A classification of bicritical rational maps with a pair of period two superattracting cycles
type: Item
instance of: scholarly article
title: A classification of bicritical rational maps with a pair of period two superattracting cycles
zbMATH Open document ID: 1343.37035
author: Thomas Sharland
author: Adam Epstein
published in: Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série VI
publication date: 2013-05-28T00:00:00Z
review text: Let \(f\) and \(g\) be unicritical monic polynomials of degree \(d\) (that is, the form \(z \longmapsto z^d+c,\) for some \(c\)). Denote by \(\widetilde{\mathbb{C}}=\mathbb{C} \cup \Big\{\infty\cdot e^{2 \pi i t},~~t \in \mathbb{R}/\mathbb{Z}\Big\}\) the complex plane compactified by the circle at infinity. We continuously extend the two polynomials to the circle at infinity by \[ f(\infty\cdot e^{2 \pi i t})=g(\infty\cdot e^{2 \pi i t})=\infty\cdot e^{2 d \pi i t}. \] The extended dynamical plane of \(f\) (respectively \(g\)) is denoted by \(\widetilde{\mathbb{C}}_f\) (respectively \(\widetilde{\mathbb{C}}_g\)). We define a topological sphere \(\Sigma_{f,g}\) by gluing the two extended planes together along the circle at infinity, that is, \[ \Sigma_{f,g}= \widetilde{\mathbb{C}}_f \cup \widetilde{\mathbb{C}}_g/\sim, \] where \(\sim\) is the relation which identifies the point \(\infty\cdot e^{2 \pi i t} \in \widetilde{\mathbb{C}}_f \) with the point \(\infty.e^{-2 \pi i t} \in \widetilde{\mathbb{C}}_g\). The formal mating is defined to be the branched covering \(f \uplus g:\Sigma_{f,g} \longrightarrow \Sigma_{f,g}\) such that \(f \uplus g|_{\widetilde{\mathbb{C}}_f}=f\) and \(f \uplus g|_{\widetilde{\mathbb{C}}_g}=g\). Assume that the filled Julia set \(K_h\) of a monic polynomial \(h :\widetilde{\mathbb{C}} \rightarrow \widetilde{\mathbb{C}}\) of degree \(d \geq 2\) is connected. Then, by Böttcher's theorem, there is a conformal isomorphism \[ \phi=\phi_h:\widetilde{\mathbb{C}}\setminus \overline{\mathbb{D}} \rightarrow \widetilde{\mathbb{C}}\setminus K_h \] which can be chosen so that it conjugates \(z \longmapsto z^d\) on \(\widetilde{\mathbb{C}}\setminus \overline{\mathbb{D}}\) with the map \(h\) on \(\widetilde{\mathbb{C}}\setminus K_h\). The external ray of angle \(t\) is given by \(R_h(t)=\phi_h(r_t),\) where \(r_t=\{r\exp(2\pi i t), r>1\}\subset \mathbb{C}\setminus \mathbb{D}\). We denote by \(\sim_h\) the smallest equivalence relation on \(\widetilde{\math…
Mathematics Subject Classification ID: 37F50
zbMATH DE Number: 6167096
zbMATH Keywords: filled Julia set
zbMATH Keywords: Julia set
zbMATH Keywords: topological mating
zbMATH Keywords: ray class, unicritical polynomial
zbMATH Keywords: periodic point
zbMATH Keywords: Möbius conjugacy
reviewed by: El Houcein El Abdalaoui
MaRDI profile type: Publication
OpenAlex ID: W2123406435
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DOI: 10.5802/AFST.1357
description: scientific article; zbMATH DE number 6167096