Hausdorff dimension of Cantor intersections and robust heterodimensional cycles for heterochaos horseshoe maps

TL;DR

This paper studies the Hausdorff dimensions of invariant sets in a class of three-dimensional dynamical systems with two horseshoes, revealing fractal intersections and robust heterodimensional cycles.

Contribution

It introduces a $C^2$-open set of diffeomorphisms with two horseshoes of different instability dimensions and proves the existence of fractal intersections and robust heterodimensional cycles.

Findings

Unstable and stable sets of the horseshoes have Hausdorff dimension nearly 2.

Their intersection contains a fractal set with Hausdorff dimension nearly 1.

The results demonstrate $C^2$-robust heterodimensional cycles.

Abstract

As a model to provide a hands-on, elementary understanding of chaotic dynamics in dimension three, we introduce a $C^2$-open set of diffeomorphisms of $\mathbb R^3$ having two horseshoes with different dimensions of instability. We prove that: the unstable set of one horseshoe and the stable set of the other are of Hausdorff dimension nearly $2$ whose cross sections are Cantor sets; the intersection of the unstable and stable sets contains a fractal set of Hausdorff dimension nearly $1$. As a corollary we detect $C^2$-robust heterodimensional cycles. Our proof employs the theory of normally hyperbolic invariant manifolds and the thicknesses of Cantor sets.