Percolation sur le syst\`eme \`a trois points

TL;DR

This paper investigates a three-dot system acting on a compact space, establishing a dichotomy related to percolation properties and rigidity of invariant measures, drawing analogies with circle doubling and tripling maps.

Contribution

It introduces a novel dichotomy for invariant measures of the three-dot system, linking measure rigidity to percolation phenomena.

Findings

Established a dichotomy for invariant measures.

Linked measure rigidity to percolation properties.

Drawn analogies with circle doubling and tripling maps.

Abstract

The three dot system is an action by homeomorphisms of $\mathbb Z^2$ on a compact space, which invariant measures are supposed to satisfy rigidity properties analoguous to the ones of invariant measures of angle doubling and tripling on the circle. For such measures, we establish a dichotomy which is related to percolation properties.