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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: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 $t\mapsto 2t(\mathrm{mod}1)$. 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.