Product recurrence and distal points

Auslander, J.; Furstenberg, H.

doi:10.1090/S0002-9947-1994-1170562-X

Product recurrence and distal points
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by J. Auslander and H. Furstenberg

Trans. Amer. Math. Soc. 343 (1994), 221-232

DOI: https://doi.org/10.1090/S0002-9947-1994-1170562-X

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Abstract:

Recurrence is studied in the context of actions of compact semigroups on compact spaces. (An important case is the action of the Stone-Čech 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 F-recurrent if $px = x$ for some $p \in F$, 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.

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