U-Gibbs measure rigidity for partially hyperbolic endomorphisms on surfaces

TL;DR

The paper establishes a measure rigidity phenomenon for $C^2$ partially hyperbolic endomorphisms of ${\mathbb T}^2$ that are strongly transitive. Using the inverse-limit construction and Pesin theory in a non-invertible setting, the authors develop leaf-wise quotient measures and a one-dimensional normal form framework to translate $u$-Gibbs properties into transverse invariances. A sophisticated Eskin--Mirzakhani style scheme, built on Y-configurations and a matching argument, reduces the problem to showing invariance under affine maps, which yields absolute continuity and the SRB property; under a non-speciality assumption and positive center exponent, this implies the uniqueness of the absolutely continuous invariant measure. The results yield open sets of partially hyperbolic endomorphisms on ${\mathbb T}^2$ with a unique $u$-Gibbs measure, and they clarify the dichotomy between special and non-special dynamics in two dimensions. The work bridges rigidity phenomena from homogeneous dynamics to smooth non-invertible systems, with potential extensions to dynamically minimal unstable leaves and related perturbations.

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.

Figures (5)

• Figure 1: Illustration of Eskin--Mirzakhani scheme.
• Figure 2: The five points in the $Y$ configuration and the dynamical relation between them.
• Figure 3: Coupled configurations on the inverse limit space ${\mathbb{T}_f^2}$.
• Figure 4: We control the movements of quotient measures along the approximating configurations in good matching so that the limit point $b$ belongs to $K\cap G(\varepsilon)$.
• Figure 5: The points $a, b, p$ and $q$ to ${\mathbb{T}^2}$ are obtained as the limit of $Y$-configurations in good matching. The points $v$ and $w$ are, respectively, the images of $p$ and $q$ under the unstable holonomy.

Theorems & Definitions (109)

• Theorem A
• Corollary 1
• Theorem 1.1: ACEPWZ
• Remark 1
• Corollary 2
• Remark 2
• Proposition 1
• Remark 3
• Proposition 2
• proof