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- Title
DEFINABLE TOPOLOGICAL DYNAMICS.
- Authors
KRUPIŃSKI, KRZYSZTOF
- 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 ${G^{\rm{ }}}/G_M^{{\rm{ }}00}$ (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 $G_{{\rm{\Delta }},M}^{{\rm{ }}00}$ and $G_{{\rm{\Delta }},M}^{{\rm{ }}000}$, and show that ${G^{\rm{ }}}/G_{{\rm{\Delta }},M}^{{\rm{ }}00}$ is the Δ-definable Bohr compactification of G. We also note that some deeper arguments from [14] can be adapted to our 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.
- Subjects
TOPOLOGICAL dynamics; DEFINABILITY theory (Mathematical logic); MORPHISMS (Mathematics); SEMIGROUPS (Algebra); MODEL theory
- Publication
Journal of Symbolic Logic, 2017, Vol 82, Issue 3, p1080
- ISSN
0022-4812
- Publication type
Article
- DOI
10.1017/jsl.2017.32