Volume growth and entropy for $C^1$ partially hyperbolic diffeomorphisms

TL;DR

This paper establishes an upper bound on the metric entropy of certain $C^1$ partially hyperbolic diffeomorphisms using volume growth and Lyapunov exponents, with implications for understanding their dynamical complexity.

Contribution

It introduces a new upper bound for metric entropy involving volume growth and Lyapunov exponents in partially hyperbolic systems with dominated splitting.

Findings

Upper bound on metric entropy using volume growth and Lyapunov exponents.

Analysis of volume growth types associated with expanding foliations.

Implications for partially hyperbolic diffeomorphisms.

Abstract

We show that the metric entropy of a $C^1$ diffeomorphism with a dominated splitting and the dominating bundle uniformly expanding is bounded from above by the integrated volume growth of the dominating (expanding) bundle plus the maximal Lyapunov exponent from the dominated bundle multiplied by its dimension. We also discuss different types of volume growth that can be associated with an expanding foliation and relationships between them, specially when there exists a closed form non-degenerate on the foliation. Some consequences for partially hyperbolic diffeomorphisms are presented.