Exact dimensionality and projections of random self-similar measures and sets (Q2922844)

The authors study the geometric properties of projections and sections of random multiplicative cascade measures on self-similar sets. Firstly, they construct the probability space underlying the random cascade measures and then obtain an ergodic random dynamical system on the space of these measures. The compact group extension theorem is applied to show that the skew product of this random dynamical system with the rotation group \(G\) is also ergodic. Later on, they take advantage of all these ergodicities and prove that almost surely a random multiplicative cascade measure along with its projections and sections is exact dimensional. Finally, these results are applied to self-similar sets and fractal percolations. New results on projections, \(\mathbb{C}^1\)-images and distance sets are also included.