in the initial cycle.

Theorem 3.

There is a residual subset ${\cal R}\subset\mbox{{\rm Diff\,}}^{1}(M)$ with the following property. Consider any diffeomorphism $f\in{\cal R}$ having a chain recurrence class $C$ with two saddles $P$ and $Q$ such that ${\mbox{\rm{ind}}^{\,{\operatorname{s}}}}(P)={\mbox{\rm{ind}}^{\,{\operatorname{s}}}}(Q)+1$ . Then $f$ has a $C^{1}$ -robust heterodimensional cycle associated to hyperbolic sets $\Lambda$ and $\Sigma$ containing $P$ and $Q$ .

Since this result follows from arguments similar to the ones in the previous sections and as robust heterodimensional cycles is not the main topic of this paper, we just give some hints for the proof.

As in the proofs above, it is enough to state a local version of the theorem for a given saddle $P$ . Then the general version follows using standard genericity arguments identical to the ones in Sections 6.2 and 6.3.

To get the local version of the theorem, note that for generic diffeomorphisms $f$ , the saddle $Q$ is robustly in the chain recurrence class $C(P,f)$ and there is a ${\operatorname{cu}}$ -blender-horseshoe $\Sigma$ associated to $Q$ . In this step, the strongly intermediate points given by Proposition 5.6 play a key role. Finally, a perturbation gives a robust cycle with a transverse intersection between $W^{\operatorname{s}}(P)$ and $W^{\operatorname{u}}(Q)$ and a robust intersection of $W^{\operatorname{u}}(P)$ with $W^{\operatorname{s}}(\Sigma)$ . This completes the brief skecth of the proof.

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Christian Bonatti (bonatti@u-bourgogne.fr)

Institut de Mathématiques de Bourgogne

B.P. 47 870

21078 Dijon Cedex

France

Abundance of C1C^{1}-robust homoclinic tangencies