On Anosov diffeomorphisms with asymptotically conformal periodic data

Details On Anosov diffeomorphisms with asymptotically conformal periodic data On Anosov diffeomorphisms with asymptotically conformal periodic data

2007-09-24

arxiv.org/abs/0709.3790v2

Mathematics

Dynamical Systems

To appear in Ergodic Theory and Dynamical Systems

Scientific paper

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.

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