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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A zero topological entropy map with recurrent points not $F_\sigma$
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by Petra Šindelářová PDF
Proc. Amer. Math. Soc. 131 (2003), 2089-2096 Request permission

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 Šindelářová
  • Affiliation: Mathematical Institute, Silesian University in Opava, Bezručovo nám. 13, 746 01 Opava, Czech Republic
  • Email: Petra.Sindelarova@math.slu.cz
  • 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
  • © Copyright 2003 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 131 (2003), 2089-2096
  • MSC (2000): Primary 26A18, 37E05
  • DOI: https://doi.org/10.1090/S0002-9939-03-06971-5
  • MathSciNet review: 1963754