Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-28T10:42:10.326Z Has data issue: false hasContentIssue false
  • English
  • Français

Parabolic-like mappings

Published online by Cambridge University Press:  03 July 2014

LUNA LOMONACO
LUNA LOMONACO*
Affiliation:
IMFUFA, Department of Science, Systems and Models, Universitetsvej 1, DK-4000 Roskilde, Denmark Departamento de Matemática Aplicada, Instituto de Matemática e Estatística da Universidade de São Paulo, Rua do Matão, 1010-CEP 05508-090, São Paulo-SP, Brazil email [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we introduce the notion of parabolic-like mapping. Such an object is similar to a polynomial-like mapping, but it has a parabolic external class, i.e. an external map with a parabolic fixed point. We define the notion of parabolic-like mapping and we study the dynamical properties of parabolic-like mappings. We prove a straightening theorem for parabolic-like mappings which states that any parabolic-like mapping of degree two is hybrid conjugate to a member of the family

$$\begin{eqnarray}\mathit{Per}_{1}(1)=\left\{[P_{A}]\,\bigg|\,P_{A}(z)=z+\frac{1}{z}+A,~A\in \mathbb{C}\right\}\!,\end{eqnarray}$$
a unique such member if the filled Julia set is connected.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 35 , Issue 7 , October 2015 , pp. 2171 - 2197
Copyright
© Cambridge University Press, 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ahlfors, L.. Lectures on Quasiconformal Mappings (AMS University Lecture Series, 38), 2nd edn. American Mathematical Society, Providence, RI, 2006.CrossRefGoogle Scholar
Douady, A. and Earle, C. J.. Conformally natural extension of homeomorphisms of the circle. Acta Math. 169(3–4) (1992), 229–325.Google Scholar
Douady, A. and Hubbard, J. H.. On the dynamics of polynomial-like mappings. Ann. Sci. Éc. Norm. Supér (4) 18 (1985), 287–343.CrossRefGoogle Scholar
Hubbard, J.. Teichmüller Theory and Applications to Geometry, Topology, and Dynamics, Volume 1: Teichmüller Theory. Matrix editions, Ithaca, NY, 2006.Google Scholar
Lomonaco, L. L. A.. Parabolic-like mappings. PhD Thesis, Roskilde Universitet, 2012.Google Scholar
Milnor, J.. Dynamics in One Complex Variable (Annals of Mathematics Studies, 160). Princeton University Press, Princeton, NJ, 2006.Google Scholar
Shishikura, M.. Bifurcation of parabolic fixed points. The Mandelbrot Set, Theme and Variations (London Mathematical Society Lecture Note Series, 274). Cambridge University Press, Cambridge, 2000, pp. 325–363.CrossRefGoogle Scholar