Entropy formula of Pesin type for noninvertible random dynamical systems
The paper is concerned with products of random \(C^2\) maps from a compact Riemannian manifold \(M\) to itself. Sufficient conditions for a formula of Pesin type, connecting entropy and Lyapunov exponents of invariant measures, are given for two cases. Firstly, the random maps are assumed to be stationary and ergodic. It is shown that if an invariant measure, which is a random variable in the space of probability measures on \(M\), is absolutely continuous (with respect to the Lebesgue measure on \(M\)) almost surely, then a formula of Pesin type holds for this invariant measure. The argument invokes Sard's theorem to obtain that points of non-invertibility are a nullset with respect to the invariant measure under consideration. Secondly, the special case of a product of independent and identically distributed maps is considered. The class of invariant measures under consideration is restricted to Markov measures. These are characterised by their expectation, which is invariant with respect to the associated Markov operator. Assuming the random maps to be local diffeomorphisms, and assuming a uniform integrability condition on the logarithm of the determinant of their derivative, a formula of Pesin type is shown to hold for invariant Markov measures whose expectation is absolutely continuous with respect to the Lebesgue measure on \(M\).
scientific article; zbMATH DE number 1336860