Submission date: 17 February 2016

Abstract:

For a group G definable in a first order structure M we develop basic topological dynamics in the category of definable G-flows. In particular, we give a description of the universal definable G-ambit and of the semigroup operation on it. We find a natural epimorphism from the Ellis group of this flow to the definable Bohr compactification of G, that is to the quotient G^*/{G^*}^{00}_M (where G^* is the interpretation of G in a monster model). More generally, we obtain these results locally, i.e. in the category of Δ-definable G-flows for any fixed set Δ of formulas of an appropriate form. In particular, we define local connected components {G^*}^{00}_{Δ,M} and {G^*}^{000}_{Δ,M}, and show that G^*/{G^*}^{00}_{Δ,M} is the Δ-definable Bohr compactification of G. We also note that some deeper arguments from the topological dynamics in the category of externally definable G-flows can be adapted to the definable context, showing for example that our epimorphism from the Ellis group to the Δ-definable Bohr compactification factors naturally yielding a continuous epimorphism from the Δ-definable generalized Bohr compactification to the Δ-definable Bohr compactification of G. Finally, we propose to view certain topological-dynamic and model-theoretic invariants as Polish structures which leads to some observations and questions.

Mathematics Subject Classification: 03C45, 54H20

Keywords and phrases: model theory, topological dynamics, definable flows, model-theoretic connected components

Full text arXiv 1602.05393: pdf, ps.