Maximal WAP and tame quotients of type spaces

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 $\varnothing$-type-definable set, and $S_X(\mathfrak{C})$ is the space of complete types over $\mathfrak{C}$ concentrated on $X$. Namely, let $F_{\text{WAP}}\subseteq S_X(\mathfrak{C})\times S_X(\mathfrak{C})$ be the finest closed, $aut(\mathfrak{C})$-invariant equivalence relation on $S_X(\mathfrak{C})$ such that the flow $(aut(\mathfrak{C}),S_X(\mathfrak{C})/F_{\text{WAP}})$ is WAP, and let $F_{\text{Tame}}\subseteq S_X(\mathfrak{C})\times S_X(\mathfrak{C})$ be the finest closed, $aut(\mathfrak{C})$-invariant equivalence relation on $S_X(\mathfrak{C})$ such that the flow $(aut(\mathfrak{C}),S_X(\mathfrak{C})/F_{\text{Tame}})$ is tame. We show good behaviour of $F_{\text{WAP}}$ and $F_{\text{Tame}}$ under changing the monster model $\mathfrak{C}$. Namely, we prove that if ${\mathfrak{C}}^{\prime }\succ \mathfrak{C}$ is a bigger monster model, $F_{\text{WAP}}^{\prime }$ and $F_{\text{Tame}}^{\prime }$ are the counterparts of $F_{\text{WAP}}$ and $F_{\text{Tame}}$ computed for ${\mathfrak{C}}^{\prime }$, and $r:S_X({\mathfrak{C}}^{\prime })\to S_X(\mathfrak{C})$ is the restriction map, then $r[F_{\text{WAP}}^{\prime }]=F_{\text{WAP}}$ and $r[F_{\text{Tame}}^{\prime }]=F_{\text{Tame}}$. Using these results, we show that the Ellis (or ideal) groups of $(aut(\mathfrak{C}),S_X(\mathfrak{C})/F_{\text{WAP}})$ and $(aut(\mathfrak{C}),S_X(\mathfrak{C})/F_{\text{Tame}})$ do not depend on the choice of the monster model $\mathfrak{C}$.