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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Structures of quasi-graphs and $\omega$-limit sets of quasi-graph maps
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by Jiehua Mai and Enhui Shi PDF
Trans. Amer. Math. Soc. 369 (2017), 139-165 Request permission

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
  • MR Author ID: 710093
  • Email: ehshi@suda.edu.cn
  • 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
  • © Copyright 2016 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 369 (2017), 139-165
  • MSC (2010): Primary 37E99, 54H20
  • DOI: https://doi.org/10.1090/tran/6627
  • MathSciNet review: 3557770