Martin boundary theory on inhomogenous fractals

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.