Complex dynamics — Julia sets and the Mandelbrot set

Parent: orbit · Prereq: discrete-dynamical-systems

The quadratic family $f_c(z) = z^2 + c$, $z, c \in \mathbb{C}$. The single organizing question: is the orbit of $z$ bounded?

Definition

• Filled Julia set $K_c = \{z \in \mathbb{C} : O_{f_c}(z) \text{ is bounded}\}$.
• Julia set $J_c = \partial K_c$ (its topological boundary) — where bounded and escaping orbits are inseparably mixed; the locus of chaos.
• Fatou set $F_c = \mathbb{C} \setminus J_c$ — the open set where nearby orbits stay comparable (formally: where the iterates form a normal family, i.e. are locally equicontinuous).

Escape radius

Let $R = \max(2, |c|)$. If $|z| > R$ then $|f_c(z)| \ge |z|^2 - |c| \ge |z|^2 - |z| = |z|(|z| - 1) > |z|$, with the excess factor growing each step — the orbit escapes to $\infty$ geometrically. Consequence: $K_c \subseteq \{|z| \le 2\}$ whenever $|c| \le 2$, and "did $|z_n|$ pass $2$?" is a finite certificate of escape (what every Mandelbrot renderer actually checks).

Mandelbrot set

$M = \{c \in \mathbb{C} : O_{f_c}(0) \text{ is bounded}\}$ — note this lives in parameter space: one point of $M$ per map, judged by the orbit of the single point $0$.

Why $0$ of all points: $0$ is the critical point of $z^2 + c$ ($f_c'(z) = 2z = 0$), and by a theorem of Fatou every attracting cycle attracts a critical point — so the critical orbit is a probe that detects every attracting cycle the map has.

Fundamental dichotomy (Fatou–Julia + Douady–Hubbard)

$c \in M \iff J_c$ is connected. Otherwise $J_c$ is a Cantor set (totally disconnected "dust") and the dynamics on it is conjugate to the full 2-shift (symbolic-dynamics).

Two exactly-solvable parameters

$c = 0$: $f_0(z) = z^2$. Orbits: $|z| < 1 \to 0$, $|z| > 1 \to \infty$, so $K_0$ = closed unit disk and $J_0$ = unit circle. On the circle $z = e^{2\pi i \theta}$, the map is angle doubling $\theta \mapsto 2\theta \bmod 1$ — literally the doubling map of ergodic-theory-of-orbits. The simplest Julia set is the ergodic-theory workhorse.

$c = -2$: $J_{-2} = [-2, 2]$, a straight segment. Conjugacy: $z = w + w^{-1}$ sends $w \mapsto w^2$ to $z \mapsto z^2 - 2$; on the segment this becomes (after $z = 2\cos(2\pi\theta)$) angle doubling again — $f_{-2}$ is the Chebyshev map, conjugate to the tent map of topological-dynamics. The two boundary cases $c = 0, -2$ are the only $c$ with smooth Julia sets; everything between is fractal.

What the pictures are

A Mandelbrot image is a map of parameter space colored by escape time of the critical orbit; a Julia image fixes $c$ and colors dynamical space. All of it is the boundedness question about orbits, made visible — and the still-open MLC conjecture (is $M$ locally connected?) is the field's Collatz.