The Library
From local to global analytic conjugacies
Tools
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

PDF
WRAP_Epstein_Loal_global.pdf  Requires a PDF viewer such as GSview, Xpdf or Adobe Acrobat Reader Download (252Kb) 
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  
Dates: 


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 
Request changes or add full text files to a record
Actions (login required)
View Item 
Downloads
Downloads per month over past year