Transitivity of real Anosov diffeomorphisms

Abstract

We prove the transitivity of real Anosov diffeomorphisms, which are Anosov diffeomorphisms where stable and unstable spaces decompose into a continuous sum of invariant one-dimensional sub-spaces with uniform contraction/expansion over the ambient manifold. We prove that if a stable/unstable curve has a well-defined length in a conformal hyperbolic distance, then it has a globally defined holonomy. We exhibit a conformal hyperbolic distance with well-defined length of stable/unstable curves for each real Anosov diffeomorphism.