An integral formula for topological entropy of $C^{\infty}$ maps

Abstract

In this paper we consider a smooth dynamical system $f$ and give estimates of the growth rates of vector fields and differential forms in the $L_p$ norm under the action of the dynamical system in terms of entropy, topological pressure and Lyapunov exponents. We prove a formula for the topological entropy $$h_{\rm top}=\lim_{n\to\infty} \frac 1n \log \int \Vert Df_x^n\,^{\wedge}\Vert \,dx,$$ where $Df_x^n\,^{\wedge}$ is a mapping between full exterior algebras of the tangent spaces. An analogous formula is given for the topological pressure.