On minimal flows and definable amenability in some distal NIP theories

Abstract

We study the definable topological dynamics (G(M),SG(M)) of a definable group acting on its type space, where M is either an o-minimal structure or a p-adically closed field, and G a definably amenable group. We focus on the problem raised in [13] of whether weakly generic types coincide with almost periodic types, showing that the answer is positive when G has boundedly many global weakly generic types. We also give two “minimal counterexamples” where G has unboundedly many global weakly generic types, extending the main results in [21] to a more general context.