Équilibre statistique pour les produits de difféomorphismes aléatoires indépendants. (Statistical equilibrium for products of independent random diffeomorphisms) (Q1819820)

Let V be a compact Riemannian variety, G be the group of diffeomorphisms on V, \(\nu\) be a probability on G, and finally \(\Pi\) (V) be the set of probabilities on V. Define a Markov kernel P, assumed ergodic, on V by \(Pf(x)=\int f(Tx)\nu (dT)\), and let m denote a P-invariant element of \(\Pi\) (V). On \(\Omega =G^ Z\) define the coordinate mappings \(T_ i\), and let \(P=\nu^ Z.\) The author investigates structural properties of the random measure \(\mu\) on V defined by \[ \mu (\omega)=\lim_{n}m[T_ 0\circ...\circ T_{- n}(\omega)]^{-1}. \] For example, let \(\lambda\) and \(\alpha\) be, respectively, the largest Lyapunov exponent and the sum of the exponents of the system; then if \(\alpha <0\), \(\mu\) is singular with respect to \(\rho\), the Riemannian volume on V, P-a.s., while if \(m\sim \rho\), then \(\alpha\leq 0\). Similarly, if \(\lambda <0\), then almost surely \(\mu\) is a finite sum of point measures of the same mass, i.e., \(\mu =(1/K)\sum^{K}_{i=1}\epsilon_{X_ i}\), where K is a finite random variable, the \(X_ i\) are random elements of V, and \(\epsilon_ x\) is the point mass at x.