Self-similar fractals as boundaries of networks

TL;DR

This paper constructs a novel network model for self-similar fractals, representing them as boundaries of reversible Markov chains, and demonstrates the convergence of random walks to these fractal boundaries.

Contribution

It introduces the first network-based boundary representation of connected self-similar fractals using finite energy functions.

Findings

Constructed weighted graphs with fractal boundaries

Proved almost sure convergence of random walks to the boundary

Established a new connection between fractals and Markov chains

Abstract

For a given pcf self-similar fractal, a certain network (weighted graph) is constructed whose ideal boundary is (homeomorphic to) the fractal. This construction is the first representation of a connected self-similar fractal as the boundary of a reversible Markov chain (i.e., a simple random walk on a network). The boundary construction is effected using certain functions of finite energy which behave like bump functions on the boundary. The random walk is shown to converge to the boundary almost surely, with respect to the standard measure on its trajectory space.