Entropy along expanding foliations

TL;DR

This paper studies the entropy of diffeomorphisms along expanding foliations, proving its upper semi-continuity and deriving consequences for Gibbs u-states, partial hyperbolicity, and robust transitivity.

Contribution

It establishes upper semi-continuity of entropy along foliations and introduces new examples of partially hyperbolic diffeomorphisms with specific dynamical properties.

Findings

Entropy varies upper semi-continuously with diffeomorphism, measure, and foliation.

Sets of partially hyperbolic diffeomorphisms with certain properties are open in the C^1 topology.

Constructs new robustly transitive diffeomorphisms with specific hyperbolic features.

Abstract

The (measure-theoretical) entropy of a diffeomorphism along an expanding invariant foliation is the rate of complexity generated by the diffeomorphism along the leaves of the foliation. We prove that this number varies upper semi-continuously with the diffeomorphism ($\C^1$ topology), the invariant measure (weak* topology) and the foliation itself in a suitable sense. This has several important consequences. For one thing, it implies that the set of Gibbs $u$-states of $\C^{1+}$ partially hyperbolic diffeomorphisms is an upper semi-continuous function of the map in the $\C^1$ topology. Another consequence is that the sets of partially hyperbolic diffeomorphisms with mostly contracting or mostly expanding center are $\C^1$ open. New examples of partially hyperbolic diffeomorphisms with mostly expanding center are provided, and the existence of physical measures for $C^1$ residual subset of diffeomorphisms are discussed. We also provide a new class of robustly transitive diffeomorphisms: every $C^2$ volume preserving, accessible partially hyperbolic diffeomorphism with one dimensional center and non-vanishing center exponent is $C^1$ robustly transitive (among neighborhood of diffeomorphisms which are not necessarily volume preserving).