A note on periodic points of dendrite maps and their induced maps

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• 作者：Haithem Abouda ^{haithem.abouda@yahoo.fr} ; Issam Naghmouchi ; ^{issam.naghmouchi@fsb.rnu.tn} ; ^{issam.nagh@gmail.com}
• 关键词：Dendrite maps ; Induced maps ; Periodic points
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：1 April 2017
• 年：2017
• 卷：448
• 期：1
• 页码：722-725
• 全文大小：331 K
• 卷排序：448

文摘

Let X   be a compact metric space and 2X2X be the hyperspace of all nonempty closed subsets of X   endowed with the Hausdorff metric. It is well known that for each continuous map f:X→Xf:X→X, the density of periodic points of f   implies the density of periodic points of the induced map 2f:2X→2X2f:2X→2X. Méndez (2010) conjectured in [6] that the converse is true when the phase space X is a dendrite. In [8], Ŝpitalský (2015) constructed a continuous transitive map F:X→XF:X→X on a dendrite X   with only two periodic points. We prove in this note that the set of periodic points of its induced map 2F2F is dense in 2X2X. This answers in the negative Méndez's conjecture.