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2003-2004 Chaos and the recurrent set.
T. H. Steele
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Real Anal. Exchange 29(1): 79-87 (2003-2004).


Let $f$ be an element of $C(I,I)$ with $R(f)=\{x\in I:x\in \omega (x,f)\}$ its recurrent set. We study the relationship between the structure of $R(f)$ and the chaotic nature of the function $f$ . We show that $R(f)$ is always a $G_{\delta }$ set whenever $f$ has zero topological entropy, although $R(f)$ is closed for the typical continuous function $f$ with zero topological entropy. We also develop necessary and sufficient conditions on $f$ for $R(f)$ to be closed.


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T. H. Steele. "Chaos and the recurrent set.." Real Anal. Exchange 29 (1) 79 - 87, 2003-2004.


Published: 2003-2004
First available in Project Euclid: 9 June 2006

zbMATH: 1069.37008
MathSciNet: MR2061294

Primary: 26A18 , 37B20 , 54H20‎

Keywords: $\omega $-limit point , recurrent point , topological entropy

Rights: Copyright © 2003 Michigan State University Press

Vol.29 • No. 1 • 2003-2004
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