U-Gibbs measure rigidity for partially hyperbolic endomorphisms on surfaces

TL;DR

This paper establishes measure rigidity results for strongly transitive, partially hyperbolic endomorphisms on the 2-torus, showing conditions under which $u$-Gibbs measures are unique or the map is special, with implications for perturbations of linear maps.

Contribution

It proves a measure rigidity theorem for partially hyperbolic endomorphisms on surfaces, characterizing when $u$-Gibbs measures are unique or the system is special.

Findings

Uniqueness of $u$-Gibbs measures for non-special perturbations of linear maps.

Identification of conditions leading to measure rigidity or special systems.

Application to a broad class of partially hyperbolic systems on the 2-torus.

Abstract

We prove that, for a $C^2$ partially hyperbolic endomorphism of the 2-torus which is strongly transitive, given an ergodic $u$-Gibbs measure that has positive center Lyapunov exponent and has full support, then either the map is special (has only one unstable direction per point), or the measure is the unique absolutely continuous invariant measure. We can apply this result in many settings, in particular obtaining uniqueness of $u$-Gibbs measures for every non-special perturbation of irreducible linear expanding maps of the torus with simple spectrum. This gives new open sets of partially hyperbolic systems displaying a unique $u$-Gibbs measure.