Phase transitions for nonsingular Bernoulli actions

TL;DR

This paper investigates phase transitions in nonsingular Bernoulli actions, extending previous Gaussian action results, and provides detailed ergodic property descriptions for actions from groups acting on trees based on their Poincaré exponent.

Contribution

It introduces new phase transition results for nonsingular Bernoulli actions and characterizes their ergodic properties in relation to group actions on trees.

Findings

Established phase transition phenomena for nonsingular Bernoulli actions.

Provided a precise ergodic classification based on the Poincaré exponent.

Extended Gaussian action phase transition results to Bernoulli actions.

Abstract

Inspired by the phase transition results for nonsingular Gaussian actions introduced in arXiv:1911.04272, we prove several phase transition results for nonsingular Bernoulli actions. For generalized Bernoulli actions arising from groups acting on trees, we are able to give a very precise description of their ergodic theoretical properties in terms of the Poincar\'e exponent of the group.