Non-uniformly hyperbolic horseshoes in the standard family

TL;DR

This paper demonstrates that non-uniformly hyperbolic horseshoes, previously studied by Palis and Yoccoz, appear in a broad set of parameters within the standard family of area-preserving diffeomorphisms on the two-torus.

Contribution

It establishes the occurrence of non-uniformly hyperbolic horseshoes in the standard family for a large measure set of parameters, expanding understanding of their prevalence.

Findings

Non-uniformly hyperbolic horseshoes occur in the standard family.

These occur for a set of parameters with positive Lebesgue measure.

The result applies to area-preserving diffeomorphisms of the two-torus.

Abstract

We show that the non-uniformly hyperbolic horseshoes of Palis and Yoccoz occur in the standard family of area-preserving diffeomorphisms of the two-torus for a set of (large) parameters of positive Lebesgue measure.