Siegel disc

Remarks

The point $z_0 = \varphi^{-1}(0)$ is a periodic point of $f$, and its multiplier is $(f^p)'(z_0) = e^{2\pi i \theta}$. So the periodic point is neutral and irrationally indifferent, and the map is linearizable near it. Example: Siegel proved in 1942 that $f(z) = e^{2\pi i \theta} z + z^2$ is linearizable at $0$ when $\theta$ is Diophantine, for instance the golden mean $(\sqrt{5} - 1)/2$. The component of the Fatou set containing $0$ is then a Siegel disc. Non-example: the unit disc for $f(z) = z^2$ fails clause 3, since $f$ is not injective there, while a rotation is. The analogous domain with an annulus in place of the disc is a Herman ring.