External ray

Remarks

The relation $\varphi \circ P = \varphi^d$ gives $P(R_\theta) = R_{d\theta}$. Example: for $P(z) = z^2$, $K$ is the closed unit disc and $\varphi$ is the identity. So $R_\theta$ is the half-line $\set{r e^{2\pi i \theta} : r > 1}$, and it lands at $e^{2\pi i \theta}$. Non-example: for $P(z) = z^2$, the spiral $\set{r e^{2\pi i (\theta + \log r)} : r > 1}$ is not an external ray, since $\varphi$ maps it to a curve whose angle varies. Douady and Hubbard showed that every ray with a rational angle lands, at a periodic or preperiodic point of the Julia set. For the quadratic family, external rays of the Mandelbrot set are defined in the same way, with the conformal map from its complement onto $\C \setminus \overline{\mathbb{D}}$.