Second Descendible Self-Mapping with Closed Periodic Points Set

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Let and *f**:**X**n**→**X**n* be a continuous map. If *f* is a second descendible map,
then *P**(**f**)* is closed if and only if
one of the following hold: 1) ; 2) For any *z ε R (f)*, there exists a yεw *(**z**,**f**) **∩ **P**(**f**)* such that every point of
the set *orb (y,f)** i*s a isolated point of the
set *w (z,f)*; 3) For any *z **ε **R**(**f**)*, the set *w (z,f)* is finite; 4) For any *z **ε **R**(**f**)*, the set *w' (z,f)* is finite. The consult give
another condition of *f* with closed periodic set
other than [1].

References

[1] R.-J. Du, Y.-G. Jin and M.-X. Li, “On the Periodic Point Set of a n-Dimensional Self-Mapping,” Journal of Chongqing Technology Business University (Natural Science Edition), Vol. 23, No. 1, 2006, pp. 12-14 (in Chinese).

[2] J.-C. Xiong, “Continous Self-Maps of the Closed Interval Whose Periodic Points Form a Closed Set,” Journal of China University of Science and Technology, Vol. 11, No. 4, 1981, pp. 14-22.

[3] G. R. Zhang, R. H. Li and C. L. Yang, “Anti-Triangular Maps with Closed Periodic Sets,” Far East Journal of Dynamical Systems, Vol. 19, No. 1, 2012, pp. 1-11.

[4] J. C. Xiong, “Dynamical Systems of Interval Maps: Non wandering Set, Topological Entropy and Chaos,” Advances in Mathematics, Vol. 17, No. 1, 1988 (in Chinese).

[5] L. D. Wang, “The Necessary and Sufficient Condition of a Closed Periodic Points Set of the Continuous Self-Maps of the Circle,” Journal of Shanxi Teacher’s University (Natural Science Edition), Vol. 4, No. 2, 1990 (in Chinese).

[6] B.-G. Yan, “The Chaotic Set and Asymptotic Periodic Point of Descendible Mapping,” Journal of Hebei University of Engineering (Natural Science Edition), Vol. 24, No. 3, 2007, pp. 102-103 (in Chinese).