Characterization of measures satisfying the Pesin entropy formula for random dynamical systems (Q1271034)

Suppose \(f:M\to M\) is a \(C^2\) diffeomorphism of a compact Riemannian manifold \(M\), \(\mu\) is an invariant probability measure for \(f\), \(h_\mu(f)\) is the associated entropy, and \(\lambda_i(x)\), \(1\leq i\leq d=\dim M\), are the Lyapunov exponents in \(x\in M\), counted with multiplicities. The Pesin formula \(h_\mu(f)= \sum_1^d \int \lambda_i^+(x) d\mu(x)\) is known to hold if and only if \(\mu\) is ``absolutely continuous along unstable directions'' [\textit{F. Ledrappier} and \textit{L.-S. Young}, Ann. Math. (2) 122, 509-539 (1985; Zbl 0605.58028)]. This had been generalized to products of independent and identically distributed \(C^2\) diffeomorphisms [\textit{Pei-Dong Liu} and \textit{M. Qian}, Smooth ergodic theory of random dynamical systems, Springer Lect. Notes Math. 1606 (1995; Zbl 0841.58041)]. Here the result is extended further to general products of \(C^2\) diffeomorphisms in stationary (and ergodic) dependence on a probability space which is assumed to be a Polish space equipped with a Borel probability measure.