Boundedness and absoluteness of some dynamical invariants in model theory

Abstract

Let $\mathfrak{C}$ be a monster model of an arbitrary theory $T$, $\bar{\hat{I}}\pm$ any tuple of bounded length of elements of $\mathfrak{C}$, and $\bar{c}$ an enumeration of all elements of $\mathfrak{C}$. By $S_{\bar{\hat{I}}\pm }(\mathfrak{C})$ denote the compact space of all complete types over $\mathfrak{C}$ extending $tp(\bar{\hat{I}}\pm /\varnothing )$, and $S_{\bar{c}}(\mathfrak{C})$ is defined analogously. Then $S_{\bar{\hat{I}}\pm }(\mathfrak{C})$ and $S_{\bar{c}}(\mathfrak{C})$ are naturally $Aut(\mathfrak{C})$-flows. We show that the Ellis groups of both these flows are of bounded size (i.e. smaller than the degree of saturation of $\mathfrak{C}$), providing an explicit bound on this size. Next, we prove that these Ellis groups do not depend on the choice of the monster model $\mathfrak{C}$; thus, we say that they are absolute. We also study minimal left ideals (equivalently subflows) of the Ellis semigroups of the flows $S_{\bar{\hat{I}}\pm }(\mathfrak{C})$ and $S_{\bar{c}}(\mathfrak{C})$. 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 $\mathfrak{C}$) 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 $S_{\bar{c}}(\mathfrak{C})$ 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 $S_{\bar{c}}(\mathfrak{C})$ to the Kim-Pillay Galois group $Ga{l}_{KP}(T)$ is an isomorphism (in particular, $T$ is G-compact). We provide some counter-examples for $S_{\bar{\hat{I}}\pm }(\mathfrak{C})$ in place of $S_{\bar{c}}(\mathfrak{C})$.