A Garden of Eden theorem for Anosov diffeomorphisms on tori

TL;DR

This paper proves a new theorem relating surjectivity and injectivity of continuous maps commuting with Anosov diffeomorphisms on tori, extending classical results in dynamical systems.

Contribution

It establishes a Garden of Eden type theorem for Anosov diffeomorphisms on tori, linking surjectivity to injectivity on homoclinicity classes.

Findings

Surjective maps are characterized by injectivity on homoclinicity classes.

The theorem generalizes classical Garden of Eden results to Anosov diffeomorphisms.

Provides a criterion for surjectivity of commuting maps in hyperbolic dynamics.

Abstract

Let $f$ be an Anosov diffeomorphism of the $n$-dimensional torus ${\mathbb{T}}^n$ and $\tau$ a continuous self-mapping of ${\mathbb{T}}^n$ commuting with $f$. We prove that $\tau$ is surjective if and only if the restriction of $\tau$ to each homoclinicity class of $f$ is injective.