A zero topological entropy map with recurrent points not
Author:
Petra Sindelárová
Journal:
Proc. Amer. Math. Soc. 131 (2003), 20892096
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), 361366]. 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/S0002993903069715
PII:
S 00029939(03)069715
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
