Intrinsic uniform structure on median algebras

TL;DR

This paper introduces the median uniformity on median algebras, explores its properties, and applies it to group actions, resulting in a new compactification framework with implications for dynamics.

Contribution

It defines the median uniformity, studies its properties, and develops the Minimal Median Compactification, extending classical topologies and analyzing group actions.

Findings

Median uniformity is Hausdorff under natural conditions.

The MMC coincides with the Roller compactification when all intervals are finite.

Finite-rank median algebras lead to tame dynamical systems.

Abstract

We introduce the median uniformity $\mathcal U_{\mathrm m}$, an intrinsic precompact convex uniform structure on a median algebra. It is Hausdorff under natural assumptions, for instance for finite-rank median algebras. In the Hausdorff case, its uniform completion yields the Minimal Median Compactification (MMC). The induced topology $\tau_{\mathrm m}$ provides a natural higher-rank analogue of the interval topology on linearly ordered sets and of the shadow topology on rank-one median algebras. When all intervals in the median algebra $X$ are finite, the MMC is the unique proper median compactification of $(X,\tau_{\mathrm m})$; in particular, it coincides with the Roller compactification. We apply this uniform framework to continuous actions of a topological group $G$ by median automorphisms. We show that the MMC is a median $G$-compactification. In the finite-rank case, the resulting compact $G$-system is Rosenthal representable and hence dynamically tame.