Symbolic blender-horseshoes and applications

TL;DR

This paper introduces symbolic blender-horseshoes in partially hyperbolic skew-product maps, providing conditions for their existence and demonstrating their applications in constructing robust non-hyperbolic dynamics.

Contribution

It defines symbolic blender-horseshoes for skew-product maps and establishes necessary conditions for their existence, enabling new constructions of robust dynamical phenomena.

Findings

Defined symbolic blender-horseshoes for skew-product maps

Established covering property as a necessary condition

Constructed robust non-hyperbolic transitive diffeomorphisms

Abstract

We study partially-hyperbolic skew-product maps over the Bernoulli shift with H\"older 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.