  We study the definable topological dynamics $(G(M), S_G(M))$ 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 definable amenable group. We focus on
the problem raised by Neweslki 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 of "On minimal flows, definably amenable groups, and
o-minimality" to a more general context.
