I am reading the book "Arithmetic Dynamics" from Joseph Silverman that has the following comment about the Fatou set:
where the Fatou of a map means the Fatou set of its iterations. I could not find or build any example of a rational function such that the set of their points where there is equicontinuity wasn't open. Does anyone know any example or a place where it is one?
Here is an example, which is not elementary (maybe there is something simpler).
Take $f(z)=e^{2i\pi \alpha}z +z^2$, where $\alpha=\frac{1+\sqrt{5}}{2}$. Then it is a theorem (by works of Siegel and Petersen) that there is a unique maximal Jordan domain $U$ containing $0$ such that $f: U \to U$ is holomorphically conjugated to $z \mapsto e^{2i\pi \alpha }z$, and that the boundary of $U$ is an invariant Jordan curve $\Gamma$ on which $f$ is topologically conjugated to the same rotation $z \mapsto e^{2i\pi \alpha}z$.
Then it is clear that $\{f^n \}$ is equicontinuous on $\overline{U}$, the closure of $U$. However, only $U$ is in the Fatou set, while $\Gamma$ is in the Julia set.
Edit
Or something much much simpler: if $z$ is a fixed point of $f$, then obviously $\{f^n\}$ is equicontinuous on $\{z\}$. However, if $z$ is a repelling fixed point, $\{f^n\}$ cannot be equicontinuous on any neighborhood of $z$.