We study the flow of trigonalizable algebraic group acting on its type space, focusing on the problem raised in [17] of whether weakly generic types coincide with almost periodic types if the group has global definable f‐generic types, equivalently whether the union of minimal subflows of a suitable type space is closed. We shall give a description of f‐generic types of trigonalizable algebraic groups, and prove that every f‐generic type is almost periodic.

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 ${G^{\rm{*}}}/G_M^{{\rm{*}}00}$. 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. (shrink)

AssumeGis a group definable in a modelMof a stable theoryT. We prove that the semigroupSG(M) of completeG-types overMis an inverse limit of some semigroups type-definable inMeq. We prove that the maximal subgroups ofSG(M) are inverse limits of some definable quotients of subgroups ofG. We consider the powers of types in the semigroupSG(M) and prove that in a way every type inSG(M) is profinitely many steps away from a type in a subgroup ofSG(M).

We interpret the basic notions of topological dynamics in the model-theoretic setting, relating them to generic types of definable group actions and their generalizations.

We consider definable topological dynamics forNIPgroups admitting certain decompositions in terms of specific classes of definably amenable groups. For such a group, we find a description of the Ellis group of its universal definable flow. This description shows that the Ellis group is of bounded size. Under additional assumptions, it is shown to be independent of the model, proving a conjecture proposed by Newelski. Finally we apply the results to new classes of groups definable in o-minimal structures, generalizing all of (...) the previous results for this setting. (shrink)

We investigate definable topological dynamics of groups definable in an o‐minimal expansion of the field of reals. Assuming that a definable group G admits a model‐theoretic analogue of Iwasawa decomposition, namely the compact‐torsion‐free decomposition, we give a description of minimal subflows and the Ellis group of its universal definable flow in terms of this decomposition. In particular, the Ellis group of this flow is isomorphic to. This provides a range of counterexamples to a question by Newelski whether the Ellis group (...) is isomorphic to. We further extend the results to universal topological covers of definable groups, interpreted in a two‐sorted structure containing the o‐minimal sort and a sort for an abelian group. (shrink)