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Let ℭ be a monster model of an arbitrary theory T , let α ̄ be any (possibly infinite) tuple of bounded length of elements of ℭ , and let c ̄ be an enumeration of all elements of ℭ (so a tuple of unbounded length). By S α ̄ (ℭ) we denote the compact space of all complete types over ℭ extending tp (α ̄ / ∅) , and S c ̄ (ℭ) is defined analogously. Then S α ̄ (ℭ) and S c ̄ (ℭ) are naturally Aut (ℭ) -flows (even Aut (ℭ) -ambits). We show that the Ellis groups of both these flows are of bounded size (i.e. smaller than the degree of saturation of ℭ), providing an explicit bound on this size. Next, we prove that these Ellis groups do not depend (as groups equipped with the so-called τ -topology) on the choice of the monster model ℭ ; thus, we say that these Ellis groups are absolute. We also study minimal left ideals (equivalently subflows) of the Ellis semigroups of the flows S α ̄ (ℭ) and S c ̄ (ℭ). We give an example of a NIP theory in which the minimal left ideals are of unbounded size. Then we show that in each of these two cases, boundedness of a minimal left ideal (equivalently, of all the minimal left ideals) is an absolute property (i.e. it does not depend on the choice of ℭ) and that whenever such an ideal is bounded, then in some sense its isomorphism type is also absolute. Under the assumption that T has NIP, we give characterizations (in various terms) of when a minimal left ideal of the Ellis semigroup of S c ̄ (ℭ) is bounded. Then we adapt the proof of Theorem 5.7 in Definably amenable NIP groups, J. Amer. Math. Soc.31 (2018) 609–641 to show that whenever such an ideal is bounded, a certain natural epimorphism (described in [K. Krupiński, A. Pillay and T. Rzepecki, Topological dynamics and the complexity of strong types, Israel J. Math.228 (2018) 863–932]) from the Ellis group of the flow S c ̄ (ℭ) to the Kim–Pillay Galois group Gal KP (T) is an isomorphism (in particular, T is G-compact). We also obtain some variants of these results, formulate some questions, and explain differences (providing a few counterexamples) which occur when the flow S c ̄ (ℭ) is replaced by S α ̄ (ℭ). [ABSTRACT FROM AUTHOR]