Product recurrence and distal points

Recurrence is studied in the context of actions of compact semigroups on compact spaces. (An important case is the action of the Stone-Cech compactification of an acting group.) If the semigroup E acts on the space X and F is a closed subsemigroup of E, then x in X is said to be Frecurrent if px = x for some p ∈ F, and product F-recurrent if whenever y is an F-recurrent point (in some space Y on which E acts) the point (x, y) in the product system is F-recurrent. The main result is that, under certain conditions, a point is product F-recurrent if and only if it is a distal point.