Distance Expanding Random Mappings, Thermodynamic Formalism, Gibbs Measures, and Fractal Geometry

In this paper we define distance expanding random dynamical systems. We develop the appropriate thermodynamic formalism of such systems. We obtain in particular the existence and uniqueness of invariant Gibbs states, the appropriate pressure function and exponentially fast convergence of iterates of Perron-Frobenius operators resulting, in particular, in an exponential decay of correlations. We also obtain the formula for the derivative of the expected value of the pressure function. Next, we define and investigate in detail conformal random expanding repellers. Applying the developed machinery of thermodynamic formalism we prove a version of Bowen's formula which identifies the Hausdorff dimension $h$ of almost all fibers with the only zero of expected value of the pressure function. We then turn to more refined fractal properties by, firstly, showing that the multifractal formalism of the Gibbs states is valid and, secondly, that the $h$--Hausdorff measure vanishes while the corresponding packing measure is infinite provided the system is not quasi-deterministic.