We study convolution semigroups of invariant/finitely satisfiable Keisler measures in
NIP groups. We show that the ideal (Ellis) subgroups are always trivial and describe
minimal left ideals in the definably amenable case, demonstrating that they
always form a Bauer simplex. Under some assumptions, we give an explicit
construction of a minimal left ideal in the semigroup of measures from a minimal
left ideal in the corresponding semigroup of types (this includes the case
of ,
which is not definably amenable). We also show that the canonical
pushforward map is a homomorphism from definable convolution
on
to classical convolution on the compact group
, and use it to
classify
-invariant
idempotent measures.
With gratitude to Ehud Hrushovski,
whose beautiful ideas have deeply influenced the
authors.
