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).

Open sets of partially hyperbolic skew products having a unique SRB measure

“…In the case that E uu has dimension one, for any ergodic f -invariant measure, we write λ uu µ to be the Lyapunov exponent of the strong unstable direction. The following result can be found in [Ya16] and [Le84].…”

“…where f −1 ξ uu (p) is the element of the partition f −1 ξ uu containing p. The definition above does not depend on the choice of the µ-measurable partition ξ uu . The notion of partial entropy along expanding foliations has been introduced in [VY17] and [Ya16] (see also…”

Partially volume expanding diffeomorphisms

“…This comes with a price: while the conclusion of Theorem 3.1 remains true for nearby maps, just because the assumptions are C 1 open, we have no control on how the number of physical measures unfolds under perturbation. This is in contrast with the case of diffeomorphisms with mostly contracting center, where a very precise bifurcation theory for physical measures exists (see [13,15,27]) that describes the number and supports of physical measures for all the perturbations of an initial map.…”

“…for any codimension-one subspace H of T x M that contains E u x . Being partially volume expanding is clearly a C 1 open property (the corresponding statement for mostly contracting cente is more subtle, and was proven by Andersson [2] and Yang [27]). Moreover, it is not difficult to find examples.…”

“…Andersson, Vásquez [3] use a slightly stronger definition, which they prove is C 2 open. The latter was improved by Yang [27], who proved C 1 openness. Bifurcation properties of the physical measures have been studied by Andersson, Vásquez [4] and Yang [28] The main novelty of our present results, with respect to those more studied cases, is perhaps that the notion of partial volume expansion requires no assumptions on the signs of the Lyapunov exponents: it allows us to focus on the more significant Gibbs u-states, with negative center exponents, and disregard all the other ones.…”