The metric entropy of diffeomorphisms

Let M be a C compact Riemannian manifold without boundary, let be a C 2 difFeomorphism of M, and let ra be an /-invariant Borel probability measure on M. The entropy of , h m (f), measures the amount of randomness with respect to ra generated by iterating . In this note we announce some results concerning the relationship between h m (f) and the geometric properties of and ra. Our main results are the characterization of measures satisfying Pesin's entropy formula (Theorem 1) and the generalization of this formula to one that applies to all invariant measures (Theorem 2). This generalized formula involves the notion of dimension and leads to other volume estimates (Theorem 3, Corollaries 4 and 5).