Random fractals and tree-indexed Markov chains

TL;DR

This paper investigates the size properties of complex fractal sets generated by tree-indexed random processes, extending classical models like Mandelbrot's fractal percolation, and introduces a generalized framework for such fractals.

Contribution

It presents a new general model of fractals based on tree-indexed random compacts and Markov chains, broadening the scope of previous recursive fractal constructions.

Findings

Analysis of size properties of the new fractal model

Extensions of Mandelbrot's fractal percolation process

Potential applications in complex fractal generation

Abstract

We study the size properties of a general model of fractal sets that are based on a tree-indexed family of random compacts and a tree-indexed Markov chain. These fractals may be regarded as a generalization of those resulting from the Moran-like deterministic or random recursive constructions considered by various authors. Among other applications, we consider various extensions of Mandelbrot's fractal percolation process.