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.
|Number of pages||21|
|Journal||Annales de l'institut Henri Poincare (B) Probability and Statistics|
|State||Published - Nov 2016|
Bibliographical noteFunding Information:
We thank Mikael de la Salle for pointing out an imprecision in the statement of Proposition 3.3 in a previous version. GA's research was supported by the Israel Science Foundation (Grant 1471) and by a grant from the GIF, the German-Israeli Foundation for Scientific Research and Development. OA was partially supported by NSERC and ENS in Paris. NMB was introduced to this subject by Anna Erschler, and thanks her for several conversations. The work of NMB was partially supported by the ERC staring Grant GA 257110 "RaWG."
© Association des Publications de l'Institut Henri Poincaré, 2016.
- Groups acting on rooted trees
- Liouville property
- Random walk entropy
- Recurrent Schreier graphs