Robust Heterodimensional Cycles and $C^1$-Generic Dynamics

Abstract

A diffeomorphism f has a heterodimensional cycle if there are (transitive) hyperbolic sets Λ and Σ having different indices (dimension of the unstable bundle) such that the unstable manifold of Λ meets the stable one of Σ and vice versa. This cycle has co-index 1 if index(Λ) = index(Σ) ± 1. This cycle is robust if, for every g close to f , the continuations of Λ and Σ for g have a heterodimensional cycle.We prove that any co-index 1 heterodimensional cycle associated with a pair of hyperbolic saddles generates C 1 -robust heterodimensioal cycles. Therefore, in dimension three, every heterodimensional cycle generates robust cycles.We also derive some consequences from this result for C 1 -generic dynamics (in any dimension). Two of such consequences are the following. For tame diffeomorphisms (generic diffeomorphisms with finitely many chain recurrence classes) there is the following dichotomy: either the system is hyperbolic or it has a robust heterodimensional cycle. Moreover, any chain recurrence class containing saddles having different indices has a robust cycle. 471Question 1.2. Let M be closed manifold. Does it exist a C 1 -open and dense subset O ⊂ Diff 1 (M ) such that every f ∈ O either verifies the Axiom A and the no-cycles condition or has a C 1 -robust heterodimensional cycle?Note that a positive answer to this question implies the C 1 -density of hyperbolic surface diffeomorphisms. See the discussion in § 1.3 about the Smale density conjecture. We will see that Theorem 1.14 gives a partial positive answer to this question for the so-called tame diffeomorphisms (diffeomorphisms finitely many homoclinic classes, see the preciseThe examples by Abraham-Smale of non-Axiom A diffeomorphisms involves a hyperbolic set Γ whose unstable manifold has dimension strictly greater than the dimension of its unstable bundle. Note that a normally hyperbolic extension of transitive Anosov diffeomorphisms on a torus T 2 gives an example of this configuration.The construction in [8] gives a slightly different mechanism for constructing non-Axiom A diffeomorphisms and robust heterodimensional cycles, based on the notion of blender. Roughly speaking, a blender is a hyperbolic set whose embedding in the ambient manifold verifies some specific geometric properties, whose effect is that, as in the Abraham-Smale example, the unstable manifold of a blender looks like a manifold of higher dimension. We review the construction and main properties of blenders in § 4.1.3. See also [15, Chapter 6.1] for a discussion of this notion.One of the goals of this paper is to show that blenders (and as a consequence robust heterodimensional cycles) appear in a natural way in the unfolding of heterodimensional cycles associated with two saddles. Definition 1.3 (heterodimensional cycle and co-index 1 cycle).A diffeomorphism f has a heterodimensional cycle (see Figure 1) associated with two hyperbolic periodic saddles P and Q of f if the saddles P and Q have different indices, the stable manifold of the orbit of P meets the unstable manifold of...

Weak* and entropy approximation of nonhyperbolic measures: a geometrical approach

“…Finally, [6, Theorem 4.1] implies that by arbitrarily small C 1 -perturbations these strong homoclinic intersections yield blender-horseshoes for some iterate of the map (stable or unstable, according to the chosen perturbation). We observe that though the terminology blender-horseshoe was not used in [6] the construction corresponds exactly to the prototypical blender-horseshoes in [7,Section 5.1]. In this way, it follows we have shown that having (stable and unstable) blender-horseshoes (for some iterate) is a C 1 -dense property in RTPH 1 (M ).…”

“…In our context, due to the nonhyperbolicity assumption, we have that C 1 -open and -densely in RTPH 1 (M ) the diffeomorphisms have simultaneously saddles of indices dim E ss and dim E ss + 1, this follows from the ergodic closing lemma in [32]. With the terminology in [6,7], the saddles of diffeomorphism in RTPH 1 (M ) have real central eigenvalues (this follows from the fact that dim E c = 1). The robust transitivity assumption and the connecting lemma [25,4] imply that C 1 -densely in RTPH 1 (M ) there are diffeomorphisms with heterodimensional cycles associated to these saddles with real central eigenvalues.…”

Robust Heteroclinic Tangencies

“…Bearing in mind the classic constructions of robust homoclinic tangencies and heterodimensional cycles ( [19,7]) via the unfolding of tangencies and cycles associated with saddles, we ask the following: Question 2. Can a diffeomorphism f having a non-transverse heterodimensional cycle associated with saddles P and Q be C r -approximated by a diffeomorphism g with a C r -robust non-transverse heterodimensional cycle associated with hyperbolic sets containing the continuations P g and Q g of P and Q?…”

Shadowable Chain Components and Hyperbolicity

“…Since, by [3], for any X ∈ R 3 , C X (γ) is robustly isolated. So, there exist a neighborhood U(X) of X and a neighborhood U of C X (γ) such that for any Y ∈ U(X),…”