Classical Expanding Random Systems

Abstract

Having treated a very general situation up to here, we now focus on more concrete random repellers and, in the next section, random maps that have been considered by Denker and Gordin. The Cantor example of Chap. 5.3 and random perturbations of hyperbolic rational functions like the examples considered by Brück and Büger are typical random maps that we consider now. We classify them into quasi-deterministic and essential systems and analyze then their fractal geometric properties. Here as a consequence of the techniques we have developed, we positively answer the question of Brück and Büger (see [9] and Question 5.4 in [8]) of whether the Hausdorff dimension of almost all (most) naturally defined random Julia sets is strictly larger than 1. We also show that in this same setting the Hausdorff dimension of almost all Julia sets is strictly less than 2.