Unstable Entropies and Variational Principle for Partially Hyperbolic Diffeomorphisms

TL;DR

This paper investigates the unstable entropy in partially hyperbolic systems, establishing a variational principle linking unstable topological and metric entropies, and explores their properties and equivalences.

Contribution

It introduces a new framework for unstable entropy, proving a variational principle and demonstrating its equivalence with volume growth and existing entropy definitions.

Findings

Unstable topological entropy equals the supremum of unstable metric entropy over invariant measures.

Unstable metric entropy matches Ledrappier-Young's definition without increasing partitions.

Properties like affineness, upper semi-continuity, and a Shannon-McMillan-Breiman type theorem are established.

Abstract

We study entropies caused by the unstable part of partially hyperbolic systems. We define unstable metric entropy and unstable topological entropy, and establish a variational principle for partially hyperbolic diffeomorphsims, which states that the unstable topological entropy is the supremum of the unstable metric entropy taken over all invariant measures. The unstable metric entropy for an invariant measure is defined as a conditional entropy along unstable manifolds, and it turns out to be the same as that given by Ledrappier-Young, though we do not use increasing partitions. The unstable topological entropy is defined equivalently via separated sets, spanning sets and open covers along a piece of unstable leaf, and it coincides with the unstable volume growth along unstable foliation. We also obtain some properties for the unstable metric entropy such as affineness, upper semi-continuity and a version of Shannon-McMillan-Breiman theorem.