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Ultrafilters and non-Cantor minimal sets in linearly ordered dynamical systems

Ultrafilters and non-Cantor minimal sets in linearly ordered dynamical systems It is well known that infinite minimal sets for continuous functions on the interval are Cantor sets; that is, compact zero dimensional metrizable sets without isolated points. On the other hand, it was proved in Alcaraz and Sanchis (Bifurcat Chaos 13:1665–1671, 2003) that infinite minimal sets for continuous functions on connected linearly ordered spaces enjoy the same properties as Cantor sets except that they can fail to be metrizable. However, no examples of such subsets have been known. In this note we construct, in ZFC, $${2^{\mathfrak{c}}}$$ non-metrizable infinite pairwise non-homeomorphic minimal sets on compact connected linearly ordered spaces. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Archive for Mathematical Logic Springer Journals

Ultrafilters and non-Cantor minimal sets in linearly ordered dynamical systems

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References (7)

Publisher
Springer Journals
Copyright
Copyright © 2008 by Springer-Verlag
Subject
Mathematics; Algebra; Mathematics, general; Mathematical Logic and Foundations
ISSN
0933-5846
eISSN
1432-0665
DOI
10.1007/s00153-008-0069-3
Publisher site
See Article on Publisher Site

Abstract

It is well known that infinite minimal sets for continuous functions on the interval are Cantor sets; that is, compact zero dimensional metrizable sets without isolated points. On the other hand, it was proved in Alcaraz and Sanchis (Bifurcat Chaos 13:1665–1671, 2003) that infinite minimal sets for continuous functions on connected linearly ordered spaces enjoy the same properties as Cantor sets except that they can fail to be metrizable. However, no examples of such subsets have been known. In this note we construct, in ZFC, $${2^{\mathfrak{c}}}$$ non-metrizable infinite pairwise non-homeomorphic minimal sets on compact connected linearly ordered spaces.

Journal

Archive for Mathematical LogicSpringer Journals

Published: Jun 17, 2008

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