Boundary Points of the Mandelbrot Set

Every attracting orbit for a polynomial map $P$ attracts at least one critical point $(f^{\prime }=0)$, Fatou’s Lemma

Since $P_C(z)$has only one critical point, and $P_{-i}(0)$ is Eventually Periodic and a periodic Source, there can be no sink for $c=-i$.

The boundary between bounded and unbounded points is the Julia Set. The filled Julia Set is $J$ and points in its interior. For disconnected $J$, the filled Julia Set is simply J,

• $J$ is invariant. If $z_0\in J$, so is $f(z_0)=z$,…
• orbits in $J$ are either periodic Sources, Eventually Periodic to periodic sources, or chaotic.
• all unstable periodic orbits of $P_C$ are in $J$
• $J$ is either totally connected or totally disconnected

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Fractal Dimension

Measuring the dimension of a linUe

In general, the number of boxes of size $ϵ$, call # $\mathbb{N}(ϵ)$, required to cover an interval is less than or equal to $\frac{c}{{ϵ}^{\prime }}$, where $c$ is a constant analogous to length. In other words, $\mathbb{N}(ϵ)$ scales as $\frac{1}{ϵ}$

More generally, a set $S\subseteq {\mathbb{R}}^m$ is said to be d-dimensional, if it can be covered by $\mathbb{N}(ϵ)=\frac{c}{{ϵ}^d}$ boxes of side length $ϵ$ in the limit as $ϵ\to 0$..o

$C$ can be as large as needed, provided the ${ϵ}^{-d}$ scaling holds as $ϵ\to 0$. $d$ need not be an integer.

$$d=\frac{\ln (\mathbb{N}(ϵ))-\ln C}{\ln \left(\frac{1}{ϵ}\right)}$$

A bounded set $S$ in ${\mathbb{R}}^n$ has box-counting dimension:

$$=\underset{ϵ\to 0}{\lim }\frac{\ln (\mathbb{N}(ϵ))}{\ln \left(\frac{1}{ϵ}\right)}$$