Buff, Xavier and Epstein, Adam L.. (2007) From local to global analytic conjugacies. Ergodic Theory and Dynamical Systems, Vol.27 (No.4). pp. 1073-1094. ISSN 0143-3857

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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 non-vanishing 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:	0143-3857
Date:	22 June 2007
Volume:	Vol.27
Number:	No.4
Page Range:	pp. 1073-1094
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. Acta Math. 171(2) (1993), 263–297. A. Eremenko. Some functional equations connected with the iteration of rational functions. Algebra i Analiz 1(4) (1989), 102–116 (transl. Leningrad Math. J. 1(4) (1990), 905–919). G. Julia. Sur une classe d’équations fonctionnelles. Ann. Sci. ´Ecole Norm. Sup. (3) 40 (1923), 97–150. L. Lattès. Sue l’itération des substitutions rationnelles et les fonctions de Poincar´e. C. R. Acad. Sci. Paris 166 (1918) 26–28. J. Milnor. Dynamics in One Complex Variable, Introductory Lectures. Friedr. Vieweg & Sohn, Braunschweig, 1999. J. Milnor. On Latt`es maps. Dynamics on the Riemann Sphere. Eds. P. Hjorth and C. L Petersen. A Bodil Branner Festschrift, European Mathematical Society, 2006. N. Steinmetz. Rational Iteration; Complex Analytic Dynamical Systems (de Gruyter Studies in Mathematics, 16). Walter de Gruyter, Berlin, 1993.
URI:	http://wrap.warwick.ac.uk/id/eprint/681

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