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

 

Relatively pointwise recurrent graph map


Author: Hattab Hawete
Journal: Proc. Amer. Math. Soc. 139 (2011), 2087-2092
MSC (2010): Primary 37B20, 37E25
Published electronically: November 9, 2010
MathSciNet review: 2775386
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Abstract: Let $ f$ be a self-continuous map of a graph $ G$. Let $ P(f)$ and $ R(f)$ denote the sets of periodic points and recurrent points respectively. We say that the map $ f$ is relatively recurrent if $ \overline{R(f)} = G$. In this paper, it is shown that $ f$ is relatively recurrent if and only if one of the following statements holds:

(a)
$ G$ is a circle and $ f$ is a homeomorphism topologically conjugate to an irrational rotation of the unit circle $ \mathbb{S}^1$;
(b)
$ \overline{P(f)} = G$.
Part (b) extends a result of Blokh.


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Additional Information

Hattab Hawete
Affiliation: Institut Supérieur d’Informatique et du Multimedia, Route de Tunis, Km 10, B.P. 242, Sfax 3021, Tunisia
Email: hattab.hawete@yahoo.fr

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10622-6
PII: S 0002-9939(2010)10622-6
Keywords: Periodic point, recurrent point, almost periodic point, minimal set, relatively pointwise recurrent map, graph map
Received by editor(s): April 10, 2010
Received by editor(s) in revised form: June 2, 2010
Published electronically: November 9, 2010
Communicated by: Yingfei Yi
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.