SRB measures for partially hyperbolic attractors of local diffeomorphisms

TL;DR

This paper studies SRB measures for partially hyperbolic attractors of local diffeomorphisms, proving their finiteness, hyperbolicity, uniqueness under transitivity, and statistical stability, with continuous entropy variation.

Contribution

It establishes the existence, finiteness, hyperbolicity, uniqueness, and stability of SRB measures for a broad class of partially hyperbolic attractors, extending thermodynamic formalism.

Findings

Finitely many SRB measures exist for these attractors.

SRB measures are hyperbolic and unique under transitivity.

SRB measures are statistically stable with continuous entropy.

Abstract

In the present paper we contribute to the thermodynamic formalism of partially hyperbolic attractors for local diffeomorphisms admitting an invariant stable bundle and a positively invariant cone field with non-uniform cone expansion at a positive Lebesgue measure set of points. These include the case of attractors for Axiom A endomorphisms and partially hyperbolic endomorphisms derived from Anosov. We prove these attractors have finitely many SRB measures, that these are hyperbolic, and that the SRB measure is unique provided the dynamics is transitive. Moreover, we show that the SRB measures are statistically stable (in the weak$^*$ topology) and that their entropy varies continuously with respect to the local diffeomorphism.