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 ($\C^1$ 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).

This is an improved version, here we add two new applications: Corollary E and Theorem F

References in corpus (2)

• Smoothness of stable holonomies inside center-stable manifolds and the $C^2$ hypothesis in Pugh-Shub and Ledrappier-Young theory
• Topological entropy of Lorenz-like flows

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• Partially volume expanding diffeomorphisms
• Unstable Entropies and Variational Principle for Partially Hyperbolic Diffeomorphisms
• Unstable Entropy and Unstable Pressure for Random Partially Hyperbolic Dynamical Systems
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• Robust minimality of strong foliations for DA diffeomorphisms: $cu$-volume expansion and new examples
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• Unstable Topological Pressure for Partially Hyperbolic Diffeomorphisms with Sub-additive Potentials