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Let ${\mathfrak C}$ be a monster model of an arbitrary theory $T$, $\bar\alpha$ any tuple of bounded length of elements of ${\mathfrak C}$, and $\barc$ an enumeration of all elements of ${\mathfrak C}$. By $S_{\bar\alpha}({\mathfrak C})$ denote the compact space of all complete types over${\mathfrak C}$ extending $tp(\bar \alpha/\emptyset)$, and $S_{\barc}({\mathfrak C})$ is defined analogously. Then $S_{\bar \alpha}({\mathfrakC})$ 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 anexplicit bound on this size. Next, we prove that these Ellis groups do notdepend on the choice of the monster model ${\mathfrak C}$; thus, we say thatthey are absolute. We also study minimal left ideals (equivalently subflows) ofthe Ellis semigroups of the flows $S_{\bar \alpha}({\mathfrak C})$ and $S_{\barc}({\mathfrak C})$. We give an example of a NIP theory in which the minimalleft 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 notdepend on the choice of ${\mathfrak C}$) and that whenever such an ideal isbounded, then its isomorphism type is also absolute.  Assuming NIP, we give characterizations of when a minimal left ideal of theEllis semigroup of $S_{\bar c}({\mathfrak C})$ is bounded. Then we adapt aproof of Chernikov and Simon to show that whenever such an ideal is bounded,the natural epimorphism (described by Krupinski, Pillay and Rzepecki) from theEllis group of the flow $S_{\bar c}({\mathfrak C})$ to the Kim-Pillay Galoisgroup $Gal_{KP}(T)$ is an isomorphism (in particular, $T$ is G-compact). Weprovide some counter-examples for $S_{\bar \alpha}({\mathfrak C})$ in place of$S_{\bar c}({\mathfrak C})$.

Boundedness and absoluteness of some dynamical invariants in model theory