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

摘要

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