Entropy of physical measures for $C^{\infty}$ Dynamical systems

Abstract

For a $C^\infty$ 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 $\Lambda\, df:\Lambda \,TM\circlearrowleft$ at $x$. This contrasts with the well-known Ruelle inequality. As a consequence we give some refinement of Tsujii’s work relating physical and Sinai-Ruelle-Bowen measures.