A topological version of a theorem of Veech and almost simple flows

Almost simple (AS) minimal flows are defined and it is shown that any factor map of an AS flow is, up to almost 1−1 equivalence, a group factor. An analogous theorem for metric, regular, point distal extensions is proved. In particular a theorem of Gottschalk is strengthened to show that any regular, point distal, metric flow is equicontinuous. When the acting group T is commutative it is shown that every proper minimal joining of an AS flow X and a minimal flow Y, is, up to almost 1−1 extensions, the relative product of X and Y over a common factor which is a group factor of X.