Markov processes associated to fractal branch groups

TL;DR

This paper studies measure-preserving dynamical systems linked to fractal branch groups, showing they are Markov processes and classifying them via invariants like Hausdorff dimension and the $f$-invariant.

Contribution

It introduces a new analysis of these systems, demonstrating they are Markov processes and establishing classification criteria based on invariants.

Findings

Computed the $f$-invariant for these systems.

Showed they are Markov processes over free semigroups.

Identified conditions for isomorphism of the processes.

Abstract

The author introduced recently a new natural construction which associates a measure-preserving dynamical system to any fractal profinite group. Here, we investigate these measure-preserving dynamical systems under the extra assumption on the groups to be branch. First, we compute their $f$-invariant, a measure-conjugacy invariant introduced by Bowen, and show that they are Markov processes over free semigroups in the sense of Bowen. Secondly, we show that fractal branch profinite groups with the same Hausdorff dimension and whose associated measure-preserving dynamical systems have the same $f$-invariant yield isomorphic Markov processes.