When all points are generic for ergodic measures

We establish connections between several properties of topological dynamical systems, such as:

– every point is generic for an ergodic measure,

– the map sending points to the measures they generate is continuous,

– the system splits into uniquely (alternatively, strictly) ergodic subsystems,

– the map sending ergodic measures to their topological supports is continuous,

– the Cesàro means of every continuous function converge uniformly.