## 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.## References

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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