Nonmetrizable topological dynamical characterization of central sets

Hong-Ting Shi; Hong-Wei Yang

Fundamenta Mathematicae (1996)

  • Volume: 150, Issue: 1, page 1-9
  • ISSN: 0016-2736

Abstract

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Without the restriction of metrizability, topological dynamical systems ( X , ⟨ T s ⟩ s ∈ G ) are defined and uniform recurrence and proximality are studied. Some well known results are generalized and some new results are obtained. In particular, a topological dynamical characterization of central sets in an arbitrary semigroup (G,+) is given and shown to be equivalent to the usual algebraic characterization.

How to cite

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Shi, Hong-Ting, and Yang, Hong-Wei. "Nonmetrizable topological dynamical characterization of central sets." Fundamenta Mathematicae 150.1 (1996): 1-9. <http://eudml.org/doc/212160>.

@article{Shi1996,
abstract = {Without the restriction of metrizability, topological dynamical systems $(X,⟨ T_s⟩_\{s ∈ G\})$ are defined and uniform recurrence and proximality are studied. Some well known results are generalized and some new results are obtained. In particular, a topological dynamical characterization of central sets in an arbitrary semigroup (G,+) is given and shown to be equivalent to the usual algebraic characterization.},
author = {Shi, Hong-Ting, Yang, Hong-Wei},
journal = {Fundamenta Mathematicae},
keywords = {topological dynamical system; enveloping semigroup; uniform recurrence; proximality; minimal idempotent; central subset},
language = {eng},
number = {1},
pages = {1-9},
title = {Nonmetrizable topological dynamical characterization of central sets},
url = {http://eudml.org/doc/212160},
volume = {150},
year = {1996},
}

TY - JOUR
AU - Shi, Hong-Ting
AU - Yang, Hong-Wei
TI - Nonmetrizable topological dynamical characterization of central sets
JO - Fundamenta Mathematicae
PY - 1996
VL - 150
IS - 1
SP - 1
EP - 9
AB - Without the restriction of metrizability, topological dynamical systems $(X,⟨ T_s⟩_{s ∈ G})$ are defined and uniform recurrence and proximality are studied. Some well known results are generalized and some new results are obtained. In particular, a topological dynamical characterization of central sets in an arbitrary semigroup (G,+) is given and shown to be equivalent to the usual algebraic characterization.
LA - eng
KW - topological dynamical system; enveloping semigroup; uniform recurrence; proximality; minimal idempotent; central subset
UR - http://eudml.org/doc/212160
ER -

References

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  1. [1] J. Auslander, Minimal Flows and their Extensions, North-Holland, Amsterdam, 1988. Zbl0654.54027
  2. [2] V. Bergelson and N. Hindman, Nonmetrizable topological dynamics and Ramsey theory, Trans. Amer. Math. Soc. 320 (1990), 293-320. Zbl0725.22001
  3. [3] J. Berglund and N. Hindman, Filters and the weak almost periodic compactification of a discrete semigroup, Trans. Amer. Math. Soc. 284 (1984), 1-38. Zbl0548.22002
  4. [4] J. Berglund, H. Junghenn and P. Milnes, Analysis on Semigroups, Wiley, New York, 1989. Zbl0727.22001
  5. [5] E. van Douwen, The Čech-Stone compactification of a discrete groupoid, Topology Appl. 39 (1991), 43-60. Zbl0758.54011
  6. [6] R. Ellis, A semigroup associated with a transformation group, Trans. Amer. Math. Soc. 94 (1960), 272-281. Zbl0094.17402
  7. [7] R. Ellis, Locally compact transformation groups, Duke Math. J. 24 (1957), 119-125. Zbl0079.16602
  8. [8] R. Ellis, Lectures on Topological Dynamics, Benjamin, New York, 1969. 
  9. [9] R. Engelking, General Topology, Polish Scientific Publishers, Warszawa, 1977. 
  10. [10] Z. Frolík, Sums of ultrafilters, Bull. Amer. Math. Soc. 73 (1967), 87-91. Zbl0166.18602
  11. [11] H. Furstenberg, Recurrence in Ergodic Theory and Combinatorical Number Theory, Princeton University Press, Princeton, 1981. 
  12. [12] L. Gillman and M. Jerison, Rings of Continuous Functions, van Nostrand, Princeton, 1960. Zbl0093.30001
  13. [13] N. Hindman, Ultrafilters and Ramsey Theory - an update, in: Set Theory and its Applications, J. Steprāns and S. Watson (eds.), Lecture Notes in Math. 1401, Springer, 1989, 97-118. 

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