The Liouville property for groups acting on rooted trees
arXiv:1307.5652 · doi:10.1214/15-AIHP697
Abstract
We show that on groups generated by bounded activity automata, every symmetric, finitely supported probability measure has the Liouville property. More generally we show this for every group of automorphisms of bounded type of a rooted tree. For automaton groups, we also give a uniform upper bound for the entropy of convolutions of every symmetric, finitely supported measure.
Major changes in the statement and proof of Theorem 1, it now holds for all groups of automorphisms of bounded type, not necessarily finite-state. Final version, to appear in Annales de l'IHP