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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The second iterate of a map with dense orbit


Author: Paul S. Bourdon
Journal: Proc. Amer. Math. Soc. 124 (1996), 1577-1581
MSC (1991): Primary 54H20; Secondary 47A15, 58F13
MathSciNet review: 1363443
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Abstract: Suppose that $X$ is a Hausdorff topological space having no isolated points and that $f:X\rightarrow X$ is continuous. We show that if the orbit of a point $x\in X$ under $f$ is dense in $X$ while the orbit of $x$ under $f\circ f$ is not, then the space $X$ decomposes into three sets relative to which the dynamics of $f$ are easy to describe. This decomposition has the following consequence: suppose that $x$ has dense orbit under $f$ and that the closure of the set of points of $X$ having odd period under $f$ has nonempty interior; then $x$ has dense orbit under $f\circ f$.


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Additional Information

Paul S. Bourdon
Affiliation: Department of Mathematics, Washington and Lee University, Lexington, Virginia 24450
Email: pbourdon@wlu.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-96-03648-9
PII: S 0002-9939(96)03648-9
Received by editor(s): June 1, 1994
Communicated by: James E. West
Article copyright: © Copyright 1996 American Mathematical Society