Physical Measures for Certain Partially Hyperbolic Attractors on 3-Manifolds

TL;DR

This paper investigates the ergodic properties of certain partially hyperbolic attractors on 3-manifolds, establishing conditions under which they admit finitely many physical measures with full measure basins.

Contribution

It introduces new conditions involving transversality and neutrality that guarantee the existence of finite physical measures for these attractors.

Findings

Finite number of physical measures established

Basins of these measures cover full Lebesgue measure

Construction of robust nonhyperbolic attractors satisfying the conditions

Abstract

In this work, we study ergodic properties of certain partially hyperbolic attractors whose central direction has a neutral behavior, the main feature is a condition of transversality between unstable leaves when projected by the stable holonomy. We prove that partial hyperbolic attractors satisfying conditions of transversality between unstable leaves via the stable holonomy, neutrality in the central direction and regularity of the stable foliation admits a finite number of physical measures, coinciding with the ergodic u-Gibbs States, whose union of the basins has full Lebesgue measure. Moreover, we describe the construction of a family of robustly nonhyperbolic attractors satisfying these properties.