Maximal WAP and tame quotients of type spaces
Krzysztof Krupiński, Adrián Portillo
TL;DR
This work establishes model-theoretic absoluteness for the Ellis groups of maximal $WAP$ and tame quotients of type spaces $S_X( rak{C})$, showing these dynamical invariants do not depend on the chosen monster model under suitable saturation and homogeneity. By introducing the finest closed, automorphism-invariant quotient relations $F_{ extrm{WAP}}$ and $F_{ extrm{Tame}}$, and developing the infinitary definability-patterns framework, the authors prove a compatibility theorem: compatible quotients yield topologically isomorphic Ellis groups across extended models. As a consequence, the Ellis groups of $( ext{Aut}( rak{C}), S_X( rak{C})/F_{ extrm{WAP}})$ and $( ext{Aut}( rak{C}), S_X( rak{C})/F_{ extrm{Tame}})$ are theory-invariant, and their relations to the classical stable/NIP quotients are clarified. The results deepen connections between stability, NIP, and topological dynamics, and suggest further generalizations to broader dynamical contexts such as Keisler measures, while highlighting open questions about the equality of certain dynamical quotients at the level of Ellis groups.
Abstract
We study maximal WAP and tame (in the sense of topological dynamics) quotients of $S_X(\mathfrak{C})$, where $\mathfrak{C}$ is a sufficiently saturated (called monster) model of a complete theory $T$, $X$ is a $\emptyset$-type-definable set, and $S_X(\mathfrak{C})$ is the space of complete types over $\mathfrak{C}$ concentrated on $X$. Namely, let $F_{\textrm{WAP}}\subseteq S_X(\mathfrak{C})\times S_X(\mathfrak{C})$ be the finest closed, $aut(\mathfrak{C})$-invariant equivalence relation on $S_X(\mathfrak{C})$ such that the flow $( aut(\mathfrak{C}), S_X(\mathfrak{C})/F_{\textrm{WAP}} )$ is WAP, and let $F_{\textrm{Tame}}\subseteq S_X(\mathfrak{C})\times S_X(\mathfrak{C})$ be the finest closed, $aut(\mathfrak{C})$-invariant equivalence relation on $S_X(\mathfrak{C})$ such that the flow $( aut(\mathfrak{C}), S_X(\mathfrak{C})/F_{\textrm{Tame}} )$ is tame. We show good behaviour of $F_{\textrm{WAP}}$ and $F_{\textrm{Tame}}$ under changing the monster model $\mathfrak{C}$. Namely, we prove that if $\mathfrak{C}'\succ \mathfrak{C}$ is a bigger monster model, $F'_{\textrm{WAP}}$ and $F'_{\textrm{Tame}}$ are the counterparts of $F_{\textrm{WAP}}$ and $F_{\textrm{Tame}}$ computed for $\mathfrak{C}'$, and $r\colon S_X(\mathfrak{C}')\to S_X(\mathfrak{C})$ is the restriction map, then $r[F'_{\textrm{WAP}}]=F_{\textrm{WAP}}$ and $r[F'_{\textrm{Tame}}]=F_{\textrm{Tame}}$. Using these results, we show that the Ellis (or ideal) groups of $( aut(\mathfrak{C}), S_X(\mathfrak{C})/F_{\textrm{WAP}} )$ and $(aut(\mathfrak{C}), S_X(\mathfrak{C})/F_{\textrm{Tame}})$ do not depend on the choice of the monster model $\mathfrak{C}$.