Ergodic Theory and Dynamical Systems



From local to global analytic conjugacies


XAVIER BUFF a1 and ADAM L. EPSTEIN a2
a1 Université Paul Sabatier, Laboratoire Emile Picard, 118, Route De Narbonne, 31062 Toulouse Cedex, France (e-mail: buff@picard.ups-tlse.fr)
a2 Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK (e-mail: adame@maths.warwick.ac.uk)

Article author query
buff x   [Google Scholar] 
epstein al   [Google Scholar] 
 

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.

(Published Online June 22 2007)
(Received November 24 2006)
(Revised February 9 2007)