The concept of generic points goes back to the work of Krylov and
Bogolyubov. In 1964, Dowker and Lederer were among the first to study
systems in which all points are generic for some invariant measure. It
turns out that combining this property with some form of topological
regularity can lead to measure-theoretic rigidity, for example,
minimality then implies unique ergodicity.

A natural alternative to minimality is to assume continuity of the map
that assigns to each point the invariant measure for which it is
generic. In this setting, recent results for abelian group actions show
that every point is generic for some ergodic measure--and even more,
each orbit closure is uniquely ergodic.

In this talk, I will show that these conclusions no longer hold for
general actions of amenable groups, providing explicit counterexamples
involving the group of orientation-preserving homeomorphisms of the unit
interval and the Lamplighter group.

This is joint work with G. Fuhrmann and T. Hauser.