Denker–Sato type Markov chains on self-similar sets

Abstract

In Denker and Sato (Potential Anal 14: 211–232, 2001; Publ RIMS 35: 769–794, 1999; Math Nachr 241: 32–55, 2002) studied certain Markov chain on the symbolic spaces of the Sierpinski gasket (SG). They showed that the Martin boundary is homeomorphic to SG, and used the potential theory on the Martin boundary to induce a harmonic structure on SG. In this paper, we consider a more general Denker–Sato type Markov chain associated with self-similar sets $$K$$ with the open set condition. The chain is defined on the augmented tree of the symbolic space. Such tree was introduced by Kaimanovich, it is hyperbolic in the sense of Gromov (Kaimanovich in Random walks on Sierpinski graphs: hyperbolicity and stochastic homogenization, Birhauser, Basel, 2003; Lau and Wang in Indiana Univ Math J 58:1777–1795, 2009). We show that the Martin boundary, the hyperbolic boundary and the self-similar set $$K$$ are homeomorphic. The hitting distribution of the chain is also obtained.