On the dimensions of attractors of random self-similar graph directed iterated function systems

TL;DR

This paper introduces a new model of random graph directed fractals, analyzing their dimensional properties and showing that Hausdorff and box counting dimensions coincide almost surely, with explicit formulas under separation conditions.

Contribution

It extends existing models of random graph directed fractals and provides new results on their dimensional properties, including cases with overlaps and separation conditions.

Findings

Hausdorff and box counting dimensions coincide almost surely

Dimensions are explicitly characterized under strong separation conditions

Model generalizes previous random graph fractal models

Abstract

In this paper we propose a new model of random graph directed fractals that extends the current well-known model of random graph directed iterated function systems, $V$-variable attractors, and fractal and Mandelbrot percolation. We study its dimensional properties for similarities with and without overlaps. In particular we show that for the two classes of $1$-variable and $\infty$-variable random graph directed attractors we introduce, the Hausdorff and upper box counting dimension coincide almost surely, irrespective of overlap. Under the additional assumption of the uniform strong separation condition we give an expression for the almost sure Hausdorff and Assouad dimension.