Ruelle's inequality for the entropy of random diffeomorphisms (Q1202831)

Let \(\nu\) be a probability measure on the space \(\Omega\) of diffeomorphisms acting on a compact Riemannian manifold \(M\) without boundary. Let \(\mu\) denote an invariant measure for the associated random transformation on \(\Omega^ \mathbb{N}\times M\) and \(h_ \mu\) its entropy [see \textit{Y. Kifer}, Ergodic theory of random transformations (1986; Zbl 0604.28014)]. The authors prove the formula \[ h_ \mu\leq\int\sum_ i\lambda^{(i)}(x)m_ i(x)\mu(dx) \] where \(\lambda^{(i)}(x)\) denote the Lyapunov exponents and \(m_ i(x)\) their multiplicity. This result has been established in the non-random case by \textit{Ya. B. Pesin} [Russ. Math. Surv. 32, No. 4, 55-114 (1977); translation from Usp. Mat. Nauk 32, No. 4(196), 55-112 (1977; Zbl 0359.58010)] and \textit{D. Ruelle} [Bol. Soc. Bras. Mat. 9, 83-87 (1978; Zbl 0432.58013)]. This problem has also been treated in Kifer [loc. cit.], but by a different approach.