We consider transitive Anosov diffeomorphisms for which every periodic orbit has only one positive and one negative Lyapunov exponent. We prove various properties of such systems, including strong pinching, C1+β smoothness of the Anosov splitting, and C1 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 T4; as a corollary, we obtain local rigidity for such diffeomorphisms. We also establish a local rigidity result for Anosov diffeomorphisms in dimension three.
All Science Journal Classification (ASJC) codes
- Applied Mathematics