The Hausdorff dimension of a class of random self-similar fractal trees

TL;DR

This paper constructs a class of random self-similar fractal trees and calculates their Hausdorff dimension using a novel percolative approach that relaxes previous geometric constraints.

Contribution

It introduces a new method to determine the Hausdorff dimension of random self-similar fractal trees without requiring uniform bounds on scaling factors.

Findings

Hausdorff dimension of the fractal trees is explicitly calculated.

A percolative argument is used to relax geometric constraints.

Results on the height of the recursive structures are obtained.

Abstract

In this article a collection of random self-similar fractal dendrites is constructed, and their Hausdorff dimension is calculated. Previous results determining this quantity for random self-similar structures have relied on geometrical properties of an underlying metric space or the scaling factors being bounded uniformly away from 0. However, using a percolative argument, and taking advantage of the tree-like structure of the sets considered here, it is shown that conditions such as these are not necessary. The scaling factors of the recursively defined structures in consideration form what is known as a multiplicative cascade, and results about the height of this random object are also obtained.