Smooth rigidity for 3-dimensional dissipative Anosov flows

Abstract

We consider two transitive $3$-dimensional Anosov flows which do not preserve volume and which are continuously conjugate to each other. Then, disregarding certain exceptional cases, such as flows with $C^1$ regular stable or unstable distributions, we prove that either the conjugacy is smooth or it sends the positive SRB measure of the first flow to the negative SRB measure of the second flow and vice versa. We give a number of corollaries of this result. In particular, we establish local rigidity on a $C^1$-open $C^\infty$-dense subspace of transitive Anosov flows; we improve the classical de la Llave-Marco-Moriy\'on rigidity theorem for dissipative Anosov diffeomorphisms on the $2$-torus by merely assuming matching of (full) Jacobian data at periodic points; we also exhibit the first evidence that the Teichm\"uller space of smooth conjugacy classes of Anosov diffeomorphisms on the $2$-torus is well-stratified according to regularity.