The dynamical hierarchy for Roelcke precompact Polish groups

Tomás Ibarlucía (1, 2)

1 ICJ - Institut Camille Jordan
2 UCBL - Université Claude Bernard Lyon 1

We study several distinguished function algebras on a Polish group $G$, under the assumption that $G$ is Roelcke precompact. We do this by means of the model-theoretic translation initiated by Ben Yaacov and Tsankov: we investigate the dynamics of $\aleph_0$-categorical metric structures under the action of their automorphism group. We show that, in this context, every strongly uniformly continuous function (in particular, every Asplund function) is weakly almost periodic. We also point out the correspondence between tame functions and NIP formulas, deducing that the isometry group of the Urysohn sphere is $\Tame\cap\UC$-trivial.

Mots clés: tame; NIP; WAP; SUC; dynamical hierarchy; Roelcke precompact

Domaines: Logique [math.LO]; Systèmes dynamiques [math.DS]

Tomás Ibarlucía. The dynamical hierarchy for Roelcke precompact Polish groups. Israel Journal of Mathematics, 2016, 215 (2), pp.965-1009. ⟨10.1007/s11856-016-1399-1⟩. ⟨hal-00992525v4⟩