Ergodic decomposition of group actions on rooted trees

TL;DR

This paper establishes a general framework for decomposing group actions on the boundaries of rooted trees into ergodic components, linking them to orbit trees and invariant measures, with applications to automaton groups.

Contribution

It introduces a novel decomposition theorem for ergodic components of group actions on rooted trees, connecting them to orbit trees and uniform measures, especially for automaton groups.

Findings

Identified ergodic components with boundary of orbit trees.

Showed the system of invariant measures matches uniform measures on minimal invariant subtrees.

Applied the results to groups like lamplighter, Sushchansky, and Universal groups.

Abstract

We prove a general result about the decomposition on ergodic components of group actions on boundaries of spherically homogeneous rooted trees. Namely, we identify the space of ergodic components with the boundary of the orbit tree associated with the action, and show that the canonical system of ergodic invariant probability measures coincides with the system of uniform measures on the boundaries of minimal invariant subtrees of the tree. A special attention is given to the case of groups generated by finite automata. Few examples, including the lamplighter group, Sushchansky group, and the, so called, Universal group are considered in order to demonstrate applications of the theorem.