The relationship between (Ellis) topological dynamics and definable groups – a topic treated by Ludomir Newelski in a series of papers – is considered here in the case of a (first-order complete) NIP theory \(T\) and a group \(G\) \(\emptyset\)-definable in its models. It is known that in this setting a canonical type-definable group \(G^{00}\) and a corresponding quotient group \(G/G^{00}\) – a compact group – can be associated to \(G\), and the latter provides a basic invariant of \(G\).
Now let \(M\) be any model of \(T\). An externally definable subset of \(G(M)\) is a set of the form \(G(M) \cap D\) where \(D\) is a subset of \(G\) definable possibly with parameters outside of \(M\). Let \(S_{G, \mathrm{ext}} (M)\) denote the Stone space of the Boolean algebra of externally definable subsets of \(G(M)\). Then \(G(M)\) acts on \(S_{G, \mathrm{ext}}(M)\) on the left, by homeomorphisms. As observed by Newelski, \(S_{G, \mathrm{ext}}(M)\) coincides with its enveloping Ellis semigroup and in particular admits a semigroup operation \(\cdot\), continuous in the first coordinate. According to the Ellis theory, if \(I\) is a minimal left ideal of \(S_{G, \mathrm{ext}}(M)\) and \(u \in I\) is an idempotent, then \(u \cdot I\) is a group. Newelski asked if this group is related to \(G/G^{00}\), in particular if these two groups coincide. A positive answer is given here when \(G\) is measure-stable.
The existence of generics for \(S_{G, \mathrm{ext}} (M)\) is also proved, indeed a one-one correspondence is established between these external generics of \(M\) and the global generic types of \(G\). Moreover it is shown that, for \(G\) a definably amenable group in a NIP theory \(T\), \(G/G^{00}\) can be recovered via the Ellis theory from a natural Ellis semigroup structure on the space of global \(f\)-generic types of \(G\).