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When do two rational functions have the same Julia set?

Author(s): G. Levin; F. Przytycki
Journal: Proc. Amer. Math. Soc. 125 (1997), 2179-2190.
MSC (1991): Primary 58F23
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Abstract: It is proved that non-exceptional rational functions $f$ and $g$ on the Riemann sphere have the same measure of maximal entropy iff there exist iterates $F$ of $f$ and $G$ of $g$ and natural numbers $M,N$ such that

\begin{equation*}(G^{-1}\circ G)\circ G^{M} = (F^{-1}\circ F) \circ F^{N}.\tag {$*$} \end{equation*}

If one assumes only that $f,g$ have the same Julia set and no singular or parabolic domains of normality for the iterates, one also proves $(*)$.

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Additional Information:

G. Levin
Affiliation: Institute of Mathematics, Hebrew University, 91904 Jerusalem, Israel
Email: levin@math.huji.ac.il

F. Przytycki
Affiliation: Institute of Mathematics, Polish Academy of Sciences, ul. Sniadeckich 8, 00-950 Warsaw, Poland
Email: feliksp@impan.gov.pl

DOI: 10.1090/S0002-9939-97-03810-0
PII: S 0002-9939(97)03810-0
Received by editor(s): January 27, 1995
Received by editor(s) in revised form: February 9, 1996
Additional Notes: The preprint version of this paper has the title Rational maps, common Julia sets, functional equations.
Communicated by: Mary Rees
Copyright of article: Copyright 1997, American Mathematical Society

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