Anosov diffeomorphism with a horseshoe that attracts almost any point

TL;DR

This paper constructs a specific C1 Anosov diffeomorphism on a two-torus featuring a horseshoe attractor with zero measure, yet with a physical measure whose basin covers almost all points.

Contribution

It provides an explicit example of a C1 Anosov diffeomorphism with a horseshoe attractor that has full measure basin but zero measure support, highlighting complex dynamical behaviors.

Findings

Existence of a C1 Anosov diffeomorphism with a zero-measure horseshoe

The basin of the physical measure has full Lebesgue measure

The support of the physical measure is a measure-zero horseshoe

Abstract

We present an example of a C1 Anosov diffeomorphism of a two-torus with a physical measure such that its basin has full Lebesgue measure and its support is a horseshoe of zero measure.