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"> 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"> 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"> 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"> 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"> 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"> 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"> 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.