On Anosov diffeomorphisms with asymptotically conformal periodic data

TL;DR

This paper studies a special class of transitive Anosov diffeomorphisms with asymptotically conformal periodic data, proving smoothness properties and rigidity results, including conjugacy to toral automorphisms and local rigidity in dimension three.

Contribution

It establishes strong smoothness and rigidity properties for Anosov diffeomorphisms with asymptotically conformal periodic data, extending understanding of their structure.

Findings

Proves C^{1+eta} smoothness of the Anosov splitting.

Shows finite covers are smoothly conjugate to toral automorphisms.

Establishes local rigidity results in dimension three.

Abstract

We consider transitive Anosov diffeomorphisms for which every periodic orbit has only one positive and one negative Lyapunov exponent. We establish various properties of such systems including strong pinching, C^{1+\beta} smoothness of the Anosov splitting, and C^1 smoothness of measurable invariant conformal structures and distributions. We apply these results to volume preserving diffeomorphisms with two-dimensional stable and unstable distributions and diagonalizable derivatives of the return maps at periodic points. We show that a finite cover of such a diffeomorphism is smoothly conjugate to an Anosov automorphism of a torus. As a corollary we obtain local rigidity for such diffeomorphisms. We also establish a local rigidity result for Anosov diffeomorphisms in dimension three.