Elliptic stars in a chaotic night

TL;DR

This paper characterizes elliptic islands within chaotic regimes of torus homeomorphisms using topological methods, showing such regimes lack Lyapunov stable points and illustrating results with a parameter family inspired by classical examples.

Contribution

It provides a new topological characterization of elliptic islands in chaotic torus dynamics and demonstrates the absence of stable points in these regimes.

Findings

Elliptic islands are characterized topologically via local rotation subsets.

Chaotic regimes with non-empty interior rotation sets contain no Lyapunov stable points.

A parameter family example illustrates the theoretical results.

Abstract

We study homeomorphisms of the two-torus, homotopic to the identity, whose rotation set has non-empty interior. For such maps, we give a purely topological characterisation of elliptic islands in a chaotic sea in terms of local rotation subsets. We further show that the chaotic regime defined in this way cannot contain any Lyapunov stable points. In order to demonstrate our results, we introduce a parameter family inspired by an example of Misiurewicz and Ziemian.