Local stable and unstable sets for random dynamical systems

Abstract

We study local stable and unstable sets for two-sided continuous bundle random dynamical systems with positive entropy. For two-sided continuous random dynamical systems and ergodic invariant measures with positive fiber measure-theoretic entropy and positive fiber maximal Lyapunov exponent, we prove that the fiber entropy is determined by the action of the random maps on the unstable sets, and establish a lower bound for the Hausdorff dimension of local unstable sets in terms of the ratio of entropy to the maximal fiber Lyapunov exponent. If the upper box dimension of the phase space is finite, we obtain a weak form of Ruelle's inequality.

27 pages