Let $D$ be a subdomain of the complex sphere and suppose $D^c$ has at least three points. Let $f$ be an analytic function which maps $D$ into itself and suppose that $f$ has an attracting fixed point $z_0$ in $D$. Show that the iterates $f^n$ converge locally uniformly to $z_0$ in $D$.
This problem, along with Montel's theorem and a few other related theorems, all have this assumption of the three points in the complement.
Are the three points in the complement necessary to show convergence on $D$, and then additional steps are needed to show convergence to $z_0$?
The condition is there to ensure that $D$ is hyperbolic. If it is not satisfied, then $D$ is either the whole sphere, the plane, or a punctured plane (which is isomorphic to the cylinder $\mathbb C / \mathbb Z$). In each case there are holomorphic self-maps of $D$ for which the property fails.
For example, you can take $f(z)=z^2$, for which $z=0$ is attracting. The map $f$ is a self-map of $D$ if $D$ is either the plane or the sphere, but in both cases there are points in $D$ that are not attracted to zero (for instance, $z=1$).
Without knowing that $D$ is hyperbolic, you can only hope to obtain local results for the dynamics of $f$ near $z_0$ (namely that there is a neighborhood of $z_0$ of points attracted to $z_0$). However if $D$ is hyperbolic and $f: D \to D$ is a holomorphic map with an attracting fixed point, a general version of Schwartz's lemma tells you that $f$ is strictly contracting for the hyperbolic metric, and then it's just Banach's fixed point theorem.