Topological entropy and Hausdorff dimension of irregular sets for non-hyperbolic dynamical systems

TL;DR

This paper explores the properties of irregular sets in non-hyperbolic dynamical systems, demonstrating they can have full topological entropy and Hausdorff dimension, with examples including partially hyperbolic systems and Lorenz flows.

Contribution

It provides systematic examples of non-hyperbolic systems with irregular sets of maximal entropy and dimension, advancing understanding of their complex structure.

Findings

Irregular sets can have full topological entropy.

Irregular sets can have full Hausdorff dimension.

Examples include partially hyperbolic systems and Lorenz flows.

Abstract

We systematically investigate examples of non-hyperbolic dynamical systems having irregular sets of full topological entropy and full Hausdorff dimension. The examples include some partially hyperbolic systems and geometric Lorenz flows. We also pose interesting questions for future research.