Unstable entropies and Dimension Theory of Partially Hyperbolic Systems

TL;DR

This paper introduces unstable topological entropy for subsets in partially hyperbolic systems, establishing foundational results and applying them to multifractal analysis, thereby extending existing theories.

Contribution

It defines a new unstable topological entropy concept for non-compact, non-invariant sets in partially hyperbolic systems and proves fundamental properties including variational principles.

Findings

Established entropy distribution and variational principles for Bowen unstable topological entropy.

Extended results on unstable topological entropy of saturated sets.

Provided new insights into multifractal analysis of partially hyperbolic systems.

Abstract

In this paper we define unstable topological entropy for any subsets (not necessarily compact or invariant) in partially hyperbolic systems as a Carath\'{e}odory dimension characteristic, motivated by the work of Bowen and Pesin etc. We then establish some basic results in dimension theory for Bowen unstable topological entropy, including an entropy distribution principle and a variational principle in general setting. As applications of this new concept, we study unstable topological entropy of saturated sets and extend some results in \cite{Bo, PS2007}. Our results give new insights to the multifractal analysis for partially hyperbolic systems.