Symbolic blender-horseshoes and applications

Abstract

We study partially-hyperbolic skew-product maps over the Bernoulli shift with Hölder dependence on the base points. In the case of contracting fiber maps, symbolic blender-horseshoe is defined as an invariant set which meets any almost horizontal disk in a robust sense. These invariant sets are understood as blenders with center stable bundle of any dimension. We then give necessary conditions (covering property) on an iterated function system such that the relevant skew-product has a symbolic blender-horseshoe. We use this local plug to yield robustly non-hyperbolic transitive diffeomorphisms and robust heterodimensional cycles of co-index equal to the dimension of the central direction.1991 Mathematics Subject Classification. Primary: 58F15, 58F17; Secondary: 53C35.

Robust tangencies of large codimension

“…In particular topologically mixing implies transitivity. Hence Φ 0 = τ × id and according to [6,Theorem 5.7] and Corollary 5.9, for any ε > 0, Φ ε is S 0 -robustly topologically mixing and has a S-robust symbolic cycle for any ε > 0. Notice that the co-index of the cycle may be chosen between zero and c. In fact, repeating the above procedure and adding more maps if necessary, the arcs may be taken in such a way so that Φ ε has S-robust cycles of all intermediate co-indices between zero and c. This completes the proof of the theorem.…”

“…We say that two s-discs, D s 1 , D s 2 ⊂ W s loc (ξ) × M are close if they are the graphs of close α-Hölder functions. This proximity between discs allows us to introduce the following: Example of s-discs are the almost horizontal discs defined as follows: given δ > 0 and a Following [29,6], we introduce symbolic cs, cu and double-blenders.…”

“…First of all, we will construct diffeomorphisms satisfying the hypothesis of Theorem 5.12. This smooth realization of robust cycles will done in a similar manner as in [6].…”

“…Note that the locally constant skew-product diffeomorphism f ε restricted to the set Λ × M is conjugated to the one-step symbolic skew-product Φ ε . In fact, according to [6,Prop. A.2] (see also, [19,24,35]), for every small enough C 1 -perturbation g of f ε , g has an invariant set ∆ g homeomorphic to Λ × M such that g| ∆ g is conjugated with a S 0 -perturbation Ψ g of Φ ε .…”

Blender-horseshoes in Center-unstable Hénon-like Families

“…Later, blenders were used in several dynamical contexts: Generation of robust heterodimensional cycles and homoclinic tangencies, stable ergodicity, Arnold diffusion, and construction of nonhyperbolic measures, among others. Each of these applications involves a specific type of blender such as blender-horseshoes [8], symbolic blenders [20,2], dynamical blenders [4] and super-blenders [1].…”