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Entropy formula of Pesin type for one-sided stationary random maps

  • PEI-DONG LIU (a1), MIN QIAN (a1) and FU-XI ZHANG (a1)

Abstract

In this paper, we consider random dynamical systems (abbreviated as RDSs) generated by compositions of one-sided stationary random endomorphisms of class C^2 of a compact manifold. We will first introduce the notions of entropy and Lyapunov exponents for such RDSs, and then prove that the entropy formula of Pesin type holds if the sample measures of an invariant measure are absolutely continuous with respect to the Lebesgue measure on the manifold. Our result covers those obtained by Ledrappier and Young [11], and Liu [12] for i.i.d. random diffeomorphisms or (non-invertible) endomorphisms, and that obtained by [12] for two-sided stationary random endomorphisms. If the phase spaces are compact and finite-dimensional manifolds without boundary, this result may be considered as the almost-final form of Pesin entropy formula for RDSs with absolutely continuous invariant or sample measures.

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