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

Abstract

In this paper we obtain C 2 -open sets of dissipative, partially hyperbolic skew products having a unique SRB measure with full support and full basin. These partially hyperbolic systems have a two dimensional center bundle which presents both expansion and contraction but does not admit any further dominated splitting of the center. These systems are non conservative perturbations of an example introduced by Berger-Carrasco.To prove the existence of SRB measures for these perturbations, we obtain a measure rigidity result for u-Gibbs measures for partially hyperbolic skew products. This is an adaptation of a measure rigidity result by A. Brown and F. Rodriguez Hertz for stationary measures of random product of surface diffeomorphisms. In particular, we classify all the possible u-Gibbs measures that may appear in a neighborhood of the example. Using this classification, and ruling out some of the possibilities, we obtain open sets of systems, in a neighborhood of the example, having a unique u-Gibbs measure which is SRB. * D.O. was partially supported by the projects ANR BEKAM : ANR-15-CE40-0001 and ERC project NUHGD.

Uniqueness of $u$-Gibbs measures for hyperbolic skew products on $\mathbb{T}^4$

“…Their result will be used by the first author with Avila et al to prove that for any Anosov C ∞ -diffeomorphism in T 3 admitting a decomposition E s ⊕ E wu ⊕ E uu either E s and E uu are jointly integrable, or every u-Gibbs measure is SRB. More related to the present work, in [Ob21], it is proved a type of classification of u-Gibbs measures for some partially hyperbolic skew products on T 4 (see Theorem 2.3 below). These results are inspired by some important measure rigidity techniques from [EM18, EL, BQ11, BRH17] to classify stationary measures of certain random products (see also [CD20] for an interesting application of [BRH17]).…”

“…This family converges exponentially fast in the C 1 -topology as n increases and the limit gives the unstable holonomy {H u p,q,f } p∈T 4 ,q∈W uu 1 (p,f ) . The speed of convergence only depends on the constants given in (1) and f C 2 (see [Br21,Ob21]). In particular, if g is C 2 -close to f , p is close to p and q is close to q, then H u p ,q ,g is C 1 -close to H u p,q,f .…”

“…Moreover, this measure coincides with the unique SRB measure of f . The proof of our Theorem A is based on the rigidity result stated in Theorem 2.3 below (see [Ob21]). This theorem states that a u-Gibbs measure is either SRB, or there is a 2-torus tangent to E ss and E uu , or there is a type of "infinitesimal" rigidity for the system.…”