Conformal hyperbolic distances and holonomies of Anosov diffeomorphisms

Abstract

We use conformal hyperbolic distances to analyze the stable/unstable holonomies of Anosov diffeomorphisms. The main result is that the lengths of the stable/unstable curves in a conformal hyperbolic distance are preserved by the stable/unstable holonomies, whenever they are well-defined. In particular, the existence of a conformal hyperbolic distance with well-defined length of stable/unstable curves ensures that the holonomies are globally defined. We exhibit a conformal hyperbolic distance with well-defined length of stable/unstable curves for each real Anosov diffeomorphism, which are Anosov diffeomorphisms with a one-dimensional finest dominated splitting where contraction/expansion in each space is constant along orbits.