We prove the existence of relative maximal entropy measures for certain random dynamical systems of the type [Formula: see text], where [Formula: see text] is an invertibe map preserving an ergodic measure [Formula: see text] and [Formula: see text] 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 [Formula: see text] with respect to [Formula: see text]. When [Formula: see text] is topologically exact and the supremum of the topological degree of [Formula: see text] is finite, the maximizing measure is unique and positive on open sets.

We consider a class of endomorphisms which contains a set of piecewise partially hyperbolic skew-products with a non-uniformly expanding base map. The aimed transformation preserves a foliation which is almost everywhere uniformly contracted with possible discontinuity sets, which are parallel to the contracting direction. We prove that the associated transfer operator, acting on suitable anisotropic normed spaces, has a spectral gap (on which we have quantitative estimation) and the disintegration of the unique invariant physical measure, along the stable leaves, is ζ-Hölder. We use this fact to obtain exponential decay of correlations on the set of ζ-Hölder functions.

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We consider a robust class of random non-uniformly expanding local homeomorphisms and Hölder continuous potentials with small variation. For each element of this class we develop the thermodynamical formalism and prove the existence and uniqueness of equilibrium states among non-uniformly expanding measures. Moreover, we show that these equilibrium states and the random topological pressure vary continuously in this setting.

The notions of expansive measures; expansive, positively expansive, measureexpansive, countably expansive and continuum-wise homeomorphisms are well known for the deterministic case. In this paper, we extend these notions to the random context and prove that some of them are particular cases of others and some of them are equivalents. Also, we show that every ergodic probability with positive entropy is positively random expansive.