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 line: the number of boxes of size $ϵ$ needed to cover an interval scales as $\frac{1}{ϵ}$, i.e. $\mathbb{N}(ϵ)\le \frac{c}{ϵ}$ for some constant $c$ analogous to length.

more generally, a set $S\subseteq {\mathbb{R}}^m$ is d-dimensional if it can be covered by $\mathbb{N}(ϵ)=\frac{c}{{ϵ}^d}$ boxes of side length $ϵ$ as $ϵ\to 0$. $c$ can be as large as needed, $d$ need not be an integer.

solving for $d$:

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

taking the limit $ϵ\to 0$ washes out $C$, so the box-counting dimension of a bounded set $S\subseteq {\mathbb{R}}^n$ is:

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

The slope of this plot is the box counting dimension.

Figure 4.16 Box-counting dimension of the H´enon attractor. A graphical report of the results of the box counts in Figures 4.13 and 4.15. The box- counting dimension is the limit of $\frac{{\log }_{2}N(ϵ)}{{\log }_2\left(\frac{1}{ϵ}\right)}$. The dimension corresponds to the limiting slope of the line shown, as $ϵ\to 0$, which is toward the right in this graph. The line shown has slope $\approx 1.27$.

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To make the Box Counting Dimension easier to use, we will make these simplifications:

1. the limit $ϵ\to 0$ need only be checked at discerete box sizes, provided sequences goes to $0$
2. boces need not sit on a grid, they can be moved around, rotated, take minimum number of boxes of size $ϵ$ required to cover the set
3. sets other than boxes ( e.g., discs, triangles, etc), will be fine.

Computing the Box counting Dimension

Example

$K_{\infty }\subseteq$

$$BCD(K_{\infty })=\underset{n\to \infty }{\lim }\left(\frac{\ln 2^n}{\ln }\right)$$

Note that the slope of the fit would be the same in any base

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In ${\mathbb{R}}^n$ there are ${ϵ}^{-n}$ boxes in the unit cube. So for example in 10 dimensions, using boxes of size $ϵ=0.01$, there are $10^{20}$ boxes to check. So we need other methods too.

To come:

• Correlation Dimension
• Lyapunov Dimension

Theorem

Let $A$ be a bounded subset of ${\mathbb{R}}^m$ with $BCD(A)=d<m$. Then $A$ is a measure zero set

Hénon Map has dimesion 1.27, thus is measure zero because it is less than its own coordinate dimension of 2.

Proof:

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

where $m-d>0$, so

$$\underset{ϵ\to 0}{\lim }\frac{m\ln ϵ+\ln N}{\ln ϵ}>0$$ $$\underset{ϵ\to 0}{\lim }\frac{\ln ({ϵ}^{m}N)}{\ln ϵ}>0$$

where ${\lim }_{ϵ\to 0}\ln ϵ=-\infty$, means:

$$\underset{ϵ\to 0}{\lim }\ln ({ϵ}^{m}N)=-\infty$$

because anything else would imply $m-d>0$ as $ϵ\to 0$.