When do two rational functions have the same Julia set?

Levin, G.; Przytycki, F.

doi:10.1090/S0002-9939-97-03810-0

When do two rational functions have the same Julia set?
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by G. Levin and F. Przytycki PDF

Proc. Amer. Math. Soc. 125 (1997), 2179-2190 Request permission

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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