Complex Hénon maps and discrete groups

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• 作者：Raluca Tanase ^{rtanase@math.stonybrook.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：37F99 ; 37C85 ; 32H50 ; 57M10 ; 57M60
• 刊名：Advances in Mathematics
• 出版年：2016
• 出版时间：4 June 2016
• 年：2016
• 卷：295
• 期：Complete
• 页码：53-89
• 全文大小：1545 K

文摘

Consider the standard family of complex Hénon maps class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816000967&_mathId=si1.gif&_user=111111111&_pii=S0001870816000967&_rdoc=1&_issn=00018708&md5=3f05f1d84b5fc29d610d1b87c27963e0" title="Click to view the MathML source">H(x,y)=(p(x)−ay,x)class="mathContainer hidden">class="mathCode">$H(x,y)=(p(x)-ay,x)$, where p is a quadratic polynomial and a   is a complex parameter. Let class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816000967&_mathId=si2.gif&_user=111111111&_pii=S0001870816000967&_rdoc=1&_issn=00018708&md5=06e9bef3cf13a338b2ea87a5bf0247aa" title="Click to view the MathML source">U⁺class="mathContainer hidden">class="mathCode">$U^{+}$ be the set of points that escape to infinity under forward iterations. The analytic structure of the escaping set class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816000967&_mathId=si2.gif&_user=111111111&_pii=S0001870816000967&_rdoc=1&_issn=00018708&md5=06e9bef3cf13a338b2ea87a5bf0247aa" title="Click to view the MathML source">U⁺class="mathContainer hidden">class="mathCode">$U^{+}$ is well understood from previous work of J. Hubbard and R. Oberste-Vorth as a quotient of class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816000967&_mathId=si128.gif&_user=111111111&_pii=S0001870816000967&_rdoc=1&_issn=00018708&md5=6ba057f1ba780a462af53ebcebeb0603">class="imgLazyJSB inlineImage" height="18" width="88" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0001870816000967-si128.gif">class="mathContainer hidden">class="mathCode">$(\mathbb{C}-\bar{\mathbb{D}})\times \mathbb{C}$ by a discrete group of automorphisms Γ isomorphic to class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816000967&_mathId=si4.gif&_user=111111111&_pii=S0001870816000967&_rdoc=1&_issn=00018708&md5=78a0283327e790cdb6a6b1e7d2a6200f" title="Click to view the MathML source">Z[1/2]/Zclass="mathContainer hidden">class="mathCode">$\mathbb{Z}[1/2]/\mathbb{Z}$. On the other hand, the boundary class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816000967&_mathId=si5.gif&_user=111111111&_pii=S0001870816000967&_rdoc=1&_issn=00018708&md5=5bce34d946968fc2f976937a97fdbeba" title="Click to view the MathML source">J⁺class="mathContainer hidden">class="mathCode">$J^{+}$ of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816000967&_mathId=si2.gif&_user=111111111&_pii=S0001870816000967&_rdoc=1&_issn=00018708&md5=06e9bef3cf13a338b2ea87a5bf0247aa" title="Click to view the MathML source">U⁺class="mathContainer hidden">class="mathCode">$U^{+}$ is a complicated fractal object on which the Hénon map behaves chaotically. We show how to extend the group action to class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816000967&_mathId=si14.gif&_user=111111111&_pii=S0001870816000967&_rdoc=1&_issn=00018708&md5=e66d903ba1764221e6f84ab65a770d3a" title="Click to view the MathML source">S¹×Cclass="mathContainer hidden">class="mathCode">${\mathbb{S}}^1\times \mathbb{C}$, in order to represent the set class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816000967&_mathId=si5.gif&_user=111111111&_pii=S0001870816000967&_rdoc=1&_issn=00018708&md5=5bce34d946968fc2f976937a97fdbeba" title="Click to view the MathML source">J⁺class="mathContainer hidden">class="mathCode">$J^{+}$ as a quotient of class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816000967&_mathId=si7.gif&_user=111111111&_pii=S0001870816000967&_rdoc=1&_issn=00018708&md5=ecf2af24c31c32f326610947e8d1be80">class="imgLazyJSB inlineImage" height="18" width="70" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0001870816000967-si7.gif">class="mathContainer hidden">class="mathCode">${\mathbb{S}}^1\times \mathbb{C}/\mathrm{\Gamma }$ by an equivalence relation. We analyze this extension for Hénon maps that are perturbations of hyperbolic polynomials with connected Julia set or polynomials with a parabolic fixed point.