Abstract: 

This is joint work with Krzysztof Krupiński.

We study maximal WAP and tame (in the sense of topological dynamics) quotients of $S_X(\C)$, where $\C$ is a sufficiently saturated (called monster) model of a complete theory $T$, $X$ is a $\emptyset$-type-definable set, and $S_X(\C)$ is the space of complete types over $\C$ concentrated on $X$. We introduce a natural condition (which we call compatibility) on closed, invariant equivalence relations $F$ on $S_X(\C)$ and $F'$ on $S_X(\C')$, guaranteeing that the Ellis groups of the quotient flows $(\aut(\C),S_X(\C)/F)$ and $(\aut(\C'),S_X(\C')/F)$ are isomorphic  as long as $\C\prec \C'$ are $\aleph_0$-saturated and strongly $\aleph_0$-homogeneous. Using these results, we show that the Ellis (or ideal) groups of $( \aut(\C), S_X(\C)/F_{WAP} )$ and  $( \aut(\C), S_X(\C)/F_{Tame})$ do not depend on the choice of the monster model $\C$, where $F_{WAP} and $F_{Tame} are the finest  closed, $\aut(\C)$-invariant equivalence relations on $S_X(\C)$ such that the quotient flows are WAP and Tame, respectively.

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