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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 lomonaco@ruc.dk
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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 

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