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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The space of $\omega$-limit sets of a continuous map
of the interval


Authors: Alexander Blokh, A. M. Bruckner, P. D. Humke and J. Smítal
Journal: Trans. Amer. Math. Soc. 348 (1996), 1357-1372
MSC (1991): Primary 26A18, 54H20, 58F03, 58F08
MathSciNet review: 1348857
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Abstract | References | Similar Articles | Additional Information

Abstract: We first give a geometric characterization of $\omega$-limit sets. We then use this characterization to prove that the family of $\omega$-limit sets of a continuous interval map is closed with respect to the Hausdorff metric. Finally, we apply this latter result to other dynamical systems.


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Additional Information

Alexander Blokh
Affiliation: Department of Mathematics, University of Alabama in Birmingham, University Station, Birmingham, Alabama 35294-2060A
Email: ablokh@math.uab.edu

A. M. Bruckner
Affiliation: Department of Mathematics, University of California at Santa Barbara, Santa Barbara, California 93106
Email: bruckner@math.ucsb.edu

P. D. Humke
Affiliation: Department of Mathematics, St. Olaf College, Northfield, Minnesota 55057
Email: humke@stolaf.edu

J. Smítal
Affiliation: Institute of Mathematics, Silesian University, 74601 Opava, Czech Republic
Email: smitalum@fpf.slu.cz

DOI: http://dx.doi.org/10.1090/S0002-9947-96-01600-5
PII: S 0002-9947(96)01600-5
Keywords: Interval maps, Hausdorff metric, $\omega$-limit sets, periodic points
Received by editor(s): January 3, 1995
Additional Notes: This work was partially done during the first author’s visit to MSRI (the research at MSRI funded in part by NSF grant DMS-9022140) whose support the first author acknowledges with gratitude. The fourth author (J. S.) was supported in part by the Grant Agency of Czech Republic, Grant No. 201/94/1088. Much of the work was accomplished during separate visits by authors 1,3, and 4 to Santa Barbara; these authors wish to express their gratitude to the Department of Mathematics at the University of California–Santa Barbara and specifically to Andy Bruckner for his kind hospitality.
Article copyright: © Copyright 1996 American Mathematical Society