Embeddability of homeomorphisms of the circle in set-valued iteration groups

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• 作者：Dorota Krassowska ^{dkrassow@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; Marek Cezary Zdun^a ; ^{mczdun@up.krakow.pl" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Iteration groups ; Flows ; Set-valued functions ; Functional equations limit sets ; Fractional iterates ; Homeomorphisms of the circle
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2016
• 出版时间：15 January 2016
• 年：2016
• 卷：433
• 期：2
• 页码：1647-1658
• 全文大小：358 K

文摘

Let class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15007908&_mathId=si1.gif&_user=111111111&_pii=S0022247X15007908&_rdoc=1&_issn=0022247X&md5=92cab181475918eb941ec9a5be90e31d" title="Click to view the MathML source">F:S¹→S¹class="mathContainer hidden">class="mathCode">$F:{\mathbb{S}}^1\to {\mathbb{S}}^1$ be a homeomorphism without periodic points. It is known that F is embeddable in a continuous iteration group if and only if F is minimal. We deal with F which is not minimal. In this case, F satisfying some additional assumptions can be embedded but only in a nonmeasurable iteration groups. There are infinitely many such nonmeasurable groups. We propose here a new approach to the problem of embeddability. For a given homeomorphism F   without periodic points we construct some substitute of an iteration group, namely the unique special set-valued iteration group class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15007908&_mathId=si104.gif&_user=111111111&_pii=S0022247X15007908&_rdoc=1&_issn=0022247X&md5=d0dfded34b5bf2dac971f82ca47bff11" title="Click to view the MathML source">{F^t:S¹→cc[S¹],t∈R}class="mathContainer hidden">class="mathCode">$\{F^t:{\mathbb{S}}^1\to \mathrm{cc}[{\mathbb{S}}^1],t\in \mathbb{R}\}$, which is regular in a sense and in which F   can be embedded i.e. class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15007908&_mathId=si3.gif&_user=111111111&_pii=S0022247X15007908&_rdoc=1&_issn=0022247X&md5=4bbe0f97415ca2a4611f034174ab9bb9" title="Click to view the MathML source">F(x)∈F¹(x)class="mathContainer hidden">class="mathCode">$F(x)\in F^1(x)$. We also determine a maximal countable and dense subgroup class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15007908&_mathId=si27.gif&_user=111111111&_pii=S0022247X15007908&_rdoc=1&_issn=0022247X&md5=f2cd4627b35cf6a4a09181578d205e6b" title="Click to view the MathML source">T⊂Rclass="mathContainer hidden">class="mathCode">$T\subset \mathbb{R}$ such that class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15007908&_mathId=si324.gif&_user=111111111&_pii=S0022247X15007908&_rdoc=1&_issn=0022247X&md5=c885878b624e70a044a73c806280c859" title="Click to view the MathML source">{F^t:S¹→cc[S¹],t∈T}class="mathContainer hidden">class="mathCode">$\{F^t:{\mathbb{S}}^1\to \mathrm{cc}[{\mathbb{S}}^1],t\in T\}$ has a continuous selection class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15007908&_mathId=si6.gif&_user=111111111&_pii=S0022247X15007908&_rdoc=1&_issn=0022247X&md5=7632aa624fc3242c3ceed0285b14ee89" title="Click to view the MathML source">{f^t:S¹→S¹,t∈T}class="mathContainer hidden">class="mathCode">$\{f^t:{\mathbb{S}}^1\to {\mathbb{S}}^1,t\in T\}$ being the best regular embedding of F  . If there exists a nonmeasurable embedding class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15007908&_mathId=si384.gif&_user=111111111&_pii=S0022247X15007908&_rdoc=1&_issn=0022247X&md5=553bd8124ea46b7836aee6b2e9f61911" title="Click to view the MathML source">{f^t:S¹→S¹,t∈R}class="mathContainer hidden">class="mathCode">$\{f^t:{\mathbb{S}}^1\to {\mathbb{S}}^1,t\in \mathbb{R}\}$ of F  , then there exists an additive function class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15007908&_mathId=si8.gif&_user=111111111&_pii=S0022247X15007908&_rdoc=1&_issn=0022247X&md5=f5cf6a4f4e11e400f28d69309ad85e91" title="Click to view the MathML source">γ:R→Tclass="mathContainer hidden">class="mathCode">$\gamma :\mathbb{R}\to T$ such that class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15007908&_mathId=si9.gif&_user=111111111&_pii=S0022247X15007908&_rdoc=1&_issn=0022247X&md5=32dfea0d7028ec86c5c3b8747ff9123c" title="Click to view the MathML source">f^t(z)∈F^γ(t)(z),t∈Rclass="mathContainer hidden">class="mathCode">$f^t(z)\in F^{\gamma (t)}(z),t\in \mathbb{R}$. We determine a unique maximal subgroup T with this property.