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)

For G a group definable in some structure M, we define notions of “definable” compactification of G and “definable” action of G on a compact space X , where the latter is under a definability of types assumption on M. We describe the universal definable compactification of G as View the MathML source and the universal definable G-ambit as the type space SG. We also point out the existence and uniqueness of “universal minimal definable G-flows”, and discuss issues of amenability (...) and extreme amenability in this definable category, with a characterization of the latter. For the sake of completeness we also describe the universal compactification and universal G-ambit in model-theoretic terms, when G is a topological group. (shrink)

We develop “local NIP group theory” in the context of pseudofinite groups. In particular, given a sufficiently saturated pseudofinite structure G expanding a group, and left invariant NIP formula δ...

We investigate some dynamical features of the actions of automorphisms in the context of model theory. We interpret a few notions such as compact systems, entropy and symbolic representations from the theory of dynamical systems in the realm of model theory. In this direction, we settle a number of characterizations of NIP theories in terms of dynamics of automorphisms and invariant measures. For example, it is shown that the property of NIP corresponds to the compactness property of some associated systems (...) and also to the zero entropy property of automorphisms. These results give a correspondence between some notions of tameness in model theory and ergodic theory. Moreover, we study the concept of symbolic representation and consider it in some well known mathematical objects such as the circle group, Bohr sets, Sturmian sequences, the structure \\), and random graphs with a model theoretic point of view in mind. We establish certain characterizations for stability theoretic dividing lines, such as independence property, order property and strict order property in terms of associated symbolic representations. At the end, we propose some applications of symbolic representations and these characterizations by giving a proof for a classical theorem by Shelah and also introducing some invariants associated to the types and elements of models. (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)

Assume$G\prec H$are groups and${\cal A}\subseteq {\cal P}(G),\ {\cal B}\subseteq {\cal P}(H)$are algebras of sets closed under left group translation. Under some additional assumptions we find algebraic connections between the Ellis [semi]groups of theG-flow$S({\cal A})$and theH-flow$S({\cal B})$. We apply these results in the model theoretic context. Namely, assumeGis a group definable in a modelMand$M\prec ^* N$. Using weak heirs and weak coheirs we point out some algebraic connections between the Ellis semigroups$S_{ext,G}(M)$and$S_{ext,G}(N)$. Assuming every minimal left ideal in$S_{ext,G}(N)$is a group we prove (...) that the Ellis groups of$S_{ext,G}(M)$are isomorphic to closed subgroups of the Ellis groups of$S_{ext,G}(N)$. (shrink)

Let [Formula: see text] be a monster model of an arbitrary theory [Formula: see text], let [Formula: see text] be any tuple of bounded length of elements of [Formula: see text], and let [Formula: see text] be an enumeration of all elements of [Formula: see text]. By [Formula: see text] we denote the compact space of all complete types over [Formula: see text] extending [Formula: see text], and [Formula: see text] is defined analogously. Then [Formula: see text] and [Formula: see (...) text] are naturally [Formula: see text]-flows. We show that the Ellis groups of both these flows are of bounded size, providing an explicit bound on this size. Next, we prove that these Ellis groups do not depend on the choice of the monster model [Formula: see text]; thus, we say that these Ellis groups are absolute. We also study minimal left ideals of the Ellis semigroups of the flows [Formula: see text] and [Formula: see text]. We give an example of a NIP theory in which the minimal left ideals are of unbounded size. Then we show that in each of these two cases, boundedness of a minimal left ideal is an absolute property and that whenever such an ideal is bounded, then in some sense its isomorphism type is also absolute. Under the assumption that [Formula: see text] has NIP, we give characterizations of when a minimal left ideal of the Ellis semigroup of [Formula: see text] is bounded. Then we adapt the proof of Theorem 5.7 in Definably amenable NIP groups, J. Amer. Math. Soc. 31 609–641 to show that whenever such an ideal is bounded, a certain natural epimorphism 863–932]) from the Ellis group of the flow [Formula: see text] to the Kim–Pillay Galois group [Formula: see text] is an isomorphism. We also obtain some variants of these results, formulate some questions, and explain differences which occur when the flow [Formula: see text] is replaced by [Formula: see text]. (shrink)