Entropy of Physical Measures for C^∞ Dynamical Systems

For a C^∞ map f on a compact manifold M we prove that for a Lebesgue randomly picked point x there is an empirical measure from x with entropy larger than or equal to the top Lyapunov exponent of Λdf:ΛTM↺ at x. This contrasts with the well-known Ruelle inequality. As a consequence we give some refinement of Tsujii’s work [23] relating physical and Sinai-Ruelle-Bowen measures.