A new criterion of physical measures for partially hyperbolic diffeomorphisms

Abstract

We show that for any C 1 partially hyperbolic diffeomorphism, there is a full volume subset, such that any Cesaro limit of any point in this subset satisfies the Pesin formula for partial entropy.This result has several important applications. First we show that for any C 1+ partially hyperbolic diffeomorphism with one dimensional center, there is a full volume subset, such that every point in this set belongs to either the basin of a physical measure with non-vanishing center exponent, or the center exponent of any limit of the sequence 1 n n−1 i=0 δ f i (x) is vanishing. We also prove that for any diffeomorphism with mostly contracting center, it admits a C 1 neighborhood such that every diffeomorphism in a C 1 residual subset of this open set admits finitely many physical measure, whose basins have full volume.Remark 1.1. As a direct consequence, if G u (f ) consists of a unique measure, then this measure is automatically a physical measure whose basin has full volume.The partially hyperbolic diffeomorphisms with mostly contracting center, which were first studied in [5], are those C 1+ partially hyperbolic diffeomorphisms whose Gibbs u-states only have negative center Lyapunov exponents.Mostly contracting diffeomorphisms contain an abundance of systems. See for example [5] and [10]. It is also shown in [27] that mostly contracting is a C 1 open property. Moreover, if f is a C 1+ partially hyperbolic volume preserving diffeomorphisms with one dimensional center, such that the volume has negative center exponent and the unstable foliation is minimal, then f is mostly contracting.

Statistical properties of physical-like measures

“…1 In [16] and [21] a similar result is obtained for the Gibbs u-states of C 1 partially hyperbolic diffeomorphisms. Here a measure is called a Gibbs u-state if it satisfies the relation hµ(f,…”

“…A program for investigating the physical measures of partially hyperbolic diffeomorphisms or diffeomorphisms with dominated splitting was initiated by Alves, Bonatti, Viana in [8,1]. Further results can be found in [17,18,2,3,4,38,39,21,16] and the references therein. All these works rely on the Pesin theory, in particular, the absolute continuity of the Pesin stable lamination.…”

“…It can be applied to a given diffeomorphism with higher regularity (C 1+α ) where it is used in combination with Pesin's theory. Examples of such applications include partially hyperbolic diffeomorphisms with mostly contracting center or with mostly expanding center, see for instance [38,21,39,20].…”

Geometrical and measure-theoretic structures of maps with mostly expanding center

“…Previously, the study of physical measures is mainly focused on maps that are sufficiently smooth, i.e., with C 1+ regularity. Recently, the new technique developed in [29,21] enables us to shows the existence of physical measure for a large family of C 1 diffeomorphisms, such as those with mostly contacting center.…”

Statistical stability for diffeomorphisms with mostly expanding and mostly contracting centers

“…The entropy h µ (f, F u ) has become a powerful tool in the study of partially hyperbolic diffeomorphisms in recent years, see [26,33,34,56] for instance. Definition 2.3.…”