Limit theorems for toral partially hyperbolic endomorphisms

TL;DR

This paper establishes limit theorems such as the CLT, Berry-Esseen, and Local Limit Theorem for a broad class of partially hyperbolic endomorphisms on the 2D torus, including skew-products and systems with multiple invariant measures.

Contribution

It extends limit theorem results to partially hyperbolic toral endomorphisms, covering cases with multiple invariant measures and perturbations.

Findings

Proves CLT, Berry-Esseen, and Local Limit Theorem under natural assumptions.

Results apply to skew-products and their perturbations.

Valid even with multiple absolutely continuous ergodic measures.

Abstract

Under natural assumptions on the observable, we prove a Central Limit Theorem, a Berry-Esseen Theorem, and a quantitative Local Limit Theorem for a broad class of partially hyperbolic endomorphisms of the two-dimensional torus. Our results apply, but are not limited to, skew-products and their perturbations, and they remain valid even when the system admits multiple, though finitely many, absolutely continuous ergodic invariant measures.