Tameness of actions on finite rank median algebras

Abstract

We show that for every finite-rank median algebra $X$, the rank of $X$ coincides with the independence number of the family of all median-preserving maps $X \to [0,1]$. In the compact topological case, the same equality holds for the family of all continuous median-preserving maps. Combined with Rosenthal's dichotomy, this yields a generalized Helly selection principle: for every finite-rank median algebra, every uniformly bounded sequence of median-preserving real-valued maps admits a pointwise convergent subsequence whose limit is again median-preserving. As a dynamical application, we generalize a joint result with Eli Glasner on dendrites and prove that every continuous action of a topological group by median automorphisms on a compact finite-rank median algebra is Rosenthal representable, and hence dynamically tame. We also apply this result to the Roller--Fioravanti compactification of finite-rank topological median $G$-algebras with compact intervals, and in particular to complete finite-rank median metric spaces under continuous isometric actions. In the metrizable cascade case this gives a new source of systems satisfying Sarnak's M\"obius disjointness conjecture, including natural compactifications arising from finite-dimensional cubical geometry.