Rigidity of measures on the torus: smooth stabilizers and entropy

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.