Definable topological dynamics

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 $\Delta$-definable $G$-flows for any fixed set $\Delta$ of formulas of an appropriate form. In particular, we define local connected components ${G^*}^{00}_{\Delta,M}$ and ${G^*}^{000}_{\Delta,M}$, and show that $G^*/{G^*}^{00}_{\Delta,M}$ is the $\Delta$-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 $\Delta$-definable Bohr compactification factors naturally yielding a continuous epimorphism from the $\Delta$-definable generalized Bohr compactification to the $\Delta$-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.