Unstable entropy of partially hyperbolic diffeomorphisms along non-compact subsets

TL;DR

This paper introduces a new concept of unstable topological entropy for partially hyperbolic diffeomorphisms on non-compact sets, extending Bowen's theorem to these settings and providing a Hausdorff dimension-like characterization.

Contribution

It defines unstable topological entropy on non-compact sets and extends Bowen's inequality to this broader context, including a Hausdorff dimension-like measure.

Findings

Unstable topological entropy is well-defined for non-compact subsets.

The measure-theoretic entropy is bounded by the unstable topological entropy on full measure sets.

A Hausdorff dimension-like characterization of unstable topological entropy is established.

Abstract

Given a partially hyperbolic diffeomorphism $f:M \rightarrow M$ defined on a compact Riemannian manifold $M$, in this paper we define the concept of unstable topological entropy of $f$ on a set $Y \subset M$ not necessarily compact and we extend a theorem of R. Bowen proving that, for an ergodic $f$-invariant measure $\mu$, the unstable measure theoretical entropy of $f$ is upper bounded by the unstable topological entropy of $f$ on any set of full $\mu$-measure. We also define a notion of unstable topological entropy of $f$ using a Hausdorff dimension like characterization.