# New Examples of Bernoulli Algebraic Actions

## TL;DR

This paper constructs explicit examples of Bernoulli actions for noncommutative free groups, demonstrating their Bernoullicity through homoclinic points, a novel achievement in algebraic dynamics.

## Contribution

It provides the first nontrivial examples of Bernoulli actions of free groups using homoclinic points, expanding understanding of algebraic actions in noncommutative settings.

## Key findings

- Explicit Bernoulli actions for free groups are constructed.
- Homoclinic points are used to establish isomorphisms.
- First known nontrivial Bernoulli examples for free group actions.

## Abstract

We give examples of principal algebraic actions of the noncommutative free group $F$ of rank two, as well as other groups, by automorphisms of a connected compact abelian group for which there is an explicit measurable isomorphism to a Bernoulli action of the group. The isomorphism is defined using homoclinic points, a method that has been used earlier to construct symbolic covers of algebraic actions. To our knowledge, these are the first nontrivial examples of the Bernoullicity of an algebraic action of $F$.