Interior points and Lebesgue measure of overlapping Mandelbrot percolation sets

TL;DR

This paper investigates the conditions under which certain random self-similar fractal sets have positive Lebesgue measure and interior points, identifying critical probabilities and parameter intervals where these properties hold.

Contribution

It provides sharp bounds on critical probabilities for positivity of Lebesgue measure and interior points in random self-similar sets, extending understanding of their geometric properties.

Findings

Identified parameter intervals with positive Lebesgue measure but empty interior.

Derived sharp bounds on critical probabilities for measure positivity.

Applied recent results to characterize the geometric structure of random fractal sets.

Abstract

We consider a special one-parameter family of d-dimensional random, homogeneous self-similar iterated function systems (IFSs) satisfying the finite type condition. The object of our study is the positivity of Lebesgue measure and the existence of interior points in these random sets and in particular the existence of an interesting parameter interval where the attractor has positive Lebesgue measure, but empty interior almost surely conditioned on the attractor not being empty. We give a sharp bound on the critical probability for the case of positivity Lebesgue measure using the theory of multitype branching processes in random environments and in some special cases on the critical probability for the existence of interior points. Using a recent result of Tom Rush, we provide a family of such random sets where there exists a parameter interval for which the corresponding attractor has a positive Lebesgue measure, but empty interior almost surely conditioned on the attractor not being empty.