Sensitivity of dendrite maps

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• 作者：Enhui Shi^a ; ^{ehshi@suda.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Suhua Wang^b ; ^{wangsuhuajust@163.com" class="auth_mail" title="E-mail the corresponding author} ; Yan Di^c ; ^{diyan5216@sina.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Dendrite ; Sensitivity ; Topological entropy ; Topological transitivity
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：1 February 2017
• 年：2017
• 卷：446
• 期：1
• 页码：908-919
• 全文大小：423 K

文摘

Suppose that X   is a dendrite and f:X→X$f:X\to X$ is a sensitive continuous map. We show that (a) fcb0fff35a64f1d0d" title="Click to view the MathML source">(X,f)$(X,f)$ contains a bilaterally transitive subsystem with nonempty interior; (b) the system fcb0fff35a64f1d0d" title="Click to view the MathML source">(X,f)$(X,f)$ satisfies only one of the following two conditions: (b1) fcb0fff35a64f1d0d" title="Click to view the MathML source">(X,f)$(X,f)$ contains a topologically transitive subsystem with nonempty interior; (b2) there exists an f-invariant nowhere dense closed subset A of X   such that the attraction basin Basin(A,f)$\mathrm{Basin}(A,f)$ contains a residual subset B   of an open set and the strong attraction basin Sbasin(A,f)$\mathrm{Sbasin}(A,f)$ is dense in B; (c) if X   is completely regular, then fcb0fff35a64f1d0d" title="Click to view the MathML source">(X,f)$(X,f)$ contains a relatively strongly mixing subsystem with nonempty interior, dense periodic points and positive topological entropy. Unlike for interval maps, we construct a sensitive dendrite map with zero topological entropy.