Volume growth and topological entropy of certain partially hyperbolic systems

TL;DR

This paper establishes a formula linking the topological entropy of certain partially hyperbolic systems to the volume growth rate of subspaces in their tangent bundle, providing a new way to quantify system complexity.

Contribution

It proves an entropy formula relating topological entropy to volume growth in partially hyperbolic systems with a specific splitting structure.

Findings

Topological entropy equals the volume growth rate of tangent subspaces.

The formula applies to systems with a partially hyperbolic splitting involving one-dimensional subbundles.

Provides a new method to compute entropy via volume growth in dynamical systems.

Abstract

Let $f$ be a $C^{1}$ diffeomorphism on a compact manifold $M$ admitting a partially hyperbolic splitting $TM=E^{s}\oplus_{\prec} E^{1}\oplus_{\prec} E^{2}\cdots \oplus_{\prec}E^{l}\oplus_{\prec} E^{u}$ where $E^{s}$ is uniformly contracting, $E^{u}$ is uniformly expanding and $\dim E^{i}=1,\,1\leq i\leq l.$ We prove an entropy formula w.r.t. the volume growth rate of subspaces in the tangent bundle: $$h_{\rm{top}}(f)=\lim_{n\to+\infty}\frac{1}{n}\log\int\max_{V\subset T_{x}M}|\det Df_{x}^{n}|_{V}|\,d x.$$