Non-uniformly hyperbolic endomorphisms

TL;DR

This paper constructs large open sets of area-preserving endomorphisms on the two-torus that are non-uniformly hyperbolic without dominated splitting, with Lyapunov exponents that vary continuously and can be made arbitrarily large.

Contribution

It demonstrates the existence of stable, non-uniformly hyperbolic endomorphisms with explicit real analytic examples, expanding the understanding of hyperbolic dynamics on the torus.

Findings

Existence of large open sets of non-uniformly hyperbolic endomorphisms

Lyapunov exponents vary continuously and can be arbitrarily large

Examples are stably ergodic and Bernoulli under certain conditions

Abstract

We show the existence of large $\mathcal C^1$ open sets of area preserving endomorphisms of the two-torus which have no dominated splitting and are non-uniformly hyperbolic, meaning that Lebesgue almost every point has a positive and a negative Lyapunov exponent. The integrated Lyapunov exponents vary continuously with the dynamics in the $\mathcal C^1$ topology and can be taken as far away from zero as desired. Explicit real analytic examples are obtained by deforming linear endomorphisms, including expanding ones. The technique works in nearly every homotopy class and the examples are stably ergodic (in fact Bernoulli), provided that the linear map has no eigenvalue of modulus one.