Boundedness and absoluteness of some dynamical invariants in model theory

By Krzysztof Krupinski (Uniwersytet Wroclawski/Mathematical Institute)
Abstract: Let C be a monster model of a theory T. With a given ∅-type-definable set X one can associate the flow (Aut(C), SX(C)), where SX(C) is the compact space of all global types concentrated on X. Then topological dynamics yields various useful notions such as the Ellis semigroup EL of this flow together with minimal left ideals of EL which in turn are decomposed as disjoint unions of certain isomorphic groups whose isomorphism type is sometimes called the Ellis group of the flow in question. These notions turned out to be essential to understand the complexity of Galois groups of first order theories and “spaces” of strong types. Now, we investigate whether these notions have a modeltheoretic nature, i.e. whether they are independent of the choice of the monster model C; if a notion does not depend on C, we call it absolute. We introduce a 3 notion of content of a tuple of types (which in a sense generalizes fundamental orders in stable theories), and, using it, we show that the Ellis group of the flow (Aut(C), SX(C)) is always absolute, and we provide an absolute bound on its size. In contrast, minimal left ideals of EL do not have to be of bounded size, but if they are, we also show the appropriate version of absoluteness. Moreover, we characterize when minimal left ideals of EL are of bounded size (which is particularly interesting in NIP theories). All of this comes from my joint paper with Ludomir Newelski and Pierre Simon.