Context: I saw the following problem on a discord server I'm in
Now, this is an obvious meme, but it's a really interesting question so I started working on it with some other people in a math discord. A fairly elementary simplification of the problem is to define the family of polynomials $p_1(c) = c$, $p_2(c) = c^2 + c$, $p_3(c) = \left(c^2 + c\right)^2 + c$, where, in general $p_n = p_{n - 1}^2 + c$.
If $c_0 \in \mathbb{C}$ is a root of any $p_n$, then $c_0$ only kills finitely many people, because iteratively applying $f$ with that choice of $c$ eventually returns you to $0$. However, this only gets you a subset of all the values of $c$ that work; in particular, this only gives you the values of $c$ which result in a loop that passes through $0$, which not all loops will. For example, choosing $c = -2$ gives the loop $0, -2, 2, 2, 2, 2, ...$, so $-2$ solves the original problem but won't be the root of any $p_i$. We can make sure we have all the solutions by considering the roots to a difference of two of the $p_i$, since if $c$ is a root to $p_n(x) - p_m(x)$ where $n \neq m$, then clearly there is a cycle because two distinct numbers of iterations of $f$ return the same output.
At this point, we got stuck, and there are two natural questions
(1) What can we say about the roots of these polynomials? Are there infinitely many distinct roots? Are these uniformly bounded, and if not, are they nicely bounded by some function of $n$? Are the roots all isolated, or are there accumulation points?
And of course, the original problem:
(2) Is there a nice way to describe the set of all $c$ which are a root of some $p_i$ or which are the root of some $p_i - p_j$ for $i \neq j$?
I've tagged the question abstract-algebra and dynamical-systems because the questions about the roots seems algebraic, and the original problem seems like a dynamical systems problem. Please let me know if there are other tags which fit better.