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

Published online by Cambridge University Press:  01 April 1998

O. S. KOZLOVSKI
Affiliation:
Department of Mathematics, University of Amsterdam, Amsterdam, The Netherlands

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.

Type
Research Article
Copyright
1998 Cambridge University Press

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