Dimension of uniformly random self-similar fractals

TL;DR

This paper determines the Hausdorff dimension of a class of randomly generated self-similar fractals and identifies conditions under which these fractals have positive Lebesgue measure.

Contribution

It provides a formula for the almost sure Hausdorff dimension of uniformly random self-similar fractals and establishes positivity of Lebesgue measure in certain cases.

Findings

Calculated the almost sure Hausdorff dimension of the fractals.

Proved the Lebesgue measure is positive under specific conditions.

Extended understanding of measure-theoretic properties of random fractals.

Abstract

We calculate the almost sure Hausdorff dimension of uniformly random self-similar fractals. These random fractals are generated from a finite family of similarities, where the linear parts of the mappings are independent uniformly distributed random variables at each step of iteration. We also prove that the Lebesgue measure of such sets is almost surely positive in some cases.