Denker–Sato type Markov chains and Harnack inequality

Abstract

In ([DS1], [DS2], [DS3]), Denker and Sato studied a Markov chain on the finite words space of the Sierpinski gasket (SG). They showed that the Martin boundary is homeomorphic to the SG. Recently, Lau and Wang (2015 Math. Z. 280 401–20) showed that the homeomorphism holds for an iterated function system with the open set condition provided that the transition probability on the finite words space is of DS-type. In this work, we continue studying this kind of transition probability on the unit interval. Using matrix expressions, we obtain a formula to calculate the Green function. By the ergodic arguments for non-negative matrices, we find that the Martin boundary is homeomorphic to the unit interval or the union of the unit interval and a countable set. This gives a good illustration for the results in Lau and Wang (2015 Math. Z. 280 401–20).