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

 

Multi-separation, centrifugality and centripetality imply chaos


Author: Jiehua Mai
Journal: Trans. Amer. Math. Soc. 351 (1999), 343-351
MSC (1991): Primary 58F03; Secondary 58F13, 26A18
MathSciNet review: 1473450
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Abstract: Let $I$ be an interval. $I$ need not be compact or bounded. Let $f:I\rightarrow I$ be a continuous map, and $(x_0, x_1, \cdots, x_n)$ be a trajectory of $f$ with $x_n\leq x_0<x_1$ or $x_1<x_0\leq x_n$. Then there is a point $v\in I$ such that $\min\{x_0, \cdots, x_n\}<v=f(v)<\max\{x_0, \cdots, x_n\}$. A point $y\in I$ is called a centripetal point of $f$ relative to $v$ if $y$ or $v<f(y)<y$, and $y$ is centrifugal if $f(y)<y<v$ or $v<y<f(y)$. In this paper we prove that if there exist $k$ centripetal points of $f$ in $\{x_0, \cdots, x_{n-1}\}, k\geq 1$, then $f$ has periodic points of some odd ($\not= 1$) period $p\leq (n-2)/k+2$. In addition, we also prove that if $(x_0, x_1, \cdots, x_{n-1}$) is multi-separated by Fix($f$), or there exists a centrifugal point of $f$ in $\{x_0, \cdots, x_{n-1}\}$, then $f$ is turbulent and hence has periodic points of all periods.


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Additional Information

Jiehua Mai
Affiliation: Institute of Mathematics, Shantou University, Shantou, Guangdong 515063 P. R. China
Email: jhmai@mailserv.stu.edu.cn

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02191-1
PII: S 0002-9947(99)02191-1
Keywords: Return trajectory, centripetal point, centrifugal point, period, turbulence, chaos
Received by editor(s): January 30, 1997
Additional Notes: This research was supported by the National Natural Science Foundation of China
Article copyright: © Copyright 1999 American Mathematical Society