📂Dynamics

The Mandelbrot Set and the Julia Set

Definition

$$P_{c} : z \mapsto z^{2} + c$$ Consider the dynamical system defined by a map and its orbit $P_{c}^{n} (z)$ in the complex plane $\mathbb{C}$ for a given $c \in \mathbb{C}$.

Mandelbrot Set ¹

The Mandelbrot set is the set $M$ of parameters $c$ for which the orbit $P_{c}^{n} (z)$ does not diverge, given an initial condition $z = 0$. $$M = \mathbb{C} \setminus \left\{ c \in \mathbb{C} : \lim_{n \to \infty} P_{c}^{n} (0) = \infty \right\}$$

Julia Set

The Julia set is the set $J_{c}$ of initial conditions $z$ for which the orbit $P_{c}^{n} (z)$ does not diverge, given a parameter $c$. $$J_{c} = \mathbb{C} \setminus \left\{ z \in \mathbb{C} : \lim_{n \to \infty} P_{c}^{n} (z) = \infty \right\}$$

Explanation

The relationship between the Mandelbrot set and the Julia set can be simply explained by whether the focus is on $c$ or $z$, with the sets being opposite in this regard.

The above animation shows the self-similarity exhibited by continuously zooming into the Mandelbrot set. As a quintessential fractal², the Mandelbrot set was not the first fractal but is regarded as the first example drawn using a computer. This was because Mandelbrot worked at IBM, providing easy access to computing resources.

Below is an image where points within the Mandelbrot set are marked in black, while points diverging to infinity are marked in white³.

The following depicts Julia sets changing according to a given $c$⁴:

• (a): The basin of the period-3 sink in $c = -.17 + .78i$ is highlighted in white.
• (b): A magnified view of (a).
• (c): The basin of the period-5 sink in $c = .38 + .32i$ is highlighted in white.
• (d): The basin of the period-11 sink in $c = .32 + .04i$ is highlighted in white.