When do two rational functions have the same Julia set?
When do two rational functions have the same Julia set?
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 …