An extended Fatou-Shishikura inequality and wandering branch continua for polynomials

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• 作者：Alexander Blokh^a ; ¹ ; ^{ablokh@math.uab.edu" class="auth_mail" title="E-mail the corresponding author} ; Doug Childers^b ; ^{dchilders@mail.usf.edu" class="auth_mail" title="E-mail the corresponding author} ; Genadi Levin^c ; ^{levin@math.huji.ac.il" class="auth_mail" title="E-mail the corresponding author} ; Lex Oversteegen^a ; ² ; ^{overstee@math.uab.edu" class="auth_mail" title="E-mail the corresponding author} ; Dierk Schleicher^d ; ³ ; ^{dierk@jacobs-university.de" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 37F10 ; secondary ; 37B45 ; 37C25 ; 37F20 ; 37F50
• 刊名：Advances in Mathematics
• 出版年：2016
• 出版时间：22 January 2016
• 年：2016
• 卷：288
• 期：Complete
• 页码：1121-1174
• 全文大小：1162 K

文摘

Let P be a polynomial of degree d   with Julia set class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815004429&_mathId=si1.gif&_user=111111111&_pii=S0001870815004429&_rdoc=1&_issn=00018708&md5=f52255d75bc768b204cbe94530923495" title="Click to view the MathML source">J_Pclass="mathContainer hidden">class="mathCode">$J_P$. Let class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815004429&_mathId=si2.gif&_user=111111111&_pii=S0001870815004429&_rdoc=1&_issn=00018708&md5=b40473e232de348b2a7ea1de51eae302">class="imgLazyJSB inlineImage" height="16" width="17" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0001870815004429-si2.gif">class="mathContainer hidden">class="mathCode">$\tilde{N}$ be the number of non-repelling cycles of P  . By the famous Fatou–Shishikura inequality class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815004429&_mathId=si3.gif&_user=111111111&_pii=S0001870815004429&_rdoc=1&_issn=00018708&md5=5697cb11dec149a14732f05064eeb04b">class="imgLazyJSB inlineImage" height="18" width="73" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0001870815004429-si3.gif">class="mathContainer hidden">class="mathCode">$\tilde{N}\le d-1$. The goal of the paper is to improve this bound. The new count includes wandering collections of non-(pre)critical branch continua  , i.e., collections of continua or points class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815004429&_mathId=si4.gif&_user=111111111&_pii=S0001870815004429&_rdoc=1&_issn=00018708&md5=3a1540f963b0ca36aa7ef60dd0947a5a" title="Click to view the MathML source">Q_i⊂J_Pclass="mathContainer hidden">class="mathCode">$Q_i\subset J_P$all   of whose images are pairwise disjoint, contain no critical points, and contain the limit sets of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815004429&_mathId=si5.gif&_user=111111111&_pii=S0001870815004429&_rdoc=1&_issn=00018708&md5=bc6a7b163e683d40722954d59e8c57c5" title="Click to view the MathML source">eval(Q_i)≥3class="mathContainer hidden">class="mathCode">$\mathrm{eval}(Q_i)\ge 3$ external rays. Also, we relate individual cycles, which are either non-repelling or repelling with no periodic rays landing, to individual critical points that are recurrent in a weak sense.

A weak version of the inequality reads

where class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815004429&_mathId=si7.gif&_user=111111111&_pii=S0001870815004429&_rdoc=1&_issn=00018708&md5=d28bf6e4b9bf82458362f9d46fe048c0" title="Click to view the MathML source">N_irrclass="mathContainer hidden">class="mathCode">$N_{\mathit{irr}}$ counts repelling cycles with no periodic rays landing at points in the cycle, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815004429&_mathId=si8.gif&_user=111111111&_pii=S0001870815004429&_rdoc=1&_issn=00018708&md5=896f172a72aae870955da68c27feb21a" title="Click to view the MathML source">{Q_i}class="mathContainer hidden">class="mathCode">$\{Q_i\}$ form a wandering collection class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815004429&_mathId=si11.gif&_user=111111111&_pii=S0001870815004429&_rdoc=1&_issn=00018708&md5=51ff89b83d43340bf10a9707a9ea712f" title="Click to view the MathML source">B_Cclass="mathContainer hidden">class="mathCode">${\mathcal{B}}_{\mathbb{C}}$ of non-(pre)critical branch continua, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815004429&_mathId=si10.gif&_user=111111111&_pii=S0001870815004429&_rdoc=1&_issn=00018708&md5=26950c8e0159c18c7a216e6dc05af198" title="Click to view the MathML source">χ=1class="mathContainer hidden">class="mathCode">$\chi =1$ if class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815004429&_mathId=si11.gif&_user=111111111&_pii=S0001870815004429&_rdoc=1&_issn=00018708&md5=51ff89b83d43340bf10a9707a9ea712f" title="Click to view the MathML source">B_Cclass="mathContainer hidden">class="mathCode">${\mathcal{B}}_{\mathbb{C}}$ is non-empty, and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815004429&_mathId=si12.gif&_user=111111111&_pii=S0001870815004429&_rdoc=1&_issn=00018708&md5=67649eb600658432c2c6be0d2cf98b87" title="Click to view the MathML source">χ=0class="mathContainer hidden">class="mathCode">$\chi =0$ otherwise.