AM  Vol.4 No.7 , July 2013
Second Descendible Self-Mapping with Closed Periodic Points Set

Let and f:XnXn 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) is 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].

Cite this paper: G. Zhang, Z. Ji and F. Zeng, "Second Descendible Self-Mapping with Closed Periodic Points Set," Applied Mathematics, Vol. 4 No. 7, 2013, pp. 969-971. doi: 10.4236/am.2013.47133.

[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).