Absolutely partially hyperbolic surface endomorphisms are dynamically coherent

TL;DR

This paper proves that absolutely partially hyperbolic surface endomorphisms on the torus always possess a center foliation, which is topologically equivalent to the linearized system, advancing understanding of their dynamical structure.

Contribution

The authors establish the existence and leaf conjugacy of the center foliation for absolutely partially hyperbolic surface endomorphisms on the torus, a previously unresolved problem.

Findings

Existence of a center foliation for such endomorphisms

Center foliation is leaf conjugate to the linearization

Advances understanding of the structure of hyperbolic surface endomorphisms

Abstract

We show that if an endomorphism $f:\mathbb{T}^2 \to \mathbb{T}^2$ is absolutely partially hyperbolic, then it has a center foliation. Moreover, the center foliation is leaf conjugate to that of its linearization.