Relative Equilibrium States and Dimensions of Fiberwise Invariant Measures for Distance Expanding Random Maps

We show that the Gibbs states (known from [9] to be unique) of Hölder continuous potentials and random distance expanding maps coincide with relative equilibrium states of those potentials, proving in particular that the latter exist and are unique. In the realm of conformal expanding random maps we prove that given an ergodic (globally) invariant measure with a given marginal, for almost every fiber the corresponding conditional measure has dimension equal to the ratio of the relative metric entropy and the Lyapunov exponent. Finally we show that there is exactly one invariant measure whose conditional measures are of full dimension. It is the canonical Gibbs state.