Entropy, Lyapunov exponents and the boundary deformation rate under the action of hyperbolic dynamical systems

Abstract

We consider an Anosov diffeomorphism of a Riemannian manifold and characterize the deformation of the boundary of a small ball in under the action of in terms of the volume of a small neighbourhood of divided by the volume of . We prove that the logarithm of this ratio divided by tends to the sum of the positive Lyapunov exponents of an arbitrary -invariant ergodic probability measure a.e. with respect to this measure, provided that increases not too fast. A statement concerning the measure-theoretic entropy of is stated as a corollary.