Product recurrence and distal points
Authors: J. Auslander and H. Furstenberg
Journal: Trans. Amer. Math. Soc. 343 (1994), 221-232
MSC: Primary 54H20
MathSciNet review: 1170562
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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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H. Furstenberg, Recurrence in ergodic theory and combinatorial number theory, Princeton Univ. Press, 1981.
---, IP systems in ergodic theory, Conference in Modern Analysis and Probability, Contemporary Math., vol. 26, Amer. Math. Soc., Providence, RI, 1984, pp. 131-148.
H. Furstenberg and B. Weiss, The finite multipliers of infinite ergodic transformations, Structure of Attractors in Dynamical Systems, Lecture Notes in Math., vol. 668, Springer, 1978, pp. 127-133.
S. Glasner and D. Maon, Rigidity in topological dynamics, Ergodic Theory Dynamical Systems 9 (1989), 309-320.
Y. Katznelson and B. Weiss, When all points are recurrent/generic, Progress in Math., vol. 10, (Proc. Special Year, Maryland 1979-1980), Birkhaüser Boston, 1981, pp. 195-210.
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