Fraenkel-Mostowski models revisited

A dynamical ideal is a group G acting on a set X, together with an ideal I on X invariant under the action. A dynamical ideal comes with an associated model of ZF set theory. I will discuss several theorems which connect dynamical properties of the ideal with fragments of axiom of choice in the associated model. This brings up many natural questions in combinatorics, topology, and other fields. A paper is available at dideals2.pdf (ufl.edu).