Maximizing entropy measures for random dynamical systems

We prove the existence of relative maximal entropy measures for certain random dynamical systems of the type F(x, y) = (theta(x), f(x)(y)), where. is an invertibe map preserving an ergodic measure P and f(x) is a local diffeomorphism of a compact Riemannian manifold exhibiting some non-uniform expansion. As a consequence of our proofs, we obtain an integral formula for the relative topological entropy as the integral of the logarithm of the topological degree of f(x) with respect to P. When F is topologically exact and the supremum of the topological degree of f(x) is finite, the maximizing measure is unique and positive on open sets.