Chaos and indecomposability

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• 作者：Udayan B. Darji^a ; ¹ ; ^{ubdarj01@louisville.edu" class="auth_mail" title="E-mail the corresponding author} ; Hisao Kato^b ; ^{hkato@math.tsukuba.ac.jp" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 37B45 ; 37B40 ; secondary ; 54H20 ; 54F15
• 刊名：Advances in Mathematics
• 出版年：2017
• 出版时间：2 January 2017
• 年：2017
• 卷：304
• 期：Complete
• 页码：793-808
• 全文大小：395 K

文摘

We use recent developments in local entropy theory to prove that chaos in dynamical systems implies the existence of complicated structure in the underlying space. Earlier Mouron proved that if X is an arc-like continuum which admits a homeomorphism f with positive topological entropy, then X contains an indecomposable subcontinuum. Barge and Diamond proved that if G   is a finite graph and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816311926&_mathId=si1.gif&_user=111111111&_pii=S0001870816311926&_rdoc=1&_issn=00018708&md5=9f97811e22419d224442632c653a23b4" title="Click to view the MathML source">f:G→Gclass="mathContainer hidden">class="mathCode">$f:G\to G$ is any map with positive topological entropy, then the inverse limit space class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816311926&_mathId=si2.gif&_user=111111111&_pii=S0001870816311926&_rdoc=1&_issn=00018708&md5=edefe0c530667abd0bba3fb713be23e5">class="imgLazyJSB inlineImage" height="19" width="65" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0001870816311926-si2.gif">class="mathContainer hidden">class="mathCode">$\underleftarrow{\lim }(G,f)$ contains an indecomposable continuum. In this paper we show that if X is a G-like continuum for some finite graph G   and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816311926&_mathId=si16.gif&_user=111111111&_pii=S0001870816311926&_rdoc=1&_issn=00018708&md5=f0fa0afd7336dc9b3b426cca31bd4df8" title="Click to view the MathML source">f:X→Xclass="mathContainer hidden">class="mathCode">$f:X\to X$ is any map with positive topological entropy, then class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816311926&_mathId=si17.gif&_user=111111111&_pii=S0001870816311926&_rdoc=1&_issn=00018708&md5=9cef9282a2f9792b444c696ccdc2785e">class="imgLazyJSB inlineImage" height="19" width="66" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0001870816311926-si17.gif">class="mathContainer hidden">class="mathCode">$\underleftarrow{\lim }(X,f)$ contains an indecomposable continuum. As a corollary, we obtain that in the case that f is a homeomorphism, X contains an indecomposable continuum. Moreover, if f has uniformly positive upper entropy, then X is an indecomposable continuum. Our results answer some questions raised by Mouron and generalize the above mentioned work of Mouron and also that of Barge and Diamond. We also introduce a new concept called zigzag pair which attempts to capture the complexity of a dynamical systems from the continuum theoretic perspective and facilitates the proof of the main result.