Möbius conjugacy

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.