# Bernoulli disjointness

## TL;DR

This paper extends Furstenberg's result to show Bernoulli flows are disjoint from all minimal flows for any infinite discrete group, revealing new structural properties of minimal functions and flows, and constructing specific free, minimal, and proximal flows.

## Contribution

It generalizes disjointness results to all infinite groups, analyzes algebraic structures of minimal functions, and constructs new types of flows with specific properties.

## Key findings

- Bernoulli flow $2^G$ is disjoint from all minimal $G$-flows for infinite groups
- The algebra generated by minimal functions is a proper subalgebra of $ell^\infty(G)$
- Existence of continuum many mutually disjoint minimal, free, metrizable $G$-flows

## Abstract

Generalizing a result of Furstenberg, we show that for every infinite discrete group $G$, the Bernoulli flow $2^G$ is disjoint from every minimal $G$-flow. From this, we deduce that the algebra generated by the minimal functions $\mathfrak{A}(G)$ is a proper subalgebra of $\ell^\infty(G)$ and that the enveloping semigroup of the universal minimal flow $M(G)$ is a proper quotient of the universal enveloping semigroup $\beta G$. When $G$ is countable, we also prove that for any metrizable, minimal $G$-flow, there exists a free, minimal flow disjoint from it and that there exist continuum many mutually disjoint minimal, free, metrizable $G$-flows. Finally, improving a result of Frisch, Tamuz, and Vahidi Ferdowsi and answering a question of theirs, we show that if $G$ is a countable icc group, then it admits a free, minimal, proximal flow.