Balanced Measures on Compact Median Algebras

TL;DR

This paper explores the dynamics of group actions on compact median algebras, revealing that invariant measures are uniform on cubes and that amenable groups fix sub-cubes, advancing understanding of median algebra symmetries.

Contribution

It introduces a systematic approach to group actions on compact median algebras and characterizes invariant measures, connecting measure invariance to geometric structure.

Findings

Invariant measures are uniform on cubes.

Amenable group actions fix sub-cubes.

Provides a new framework for median algebra dynamics.

Abstract

We initiate a systematic investigation of group actions on compact medain algebras via the corresponding dynamics on their spaces of measures. We show that a probability measure which is invariant under a natural push forward operation must be a uniform measure on a cube and use this to show that every amenable group action on a locally convex compact median algebra fixed a sub-cube.