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

2020/11/24 by Yoshitaka Saiki, Hiroki Takahasi, Saiki, Yoshitaka +3
Mathematics · Physics and Astronomy · #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems #Chaos control and synchronization

paper · pdf · doi:10.48550/arxiv.2011.12264

Abstract

As a model to provide a hands-on, elementary understanding of chaotic dynamics in dimension three, we introduce a C²-open set of diffeomorphisms of \mathbb R³ 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 C²-robust heterodimensional cycles. Our proof employs the theory of normally hyperbolic invariant manifolds and the thicknesses of Cantor sets.

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