Transitivity of conservative toral endomorphisms

TL;DR

This paper proves that certain non-invertible, area-preserving maps on the 2-torus, homotopic to expanding or hyperbolic linear maps, are necessarily topologically transitive, providing a full classification of such maps.

Contribution

It establishes a complete characterization of conservative toral endomorphisms homotopic to expanding or hyperbolic maps that are transitive.

Findings

Non-invertible area-preserving maps homotopic to expanding/hyperbolic endomorphisms are transitive.

Provides a full classification of transitive conservative endomorphisms on the torus.

Results hold across all smoothness categories.

Abstract

It is shown that if a non-invertible area preserving local homeomorphism on $\mathbb{T}^2$ is homotopic to a linear expanding or hyperbolic endomorphism, then it must be topologically transitive. This gives a complete characterization, in any smoothness category, of those homotopy classes of conservative endomorphisms that consist entirely of transitive maps.