Tameness of actions on finite rank median algebras
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911607262543872 |
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| author | Megrelishvili, Michael |
| author_facet | Megrelishvili, Michael |
| contents | 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 E. 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, and in particular to complete finite-rank median metric spaces under continuous isometric actions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_01681 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Tameness of actions on finite rank median algebras Megrelishvili, Michael Dynamical Systems Functional Analysis General Topology 37B05, 54H20, 52A01, 20F65 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 E. 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, and in particular to complete finite-rank median metric spaces under continuous isometric actions. |
| title | Tameness of actions on finite rank median algebras |
| topic | Dynamical Systems Functional Analysis General Topology 37B05, 54H20, 52A01, 20F65 |
| url | https://arxiv.org/abs/2601.01681 |