Exact dimensionality and projections of random self-similar measures and sets

TL;DR

This paper investigates the geometric properties of random self-similar measures, demonstrating their exact-dimensionality and analyzing their projections and sections, with applications to fractal sets and percolation.

Contribution

It extends existing results on self-similar measures to the random case, removing separation conditions and applying to a broader class of fractal measures.

Findings

Random self-similar measures are almost surely exact-dimensional.

Projections and sections of these measures retain exact-dimensionality.

New results on projections, $C^1$-images, and distance sets for fractal measures.

Abstract

We study the geometric properties of random multiplicative cascade measures defined on self-similar sets. We show that such measures and their projections and sections are almost surely exact-dimensional, generalizing Feng and Hu's result \cite{FeHu09} for self-similar measures. This, together with a compact group extension argument, enables us to generalize Hochman and Shmerkin's theorems on projections of deterministic self-similar measures \cite{HoSh12} to these random measures without requiring any separation conditions on the underlying sets. We give applications to self-similar sets and fractal percolation, including new results on projections, $C^1$-images and distance sets.