The second iterate of a map with dense orbit
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- by Paul S. Bourdon PDF
- Proc. Amer. Math. Soc. 124 (1996), 1577-1581 Request permission
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 $f\circ f$.References
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Additional Information
- Paul S. Bourdon
- Affiliation: Department of Mathematics, Washington and Lee University, Lexington, Virginia 24450
- Email: pbourdon@wlu.edu
- Received by editor(s): June 1, 1994
- Communicated by: James E. West
- © Copyright 1996 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 124 (1996), 1577-1581
- MSC (1991): Primary 54H20; Secondary 47A15, 58F13
- DOI: https://doi.org/10.1090/S0002-9939-96-03648-9
- MathSciNet review: 1363443