Abstract: 

This is joint work with Krzysztof Krupiński.

We study maximal WAP and tame (in the sense of topological dynamics) quotients of $S_X(\text{C})$, where $\text{C}$ is a sufficiently saturated (called monster) model of a complete theory $T$, $X$ is a $\mathrm{\varnothing }$-type-definable set, and $S_X(\text{C})$ is the space of complete types over $\text{C}$ concentrated on $X$. We introduce a natural condition (which we call compatibility) on closed, invariant equivalence relations $F$ on $S_X(\text{C})$ and $F^{\prime }$ on $S_X({\text{C}}^{\prime })$, guaranteeing that the Ellis groups of the quotient flows $(\text{aut}(\text{C}),S_X(\text{C})/F)$ and $(\text{aut}({\text{C}}^{\prime }),S_X({\text{C}}^{\prime })/F)$ are isomorphic  as long as $\text{C}\prec {\text{C}}^{\prime }$ are ${\mathrm{\aleph }}_0$-saturated and strongly ${\mathrm{\aleph }}_0$-homogeneous. Using these results, we show that the Ellis (or ideal) groups of $(\text{aut}(\text{C}),S_X(\text{C})/F_{WAP})$ and  $(\text{aut}(\text{C}),S_X(\text{C})/F_{Tame})$ do not depend on the choice of the monster model $\text{C}$, where $F_{WAP}and$F_{Tame} are the finest  closed, $\text{aut}(\text{C})$-invariant equivalence relations on $S_X(\text{C})$ such that the quotient flows are WAP and Tame, respectively.

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