## Product recurrence and distal points

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- by J. Auslander and H. Furstenberg PDF
- Trans. Amer. Math. Soc.
**343**(1994), 221-232 Request permission

## 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.

## References

- Joseph Auslander,
*Minimal flows and their extensions*, North-Holland Mathematics Studies, vol. 153, North-Holland Publishing Co., Amsterdam, 1988. Notas de Matemática [Mathematical Notes], 122. MR**956049** - Robert Ellis,
*Lectures on topological dynamics*, W. A. Benjamin, Inc., New York, 1969. MR**0267561** - H. Furstenberg,
*Recurrence in ergodic theory and combinatorial number theory*, Princeton University Press, Princeton, N.J., 1981. M. B. Porter Lectures. MR**603625** - Hillel Furstenberg,
*IP-systems in ergodic theory*, Conference in modern analysis and probability (New Haven, Conn., 1982) Contemp. Math., vol. 26, Amer. Math. Soc., Providence, RI, 1984, pp. 131–148. MR**737395**, DOI 10.1090/conm/026/737395 - Hillel Furstenberg and Benjamin Weiss,
*The finite multipliers of infinite ergodic transformations*, The structure of attractors in dynamical systems (Proc. Conf., North Dakota State Univ., Fargo, N.D., 1977) Lecture Notes in Math., vol. 668, Springer, Berlin, 1978, pp. 127–132. MR**518553** - S. Glasner and D. Maon,
*Rigidity in topological dynamics*, Ergodic Theory Dynam. Systems**9**(1989), no. 2, 309–320. MR**1007412**, DOI 10.1017/S0143385700004983 - Yitzhak Katznelson and Benjamin Weiss,
*When all points are recurrent/generic*, Ergodic theory and dynamical systems, I (College Park, Md., 1979–80), Progr. Math., vol. 10, Birkhäuser, Boston, Mass., 1981, pp. 195–210. MR**633765**

## Additional Information

- © Copyright 1994 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**343**(1994), 221-232 - MSC: Primary 54H20
- DOI: https://doi.org/10.1090/S0002-9947-1994-1170562-X
- MathSciNet review: 1170562