Holomorphic conjugacy

Holomorphic maps $R$ and $S$ are holomorphically conjugate if a biholomorphic coordinate change $h$ satisfies $h\circ R=S\circ h$. Iterating gives $h\circ R^n=S^n\circ h$, so periodic points and their exact periods correspond. Differentiating at a periodic point gives equality of return multipliers, since the nonzero derivative of $h$ cancels. Locally uniform convergence and normality transfer through the coordinate change, so it also transports Fatou sets and Julia sets. Between entire Riemann spheres the biholomorphic change is a Möbius transformation; an affine change is sufficient for many polynomial normalizations.

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.