 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.