Conservative Anosov diffeomorphisms of the two torus without an absolutely continuous invariant measure

TL;DR

This paper constructs examples of $C^{1}$ Anosov diffeomorphisms on the 2-torus that are of Krieger type ${ m III}_{1}$, demonstrating that the smoothness condition is crucial for the existence of absolutely continuous invariant measures.

Contribution

It provides the first known examples of $C^{1}$ Anosov diffeomorphisms without absolutely continuous invariant measures on the 2-torus.

Findings

Existence of $C^{1}$ Anosov diffeomorphisms of Krieger type ${ m III}_{1}$

Counterexample to the Gurevic Oseledec phenomena in $C^{1}$ setting

Absence of smooth invariant measures in these examples

Abstract

We construct examples of $C^{1}$ Anosov diffeomorphisms on $\mathbb{T}^{2}$ which are of Krieger type ${\rm III}_{1}$ with respect to Lebesgue measure. This shows that the Gurevic Oseledec phenomena that conservative $C^{1+\alpha}$ Anosov diffeomorphisms have a smooth invariant measure does not hold true in the $C^{1}$ setting.