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Structures of quasi-graphs and $ \omega$-limit sets of quasi-graph maps


Authors: Jiehua Mai and Enhui Shi
Journal: Trans. Amer. Math. Soc. 369 (2017), 139-165
MSC (2010): Primary 37E99, 54H20
DOI: https://doi.org/10.1090/tran/6627
Published electronically: March 21, 2016
MathSciNet review: 3557770
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Abstract: An arcwise connected compact metric space $ X$ is called a quasi-graph if there is a positive integer $ N$ with the following property: for every arcwise connected subset $ Y$ of $ X$, the space $ \overline {Y}-Y$ has at most $ N$ arcwise connected components. If a quasi-graph $ X$ contains no Jordan curve, then $ X$ is called a quasi-tree. The structures of quasi-graphs and the dynamics of quasi-graph maps are investigated in this paper. More precisely, the structures of quasi-graphs are explicitly described; some criteria for $ \omega $-limit points of quasi-graph maps are obtained; for every quasi-graph map $ f$, it is shown that the pseudo-closure of $ R(f)$ in the sense of arcwise connectivity is contained in $ \omega (f)$; it is shown that $ \overline {P(f)}=\overline {R(f)}$ for every quasi-tree map $ f$. Here $ P(f)$, $ R(f)$ and $ \omega (f)$ are the periodic point set, the recurrent point set and the $ \omega $-limit set of $ f$, respectively. These extend some well-known results for interval dynamics.


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Additional Information

Jiehua Mai
Affiliation: Institute of Mathematics, Shantou University, Shantou, Guangdong 515063, People’s Republic of China
Email: jhmai@stu.edu.cn

Enhui Shi
Affiliation: Department of Mathematics, Soochow University, Suzhou, Jiangsu, 215006, People’s Republic of China
Email: ehshi@suda.edu.cn

DOI: https://doi.org/10.1090/tran/6627
Keywords: Quasi-graph, graph, periodic point, recurrent point, $\omega$-limit point
Received by editor(s): February 23, 2013
Received by editor(s) in revised form: July 23, 2014, and December 9, 2014
Published electronically: March 21, 2016
Additional Notes: The second author is the corresponding author
Article copyright: © Copyright 2016 American Mathematical Society