I am attempting to solve a problem involving a recursive sequence defined as follows: given $a_0 = 0$ and $a_n = a_{n-1}^2 + c$, where $c$ is a complex number. The objective is to find the supremum of the absolute value of $c$ such that there exists an integer $T$ satisfying $\forall n \in \mathbb{N}, a_{n+T} = a_n$.
In order to determine this supremum, I sought to find the value of $c$ for which $a_T = 0$. However, upon attempting to solve this problem using MATLAB, I found something interesting. The maximum absolute value obtained was tending to 2, with a sequence of values leading to it: $0 \rightarrow 1 \rightarrow 1.7549 \rightarrow 1.9408 \rightarrow 19854 \rightarrow 1.9964 \rightarrow 1.9991$.
I conjectured that the supremum might be 2, but I have not been able to conclusively solve this problem. I am seeking guidance on how to proceed further.
Thank you for your assistance.
The supremum is $2$ and attained at $c=-2$. For the proof, see any text on Mandelbrot set (beginning with Wikipedia article).