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We consider transitive Anosov diffeomorphisms for which every periodic orbithas only one positive and one negative Lyapunov exponent. We establish variousproperties of such systems including strong pinching, C^{1+\beta} smoothness ofthe Anosov splitting, and C^1 smoothness of measurable invariant conformalstructures and distributions. We apply these results to volume preservingdiffeomorphisms with two-dimensional stable and unstable distributions anddiagonalizable derivatives of the return maps at periodic points. We show thata finite cover of such a diffeomorphism is smoothly conjugate to an Anosovautomorphism of a torus. As a corollary we obtain local rigidity for suchdiffeomorphisms. We also establish a local rigidity result for Anosovdiffeomorphisms in dimension three.

On Anosov diffeomorphisms with asymptotically conformal periodic data