The metric entropy of random dynamical systems in a Hilbert space: Characterization of invariant measures satisfying Pesin's entropy formula

Abstract

Consider a random cocycle $\Phi$ on a separableinfinite-dimensional Hilbert space preserving a probability measure$\mu$, which is supported on a random compact set $K$. We show thatif $\Phi$ is $C^2$ (over $K$) and satisfies some mild integrableconditions of the differentials, then Pesin's entropy formula holdsif $\mu$ has absolutely continuous conditionalmeasures on the unstable manifolds. The converse is also true under an additional condition on $K$ when the system has no zero Lyapunov exponent.