  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.
