= Complex dynamics
{wiki}
<Complex dynamics> studies <iterations of a map> $z_{n+1}=f(z_n)$ when $f$ is a <holomorphic map>. A central case is a <rational map> of the <Riemann sphere> of <degree of a rational map of the Riemann sphere> at least two. One asks which <orbits> approach an <attracting periodic point>, which escape, and where nearby starting points have unstable long-term behavior.
The <polynomial function> $f(z)=z^2$ is a complete first example:
$$
f^n(z)=z^{2^n}.
$$
For $|z|<1$ these <iterations of a map> tend to zero; for $|z|>1$ they tend to infinity on the <Riemann sphere>. On the <unit circle>, writing $z=e^{2\pi it}$ turns the <iteration of a map> into angle doubling $t\mapsto2t\pmod1$. The two open regions make up the <Fatou set>, where the <iterations of a map> form a <normal family>; their shared <unit circle> is the <Julia set>.
The <quadratic polynomial> family $f_c(z)=z^2+c$ illustrates two different questions. For a fixed $c$, its <Julia set> lives in the dynamical $z$-plane. The <Mandelbrot set> lives in the parameter $c$-plane and records when the <critical orbit> starting at zero stays <bounded>. Varying a parameter and varying an initial point are separate operations.
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