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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305236&_mathId=si1.gif&_user=111111111&_pii=S0022247X16305236&_rdoc=1&_issn=0022247X&md5=3fa01e72af82248a5684c556f0808d69" title="Click to view the MathML source">f:X→Xclass="mathContainer hidden">class="mathCode">$f:X\to X$ is a sensitive continuous map. We show that (a) class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305236&_mathId=si2.gif&_user=111111111&_pii=S0022247X16305236&_rdoc=1&_issn=0022247X&md5=926ce935a82f96bfcb0fff35a64f1d0d" title="Click to view the MathML source">(X,f)class="mathContainer hidden">class="mathCode">$(X,f)$ contains a bilaterally transitive subsystem with nonempty interior; (b) the system class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305236&_mathId=si2.gif&_user=111111111&_pii=S0022247X16305236&_rdoc=1&_issn=0022247X&md5=926ce935a82f96bfcb0fff35a64f1d0d" title="Click to view the MathML source">(X,f)class="mathContainer hidden">class="mathCode">$(X,f)$ satisfies only one of the following two conditions: (b1) class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305236&_mathId=si2.gif&_user=111111111&_pii=S0022247X16305236&_rdoc=1&_issn=0022247X&md5=926ce935a82f96bfcb0fff35a64f1d0d" title="Click to view the MathML source">(X,f)class="mathContainer hidden">class="mathCode">$(X,f)$ 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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305236&_mathId=si3.gif&_user=111111111&_pii=S0022247X16305236&_rdoc=1&_issn=0022247X&md5=84720deb2f40d3e47b5d002e43fd4487" title="Click to view the MathML source">Basin(A,f)class="mathContainer hidden">class="mathCode">$\mathrm{Basin}(A,f)$ contains a residual subset B   of an open set and the strong attraction basin class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305236&_mathId=si4.gif&_user=111111111&_pii=S0022247X16305236&_rdoc=1&_issn=0022247X&md5=cdc37477d81426c91ff9afa9e56b23a6" title="Click to view the MathML source">Sbasin(A,f)class="mathContainer hidden">class="mathCode">$\mathrm{Sbasin}(A,f)$ is dense in B; (c) if X   is completely regular, then class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305236&_mathId=si2.gif&_user=111111111&_pii=S0022247X16305236&_rdoc=1&_issn=0022247X&md5=926ce935a82f96bfcb0fff35a64f1d0d" title="Click to view the MathML source">(X,f)class="mathContainer hidden">class="mathCode">$(X,f)$ 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.