Chain recurrent sets of generic mappings on compact spaces

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• 作者：Paweł Krupski ; ^{pawel.krupski@uwr.edu.pl" class="auth_mail" title="E-mail the corresponding author} ; Krzysztof Omiljanowski ^{Krzysztof.Omiljanowski@math.uni.wroc.pl" class="auth_mail" title="E-mail the corresponding author} ; Konrad Ungeheuer ^{konrad.ungeheuer@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 54H20 ; secondary ; 37B45 ; 37B20
• 刊名：Topology and its Applications
• 出版年：2016
• 出版时间：1 April 2016
• 年：2016
• 卷：202
• 期：Complete
• 页码：251-268
• 全文大小：459 K

文摘

Let 0-CR denote the class of all metric compacta X   such that the set of maps pan id="mmlsi1" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864116000316&_mathId=si1.gif&_user=111111111&_pii=S0166864116000316&_rdoc=1&_issn=01668641&md5=db363b69f7d591a6b194b49e53c0b830" title="Click to view the MathML source">f:X→Xpan>pan class="mathContainer hidden">pan class="mathCode">$f:X\to X$pan>pan>pan> with 0-dimensional sets pan id="mmlsi2" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864116000316&_mathId=si2.gif&_user=111111111&_pii=S0166864116000316&_rdoc=1&_issn=01668641&md5=142656390fa9f101322920f377e2cbb5" title="Click to view the MathML source">CR(f)pan>pan class="mathContainer hidden">pan class="mathCode">$\mathit{CR}(f)$pan>pan>pan> of chain recurrent points is a dense pan id="mmlsi3" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864116000316&_mathId=si3.gif&_user=111111111&_pii=S0166864116000316&_rdoc=1&_issn=01668641&md5=af686506feb0ecec56886d79bb5b994c" title="Click to view the MathML source">G_δpan>pan class="mathContainer hidden">pan class="mathCode">$G_{\delta }$pan>pan>pan>-subset of the mapping space pan id="mmlsi38" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864116000316&_mathId=si38.gif&_user=111111111&_pii=S0166864116000316&_rdoc=1&_issn=01668641&md5=4b245b5520cfd6053a279deebd7646f0" title="Click to view the MathML source">C(X,X)pan>pan class="mathContainer hidden">pan class="mathCode">$C(X,X)$pan>pan>pan> (with the uniform convergence). We prove, among others, that countable products of polyhedra or locally connected curves belong to 0-CR  . Compacta that admit, for each pan id="mmlsi200" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864116000316&_mathId=si200.gif&_user=111111111&_pii=S0166864116000316&_rdoc=1&_issn=01668641&md5=d296f9076e2722bd2ecea809f1e50090" title="Click to view the MathML source">ϵ>0pan>pan class="mathContainer hidden">pan class="mathCode">$ϵ>0$pan>pan>pan>, an ϵ-retraction onto a subspace from 0-CR belong to 0-CR themselves. Perfect ANR-compacta or n  -dimensional pan id="mmlsi198" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864116000316&_mathId=si198.gif&_user=111111111&_pii=S0166864116000316&_rdoc=1&_issn=01668641&md5=2e6b97eb5c32befae40691ac1eb09c2e" title="Click to view the MathML source">LCⁿ⁻¹pan>pan class="mathContainer hidden">pan class="mathCode">${\mathit{LC}}^{n-1}$pan>pan>pan>-compacta have perfect pan id="mmlsi2" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864116000316&_mathId=si2.gif&_user=111111111&_pii=S0166864116000316&_rdoc=1&_issn=01668641&md5=142656390fa9f101322920f377e2cbb5" title="Click to view the MathML source">CR(f)pan>pan class="mathContainer hidden">pan class="mathCode">$\mathit{CR}(f)$pan>pan>pan> for a generic self-map f. In the cases of polyhedra, compact Hilbert cube manifolds, local dendrites and their finite products, a generic f   has pan id="mmlsi2" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864116000316&_mathId=si2.gif&_user=111111111&_pii=S0166864116000316&_rdoc=1&_issn=01668641&md5=142656390fa9f101322920f377e2cbb5" title="Click to view the MathML source">CR(f)pan>pan class="mathContainer hidden">pan class="mathCode">$\mathit{CR}(f)$pan>pan>pan> being a Cantor set and the set of periodic points of f   of arbitrarily large periods is dense in pan id="mmlsi2" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864116000316&_mathId=si2.gif&_user=111111111&_pii=S0166864116000316&_rdoc=1&_issn=01668641&md5=142656390fa9f101322920f377e2cbb5" title="Click to view the MathML source">CR(f)pan>pan class="mathContainer hidden">pan class="mathCode">$\mathit{CR}(f)$pan>pan>pan>. The results extend some known facts about pan id="mmlsi2" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864116000316&_mathId=si2.gif&_user=111111111&_pii=S0166864116000316&_rdoc=1&_issn=01668641&md5=142656390fa9f101322920f377e2cbb5" title="Click to view the MathML source">CR(f)pan>pan class="mathContainer hidden">pan class="mathCode">$\mathit{CR}(f)$pan>pan>pan> of generic self-maps f on PL-manifolds.