Martin boundary theory on inhomogenous fractals

TL;DR

This paper extends Martin boundary theory to inhomogeneous fractals generated by probabilistic iterated function schemes, showing that the boundary remains unchanged despite weighting variations.

Contribution

It redefines transition probabilities considering weights and proves the Martin boundary coincides with the homogeneous case for inhomogeneous fractals.

Findings

Martin boundary remains the same in inhomogeneous case

Transition probabilities can be redefined with weights

Homogeneous and inhomogeneous boundaries are homeomorphic

Abstract

We want to consider fractals generated by a probabilistic iterated function scheme with open set condition and we want to interpret the probabilities as weights for every part of the fractal. In the homogenous case, where the weights are not taken into account, Denker and Sato introduced in 2001 a Markov chain on the word space and proved, that the Martin boundary is homeomorphic to the fractal set. Our aim is to redefine the transition probability with respect to the weights and to calculate the Martin boundary. As we will see, the inhomogenous Martin boundary coincides with the homogenous case.