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Stirring our way to Sharkovsky's theorem

Published online by Cambridge University Press:  17 April 2009

Seth Patinkin
Seth Patinkin
Affiliation:
Department of MathematicsIndiana UniversityBloomington IN 47406United States of America e-mail: spatinki@indiana.edu
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Abstract

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The periodic-point or cycle structure of a continuous map of a topological space has been a subject of great interest since A.N. Sharkovsky completely explained the hierarchy of periodic orders of a continuous map f: R → R, where R is the real line. In this paper the topological idea of “stirring” is invoked in an effort to obtain a transparent proof of a generalisation of Sharkovsky's Theorem to continuous functions f: I → I where I is any interval. The stirring approach avoids all graph-theoretical and symbolic abstraction of the problem in favour of a more concrete intermediate-value-theorem-oriented analysis of cycles inside an interval.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 56 , Issue 3 , December 1997 , pp. 453 - 458
Copyright
Copyright © Australian Mathematical Society 1997

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