  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 $\bar
c$ 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 $S_{\bar
c}({\mathfrak C})$ is defined analogously. Then $S_{\bar \alpha}({\mathfrak
C})$ 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 an
explicit bound on this size. Next, we prove that these Ellis groups do not
depend on the choice of the monster model ${\mathfrak C}$; thus, we say that
they are absolute. We also study minimal left ideals (equivalently subflows) of
the Ellis semigroups of the flows $S_{\bar \alpha}({\mathfrak C})$ and $S_{\bar
c}({\mathfrak C})$. We give an example of a NIP theory in which the minimal
left 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 not
depend on the choice of ${\mathfrak C}$) and that whenever such an ideal is
bounded, then its isomorphism type is also absolute.
  Assuming NIP, we give characterizations of when a minimal left ideal of the
Ellis semigroup of $S_{\bar c}({\mathfrak C})$ is bounded. Then we adapt a
proof of Chernikov and Simon to show that whenever such an ideal is bounded,
the natural epimorphism (described by Krupinski, Pillay and Rzepecki) from the
Ellis group of the flow $S_{\bar c}({\mathfrak C})$ to the Kim-Pillay Galois
group $Gal_{KP}(T)$ is an isomorphism (in particular, $T$ is G-compact). We
provide some counter-examples for $S_{\bar \alpha}({\mathfrak C})$ in place of
$S_{\bar c}({\mathfrak C})$.
