H\'enon-like families and blender-horseshoes at non-transverse heterodimensional cycles

TL;DR

This paper develops a renormalisation scheme for certain three-dimensional dynamical systems with heterodimensional cycles, demonstrating how blender-horseshoes can be approximated near these cycles, revealing complex chaotic structures.

Contribution

It introduces a novel renormalisation approach at heterodimensional tangencies, connecting heterodimensional cycles with blender-horseshoes in three-dimensional dynamics.

Findings

Construction of a renormalisation scheme at heterodimensional tangencies.

Approximation of heterodimensional cycles by diffeomorphisms with blender-horseshoes.

Identification of Hénon-like families exhibiting blender-horseshoes near cycles.

Abstract

In dimension three and under certain regularity assumptions, we construct a renormalisation scheme at the heterodimensional tangency of a non-transverse heterodimensional cycle associated with a pair of saddle-foci whose limit dynamic is a center-unstable H\'enon-like family displaying blender-horseshoes. As a consequence, the initial cycle can be approximated in higher regularity topologies by diffeomorphisms having blender-horseshoes.