= Möbius conjugacy
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Two rational sphere maps are Möbius conjugate if $S=h\circ R\circ h^{-1}$ for a <Möbius transformation> $h$. This is precisely global <holomorphic conjugacy> between Riemann-sphere self-maps. It preserves degree, <local degrees of a holomorphic map>, exact periods and return <periodic-orbit multipliers>, and transports the <Fatou set> and <Julia set> by $h$. Since a Möbius map and its inverse are Lipschitz on the <compact> sphere, this last assertion follows directly from <equicontinuity> of the iterate families. An affine <holomorphic conjugacy> is the special case where $h$ fixes infinity.
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