Existence of robust non-uniformly hyperbolic endomorphism in homotopy classes

TL;DR

This paper proves the existence of a broad class of non-uniformly hyperbolic, ergodic, area-preserving maps on the 2-torus within various homotopy classes, extending previous results to include maps with certain degrees.

Contribution

It demonstrates that any homothety on the 2-torus with degree at least 25 can be homotoped to a non-uniformly hyperbolic map, expanding the known classes of such systems.

Findings

Existence of a C^1 open set of non-uniformly hyperbolic systems

These systems intersect all homotopy classes in the 2-torus

Lyapunov exponents vary continuously within this set

Abstract

We extend the results of arXiv:2206.08295v2 by showing that any homothety in $\mathbb T^2$ is homotopic to a non-uniformly hyperbolic ergodic area preserving map, provided that its degree is at least $5^2$. We also address other small topological degree cases not considered in the previous article. This proves the existence of a $\mathcal C^1$ open set of non-uniformly hyperbolic systems, that intersects essentially every homotopy classes in $\mathbb T^2$, where the Lyapunov exponents vary continuously.