Blender-horseshoes in center-unstable H\'enon-like families

TL;DR

This paper demonstrates the emergence of blender-horseshoes, a special type of hyperbolic set with unique stable manifold properties, in a family of three-dimensional Hénon-like maps, highlighting their role in robust non-hyperbolic dynamics.

Contribution

It provides a rigorous construction of blender-horseshoes in a specific parameter range for center-unstable Hénon-like endomorphisms in three dimensions.

Findings

Blender-horseshoes occur naturally in the studied family.

The stable manifold behaves as a higher-dimensional manifold.

These structures persist under perturbations.

Abstract

A blender-horseshoe is a locally maximal transitive hyperbolic set that appears in dimension at least three carrying a distinctive geometrical property: its local stable manifold "behaves" as a manifold of topological dimension greater than the expected one (the dimension of the stable bundle). This property persists under perturbations turning this kind of dynamics an important piece in the global description of robust non-hyperbolic systems. In this paper, we consider a parameterized family of center-unstable H\'enon-like of endomorphisms in dimension three and show how blender-horseshoes naturally occur in a specific parameter range.