Entropy along expanding foliations

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 (C1 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 C1+ partially hyperbolic diffeomorphisms is an upper semi-continuous function of the map in the C1 topology. Another consequence is that the sets of partially hyperbolic diffeomorphisms with mostly contracting or mostly expanding center are C1 open. New examples of partially hyperbolic diffeomorphisms with mostly expanding center are provided, and the existence of physical measures for C1 residual subset of diffeomorphisms is discussed.We also provide a new class of robustly transitive diffeomorphisms: every C2 volume preserving, accessible partially hyperbolic diffeomorphism with one dimensional center and non-vanishing center exponent is C1 robustly transitive (among neighborhood of diffeomorphisms which are not necessarily volume preserving).