Boundary Theory on the Hata Tree

Ka-Sing Lau, The Chinese University of Hong Kong
Sze-Man Ngai, Georgia Southern University

Abstract

We prove that for a certain Markov chain on the symbolic space of the Hata tree K, the Martin boundary M is homeomorphic to the trunk of the Hata tree, and the minimal Martin boundary is the post-critical set {12̇,1̇,2̇}, which corresponds to the three vertices of the trunk. Moreover, the class of P-harmonic functions on M coincides with Kigami’s class of harmonic functions on K.