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From local to global analytic conjugacies
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Buff, Xavier and Epstein, Adam L.. (2007) From local to global analytic conjugacies. Ergodic Theory and Dynamical Systems, Vol.27 (No.4). pp. 10731094. ISSN 01433857

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Official URL: http://dx.doi.org/10.1017/S0143385707000041
Abstract
Let $f_1$ and $f_2$ be rational maps with Julia sets $J_1$ and $J_2$, and let $\Psi:J_1\to \mathbb{P}^1$ be any continuous map such that $\Psi\circ f_1=f_2\circ \Psi$ on $J_1$. We show that if $\Psi$ is $\mathbb{C}$differentiable, with nonvanishing derivative, at some repelling periodic point $z_1\in J_1$, then $\Psi$ admits an analytic extension to $\mathbb{P}^1\setminus {\mathcal E}_1$, where ${\mathcal E}_1$ is the exceptional set of $f_1$. Moreover, this extension is a semiconjugacy. This generalizes a result of Julia (Ann. Sci. École Norm. Sup. (3) 40 (1923), 97–150). Furthermore, if ${\mathcal E}_1=\emptyset$ then the extended map $\Psi$ is rational, and in this situation $\Psi(J_1)=J_2$ and $\Psi^{1}(J_2)=J_1$, provided that $\Psi$ is not constant. On the other hand, if ${\mathcal E}_1\neq \emptyset$ then the extended map may be transcendental: for example, when $f_1$ is a power map (conjugate to $z\mapsto z^{\pm d}$) or a Chebyshev map (conjugate to $\pm \text{Х}_d$ with $\text{Х}_d(z+z^{1}) = z^d+z^{d}$), and when $f_2$ is an integral Lattès example (a quotient of the multiplication by an integer on a torus). Eremenko (Algebra i Analiz 1(4) (1989), 102–116) proved that these are the only such examples. We present a new proof.
Item Type:  Journal Article 

Subjects:  Q Science > QA Mathematics 
Divisions:  Faculty of Science > Mathematics 
Library of Congress Subject Headings (LCSH):  Julia sets, Fractional calculus, Analytic functions 
Journal or Publication Title:  Ergodic Theory and Dynamical Systems 
Publisher:  Cambridge University Press 
ISSN:  01433857 
Official Date:  22 June 2007 
Volume:  Vol.27 
Number:  No.4 
Page Range:  pp. 10731094 
Identification Number:  10.1017/S0143385707000041 
Status:  Peer Reviewed 
Access rights to Published version:  Open Access 
References:  A. Douady and J. H. Hubbard. A proof of Thurston’s topological characterization of rational functions. 
URI:  http://wrap.warwick.ac.uk/id/eprint/681 
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