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

 

A zero topological entropy map with recurrent points not $F_\sigma$


Author: Petra Sindelárová
Journal: Proc. Amer. Math. Soc. 131 (2003), 2089-2096
MSC (2000): Primary 26A18, 37E05
Published electronically: February 5, 2003
MathSciNet review: 1963754
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Abstract: We show that there is a continuous map $\chi$ of the unit interval into itself of type $2^\infty$ which has a trajectory disjoint from the set $ \operatorname{Rec}(\chi )$ of recurrent points of $\chi$, but contained in the closure of $ \operatorname{Rec}(\chi )$. In particular, $ \operatorname{Rec}(\chi )$ is not closed. A function $\psi$ of type $2^\infty$, with nonclosed set of recurrent points, was found by H. Chu and J. Xiong [Proc. Amer. Math. Soc. 97 (1986), 361-366]. However, there is no trajectory contained in $\overline {\operatorname{Rec} (\psi)}\setminus \operatorname{Rec}(\psi)$, since any point in $\overline { \operatorname{Rec}(\psi)}$ is eventually mapped into $\operatorname{Rec} (\psi)$. Moreover, our construction is simpler.

We use $\chi$ to show that there is a continuous map of the interval of type $2^\infty$ for which the set of recurrent points is not an $F_\sigma$set. This example disproves a conjecture of A. N. Sharkovsky et al., from 1989. We also provide another application of $\chi$.


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Additional Information

Petra Sindelárová
Affiliation: Mathematical Institute, Silesian University in Opava, Bezručovo nám. 13, 746 01 Opava, Czech Republic
Email: Petra.Sindelarova@math.slu.cz

DOI: http://dx.doi.org/10.1090/S0002-9939-03-06971-5
PII: S 0002-9939(03)06971-5
Keywords: Topological entropy, recurrent points, periodic points, $\omega$-limit set
Received by editor(s): January 7, 2002
Published electronically: February 5, 2003
Additional Notes: This research was supported, in part, by contracts 201/00/0859 from the Grant Agency of Czech Republic and CEZ:J10/98:192400002 from the Czech Ministry of Education. The support of these institutions is gratefully acknowledged
Dedicated: Dedicated to Professor Jaroslav Smítal on the occasion of his 60th birthday
Communicated by: Michael Handel
Article copyright: © Copyright 2003 American Mathematical Society