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Preprint Number 1214

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1214. Krzysztof Krupinski, Ludomir Newelski, and Pierre Simon
Boundedness and absoluteness of some dynamical invariants in model theory
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Submission date: 29 April 2017


Let C be a monster model of an arbitrary theory T, a any tuple of bounded length of elements of C, and c an enumeration of all elements of C. By S_a(C) denote the compact space of all complete types over C extending tp(a/∅), and S_c(C) is defined analogously. Then S_a(C) and S_c(C) are naturally Aut(C)-flows. We show that the Ellis groups of both these flows are of bounded size (i.e. smaller than the degree of saturation of C), providing an explicit bound on this size. Next, we prove that these Ellis groups do not depend on the choice of the monster model C; thus, we say that they are absolute. We also study minimal left ideals (equivalently subflows) of the Ellis semigroups of the flows S_a(C) and S_c(C). We give an example of a NIP theory in which the minimal left ideals are of unbounded size. We show that in each of these two cases, boundedness of a minimal left ideal is an absolute property (i.e. it does not depend on the choice of C) and that whenever such an ideal is bounded, then its isomorphism type is also absolute.

Assuming NIP, we give characterizations of when a minimal left ideal of the Ellis semigroup of S_c(C) is bounded. Then we adapt a proof of Chernikov and Simon to show that whenever such an ideal is bounded, the natural epimorphism (described by Krupinski, Pillay, and Rzepecki) from the Ellis group of the flow S_c(C) to the Kim-Pillay Galois group Gal_{KP}(T) is an isomorphism (in particular, T is G-compact). We provide some counter-examples for S_a(C) in place of S_c(C).

Mathematics Subject Classification: 03C45, 54H20, 37B05

Keywords and phrases: Group of automorphisms, Ellis group, minimal flow, boundedness, absoluteness

Full text arXiv 1705.00159: pdf, ps.

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