Let $f : \mathbb{C} \to \mathbb{C}$ be a polynomial map of degree $q > 1$. Consider $E\_n \subset \mathbb{C}$ the set of periodic points with period (dividing) $n$; generally, $|E\_n| = q^n$. Since all the peridic points are limited to a bounded subset of $\mathbb{C}$, we know that there are always two among them within a distance of $O(q^{-n/2})$.


**Question.** (i) *How close can two distinct elements of $E\_n$ get? In the asymptotic where $n \to \infty$ along a sequence, can it happen that $E\_n$ contains a pair of points within a distance of $o(1/q^n)$?* 


(ii) *Given any point $z\_0 \in \mathbb{C}$ and $C$, does the disk $|z-z\_0| < C/q^n$ contain only $O\_{C,z\_0}(1)$ points from $E\_n$?*



Clustering of periodic points for a polynomial iteration of $\mathbb{C}$
Consider $f(x) = x^2 - 2$. This has the property that $f(2 \cos (x)) = 2 \cos(2x)$, so that $f\_n(2 \cos (x)) = 2 \cos(2^n x)$ (where $f\_n$ is $f$ iterated $n$ times). Thus $E\_n = \{2 \cos (x): \cos(x) = \cos(2^n x)\}$, and the two
greatest members of $E\_n$ are both greater than $2 \cos(\pi 2^{1-n})\approx 2 - \pi^2 2^{2-2n}$ 


