We end by complex dynamics, a fascinating field of mathematics, widely known because of the famous Mandelbrot set. We consider the simplest example. Consider the quadratic map $f_c(z)=z^2+c$. Then define

$$ z_{n+1}=f_c(z_n)=z_n^2+c $$

where $z_n$ and $c$ are complex numbers. Given an initial condition $z_0$, we can compute its image $z_1=f_c(z_0)$, next the image of its image, namely $z_2=f_c(z_1)=f_c^2(z_0)$, and so forth and so on. Here $f_c^2$ means $f_c$ composed with itself, i.e., $f_c^2=f_c\circ f_c$ , and more generally $f_c^n(z_0)$ is the $n$-th iterate of $z_0$. The infinite sequence $(z_0,z_1,\ldots)$ is called the trajectory of $z_0$.

The Julia set associated with a quadratic polynomial $f_c$ is, informally speaking, the set of initial conditions $z_0$ whose trajectories do not tend to infinity. Depending on the value of $c$, you get different Julia sets, the beauty of which is hard to deny.