A zero topological entropy map with recurrent points not

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 of the unit interval into itself of type which has a trajectory disjoint from the set of recurrent points of , but contained in the closure of . In particular, is not closed. A function of type , 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 , since any point in is eventually mapped into . Moreover, our construction is simpler.

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

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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

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