Le Jan, Yves Équilibre statistique pour les produits de difféomorphismes aléatoires indépendants (Statistical equilibrium for products of random diffeomorphisms). (French) Zbl 0589.60053 C. R. Acad. Sci., Paris, Sér. I 302, 351-354 (1986). This note is concerned with properties of invariant measures of a Markov chain on a compact manifold V induced by i.i.d. diffeomorphisms. The two main assertions are: if the sum of the Lyapunov exponents associated to an invariant measure m is negative then a.s. the measure \(\mu_{\omega}\) is singular with respect to Riemannian volumes \((d\mu_{\omega}(x)dP(\omega)\) is the associated invariant measure for the skew product transformation on \(V\times \Omega)\). If furthermore the largest Lyapunov exponent is negative then a.s. \(\mu_{\omega}\) is a convex combination of Dirac measures with equal weights. Reviewer: H.Crauel Cited in 2 Documents MSC: 60H99 Stochastic analysis 37A99 Ergodic theory 58J65 Diffusion processes and stochastic analysis on manifolds 60J60 Diffusion processes Keywords:invariant measures of a Markov chain; compact manifold; diffeomorphisms; Riemannian volumes; skew product transformation; largest Lyapunov exponent; convex combination of Dirac measures PDFBibTeX XMLCite \textit{Y. Le Jan}, C. R. Acad. Sci., Paris, Sér. I 302, 351--354 (1986; Zbl 0589.60053)