Abstract
In this chapter we consider a uniform action φ: T × X → X with X compact, and compare the semigroup and family viewpoints. Recall that by Lemma 1.2 any topological action of a uniform monoid on a compact space is a uniform action. The focus of our comparison is the uniform Stone-Čtech compactification of the uniform monoid T. In itself ß u T combines three different phenomena. First T acts uniformly on the compact space ß u T, which is the orbit closure of j u (0) in ß u T. Furthermore using the maps Φ x , we see that this action is the universal compact T action ambit [see (6.18)]. Next ß u T is an Ellis semigroup mapping onto the enveloping semigroup, S φ , by the homomorphism Φ# [see (6.13)]. Finally we recall that ß u T can be regarded as the space of maximal open filters on T as in Theorem 5.2. This connects ß u T with all of the family constructions in Chapters 3 and 4.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1997 Springer Science+Business Media New York
About this chapter
Cite this chapter
Akin, E. (1997). Semigroups and Families. In: Recurrence in Topological Dynamics. The University Series in Mathematics. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-2668-8_8
Download citation
DOI: https://doi.org/10.1007/978-1-4757-2668-8_8
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4419-3272-3
Online ISBN: 978-1-4757-2668-8
eBook Packages: Springer Book Archive