Bibliographic Details
| 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 |