Mandelbrot Set

Mandelbrot Set

We define the Mandelbrot set $M$ as: $M=\{c\in \mathbb{C}\mid J_c\text{ is connected}\}$ Equivalently, $M=\{c\in \mathbb{C}\mid \text{the orbit of }0\text{ under }{f}_c(z)=z^2+c\text{ is bounded}\}$

Proposition

• Contains the origin: $c=0\in M$ because $f_0(z)=z^2$ keeps $0$ bounded within its filled Julia set ($K_0$).
• Boundedness: $M$ is entirely contained within the closed disk of radius 2 ($|c|\le 2$).
• Topological features: $M$ is compact, connected, and crucially, simply-connected (meaning it has no “holes”).

The Main Cardioid ($MC$)

The largest central region of the Mandelbrot set is called the Main Cardioid ($MC$). Your notes explore the dynamical behavior of points within this specific region.

Main Cardioid

Consider the function that maps the complex unit disk to the complex plane: $g(\mu )=\frac{\mu }{2}-\frac{{\mu }^2}{4}$ If we restrict our input $\mu$ to the open unit disk ($|\mu |<1$), the image of this function $g$ traces out the exact shape of the Main Cardioid. Therefore, any point $c\in MC$ can be written as $c=\frac{\mu }{2}-\frac{{\mu }^2}{4}$ for some $|\mu |<1$.

Dynamical Property inside the Main Cardioid:

The primary claim is: For any $c\in MC$, the dynamical system $f_c(z)=z^2+c$ has a stable (attracting) fixed point, and the orbit of 0 approaches it.

Proof Outline:

1. Find the fixed points by setting $f(z)=z$: $z^2+c=z⟹z=\frac{1\pm \sqrt{1-4c}}{2}$
2. To check for stability, we find the derivative (multiplier) at the fixed point $z^{\ast }$: $f^{\prime }(z^{\ast })=2z^{\ast }$
3. Let this multiplier be denoted by $\mu =2z^{\ast }$. For the fixed point to be stable, we need $|\mu |<1$.
4. Since $z^{\ast }=\frac{\mu }{2}$, we can substitute this back into the fixed point equation $c=z^{\ast }-(z^{\ast }{)}^2$ to find the corresponding parameter $c$: $c=\frac{\mu }{2}-{\left(\frac{\mu }{2}\right)}^2=\frac{\mu }{2}-\frac{{\mu }^2}{4}$
5. This is exactly the equation for the Main Cardioid. Because we required $|\mu |<1$ for stability, any $c$ generated by this equation guarantees that $z^{\ast }=\frac{\mu }{2}$ is a stable fixed point.

Bulbs of the Mandelbrot Set

The notes conclude with an “upshot” regarding the boundary of the Main Cardioid (where $|\mu |=1$).

For every rational angle $p/q$ (where $0<p/q<1$) around the boundary of the Main Cardioid, there is a corresponding “bulb” (a smaller cardioid or circular shape) attached to the $MC$ at exactly one point. This explains the fractal, budding structure surrounding the main body of the set.