Rigidity of measures on the torus: smooth stabilizers and entropy

TL;DR

This paper investigates the structure of measure-preserving diffeomorphisms on the 2-torus under nonlinear Anosov diffeomorphisms, revealing conditions under which these groups are cyclic or virtually cyclic, with implications for entropy and stability.

Contribution

It provides new examples and results characterizing the rigidity of measures on the torus, especially regarding the structure of their stabilizer groups and entropy properties.

Findings

Group of $mbda$-preserving diffeomorphisms is cyclic up to zero-entropy transformations.

For equilibrium states, the group is shown to be virtually cyclic.

Results demonstrate rigidity phenomena for measures under nonlinear Anosov diffeomorphisms.

Abstract

For a nonlinear Anosov diffeomorphism of the 2-torus, we present examples of measures so that the group of $\mu$-preserving diffeomorphisms is, up to zero-entropy transformations, cyclic. For families of equilibrium states $\mu$, we strengthen this to show that the group of $\mu$-preserving diffeomorphism is virtually cyclic.