Topological dynamics and definable groups

We give a commentary on Newelski’s suggestion or conjecture [8] that topological dynamics, in the sense of Ellis [3], applied to the action of a definable group G(M) on its type space SG(M), can explain, account for, or give rise to, the quotient G/G00, at least for suitable groups in NIP theories. We give a positive answer for measure-stable (or fsg) groups in NIP theories. As part of our analysis we show the existence of “externally definable” generics of G(M) for measurestable groups. We also point out that for G definably amenable (in a NIP theory) G/G00 can be recovered, via the Ellis theory, from a natural Ellis semigroup structure on the space of global f -generic types.