On tame amenable flows and generalized Newelski's conjecture

Abstract

We are developing some topological dynamics below a given ambit. We define and study Bohr compactifications below ambits and other related objects. We describe a natural epimorphism from the Ellis group of an ambit to the Bohr compactification below its Ellis semigroup ambit. The first main result says that, assuming tameness and hereditary amenability, this natural map is an isomorphism of topological groups. This is an abstract dynamical generalization of Newelski's conjecture, or rather a theorem of Chernikov and Simon that a natural map from the Ellis group of the flow $(S_G(M),G)$ to the definable Bohr compactification of $G$ is an isomorphism when $G$ is a definably amenable group definable in a NIP structure $M$. We give two proofs of our main result: a shorter dynamical proof and an analytic proof. In the course of the dynamical proof, we prove some results on the universal compactification flow and on the generalized Bohr compactification below a given ambit which admits a left continuous semigroup operation extending the action of $G$. The considerations around the analytic proof yield a description of the measured structure of amenable tame minimal flows, which is the second main result of this paper. We also study specializations of the above abstract topological dynamics context to model theory contexts to recover some known and to obtain some new notions and results. The main new results in the model theory contexts are: (i) a description of the Bohr compactification of the group of automorphisms of a sufficiently saturated model $\mathfrak{C}$ of a first order theory $T$ below a certain fundamental ambit associated with $\mathfrak{C}$; (ii) transfer of amenability of a NIP theory $T$ to amenability of the Ellis semigroup ambit; (iii) a version of Newelski's conjecture for spaces of local types with a description of Bohr compactifications below such spaces.