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Mating Kleinian groups isomorphic to \(C_2\ast C_5\) with quadratic polynomials

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Title: Mating Kleinian groups isomorphic to \(C_2\ast C_5\) with quadratic polynomials
Authors: Freiberger, Marianne
Publisher Information: American Mathematical Society (AMS), Providence, RI
Subject Terms: rational maps, holomorphic correspondence, quasi-conformal surgery, Quasiconformal methods and Teichmüller theory, etc. (dynamical systems), Holomorphic families of dynamical systems, the Mandelbrot set, bifurcations, Dynamical systems involving relations and correspondences in one complex variable, Kleinian groups, Dynamics of complex polynomials, rational maps, entire and meromorphic functions, Fatou and Julia sets
Description: Summary: Given a quadratic polynomial \(q:\widehat\mathbb{C}\to\mathbb{C}\) and a representation \(G\): \(\widehat\mathbb{C}\to\widehat\mathbb{C}\) of \(C_2\) \(C_5\) in \(\text{PSL}(2,\mathbb{C})\) satisfying certain conditions we will construct a \(4:4\) holomorphic correspondence on the sphere (given by a polynomial relation \(p(z,w))\) that mates the two actions: The sphere will be partitioned into two completely invariant sets \(\Omega\) and \(\Lambda\). The set \(\Omega\) consists of the disjoint union of two sets, each of which is conformally homeomorphic to the filled Julia set of a degree 4 polynomial \(P\). This filled Julia set contains infinitely many copies of the filled Julia set of \(q\). Suitable restrictions of the correspondence are conformally conjugate to \(P\). The set \(\Lambda\) will not be connected, but it can be joined up using a family \({\mathcal C}\) of completely invariant curves. The action of the correspondence on the complement of \(\cup{\mathcal C}\) will then be conformally conjugate to the action of \(G\) on a simply connected subset of its regular set.
Document Type: Article
File Description: application/xml
DOI: 10.1090/s1088-4173-03-00087-0
Access URL: https://zbmath.org/2116080
Accession Number: edsair.c2b0b933574d..d249f199bdad4ccb636e77c9a90be82f
Database: OpenAIRE
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  Data: Mating Kleinian groups isomorphic to \(C_2\ast C_5\) with quadratic polynomials
– Name: Author
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  Data: <searchLink fieldCode="AR" term="%22Freiberger%2C+Marianne%22">Freiberger, Marianne</searchLink>
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  Data: American Mathematical Society (AMS), Providence, RI
– Name: Subject
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  Data: <searchLink fieldCode="DE" term="%22rational+maps%22">rational maps</searchLink><br /><searchLink fieldCode="DE" term="%22holomorphic+correspondence%22">holomorphic correspondence</searchLink><br /><searchLink fieldCode="DE" term="%22quasi-conformal+surgery%22">quasi-conformal surgery</searchLink><br /><searchLink fieldCode="DE" term="%22Quasiconformal+methods+and+Teichmüller+theory%2C+etc%2E+%28dynamical+systems%29%22">Quasiconformal methods and Teichmüller theory, etc. (dynamical systems)</searchLink><br /><searchLink fieldCode="DE" term="%22Holomorphic+families+of+dynamical+systems%22">Holomorphic families of dynamical systems</searchLink><br /><searchLink fieldCode="DE" term="%22the+Mandelbrot+set%22">the Mandelbrot set</searchLink><br /><searchLink fieldCode="DE" term="%22bifurcations%22">bifurcations</searchLink><br /><searchLink fieldCode="DE" term="%22Dynamical+systems+involving+relations+and+correspondences+in+one+complex+variable%22">Dynamical systems involving relations and correspondences in one complex variable</searchLink><br /><searchLink fieldCode="DE" term="%22Kleinian+groups%22">Kleinian groups</searchLink><br /><searchLink fieldCode="DE" term="%22Dynamics+of+complex+polynomials%2C+rational+maps%2C+entire+and+meromorphic+functions%22">Dynamics of complex polynomials, rational maps, entire and meromorphic functions</searchLink><br /><searchLink fieldCode="DE" term="%22Fatou+and+Julia+sets%22">Fatou and Julia sets</searchLink>
– Name: Abstract
  Label: Description
  Group: Ab
  Data: Summary: Given a quadratic polynomial \(q:\widehat\mathbb{C}\to\mathbb{C}\) and a representation \(G\): \(\widehat\mathbb{C}\to\widehat\mathbb{C}\) of \(C_2\) \(C_5\) in \(\text{PSL}(2,\mathbb{C})\) satisfying certain conditions we will construct a \(4:4\) holomorphic correspondence on the sphere (given by a polynomial relation \(p(z,w))\) that mates the two actions: The sphere will be partitioned into two completely invariant sets \(\Omega\) and \(\Lambda\). The set \(\Omega\) consists of the disjoint union of two sets, each of which is conformally homeomorphic to the filled Julia set of a degree 4 polynomial \(P\). This filled Julia set contains infinitely many copies of the filled Julia set of \(q\). Suitable restrictions of the correspondence are conformally conjugate to \(P\). The set \(\Lambda\) will not be connected, but it can be joined up using a family \({\mathcal C}\) of completely invariant curves. The action of the correspondence on the complement of \(\cup{\mathcal C}\) will then be conformally conjugate to the action of \(G\) on a simply connected subset of its regular set.
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  Data: 10.1090/s1088-4173-03-00087-0
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  Data: <link linkTarget="URL" linkTerm="https://zbmath.org/2116080" linkWindow="_blank">https://zbmath.org/2116080</link>
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        Value: 10.1090/s1088-4173-03-00087-0
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    Subjects:
      – SubjectFull: rational maps
        Type: general
      – SubjectFull: holomorphic correspondence
        Type: general
      – SubjectFull: quasi-conformal surgery
        Type: general
      – SubjectFull: Quasiconformal methods and Teichmüller theory, etc. (dynamical systems)
        Type: general
      – SubjectFull: Holomorphic families of dynamical systems
        Type: general
      – SubjectFull: the Mandelbrot set
        Type: general
      – SubjectFull: bifurcations
        Type: general
      – SubjectFull: Dynamical systems involving relations and correspondences in one complex variable
        Type: general
      – SubjectFull: Kleinian groups
        Type: general
      – SubjectFull: Dynamics of complex polynomials, rational maps, entire and meromorphic functions
        Type: general
      – SubjectFull: Fatou and Julia sets
        Type: general
    Titles:
      – TitleFull: Mating Kleinian groups isomorphic to \(C_2\ast C_5\) with quadratic polynomials
        Type: main
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          Name:
            NameFull: Freiberger, Marianne
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