I know that the sole real solution to $\cos x = x$, which I'll call $\textbf{d}$, is a universal attractor of the cosine function for all the real numbers. Meaning that:
$$\forall x\in\mathbb{R} \left(\lim_{n\to\infty} \underbrace{\cos\circ\cos\circ\cdots\circ\cos}_{n\text{ iterations}}\ x=\textbf{d}\right)$$
I also know that this is true for some subset of $\mathbb{C}$. But what is that subset? How would I find that subset? How would I even prove the above for the real numbers?
Here is a proof in the real case that the basin of attraction of $d$ for $\cos$ is the whole real line.
$\cos\circ\cos$ maps the real line to the interval $I=[\cos(1),1] \supseteq [0.53,1] \ni d \approx 0.739085$.
The derivative of $\cos$ in $I$ ranges in $[\sin(\cos(1),\sin(1)] \subseteq [0.52,0.85] \subset [0,1)$.
Thus, $\cos$ is a contraction in $I$ and so $I$ is contained in the basin of attraction of $\cos$. Therefore, so is the whole real line.