Martin boundary theory on inhomogeneous fractals

We consider fractals generated by a probabilistic iterated function scheme with open set condition and we interpret the probabilities as weights for every part of the fractal. In the homogeneous 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 inhomogeneous Martin boundary coincides with the homogeneous case.

Keywords

Martin boundary, Markov chains, Green function, fractals

2010 Mathematics Subject Classification

28A80, 31C35, 60J10

Full Text (PDF format)

Dedicated to the memory of Professor Ka-Sing Lau

Received 22 January 2023

Accepted 12 October 2023

Published 12 July 2024