Ravings of a Mathematician: 3rd year essay

I have been mentioning Julia Sets in my essay updates a lot, and yet I still have not really explained what they are in great detail, or given you any examples. This post aims to explain Julia Sets and some related results in a manner which is graspable by a first year mathematician or so. The prerequisite knowledge really for a basic understanding is Complex Numbers and Functions.

To find a Julia Set, we first need the function for which it is a Julia Set of. Although this concept can be extended to other cases (or reduced to in the case of $\mathbb{R}$ ), the most common case is that of a function $f:\mathbb{C}\rightarrow\mathbb{C}$ . The function $f$ also needs to be quite smooth, specifically it needs to be holomorphic (you do not really need to understand this greatly for a basic understanding, but if you're interested wikipedia has the definition behind the link).

Now, the concept of a Julia Set is based upon the long term effect of repeated iterations of $f$ on points in the plane. That is to say, given a point $z\in\mathbb{C}$ , we want to find out what happens as we take repeated iterations of $z$ , that is $f(z),\,f^2(z)=f(f(z)),\ldots f^n(z)\ldots$ as $n\rightarrow\infty$ . There are, in a simplistic case, 2 different things that can happen. The first is that the point tends off to infinity upon repeated iteration of $f$ . The second is that it doesn't, and remains bounded. Now, this may not sound particularly exciting, but this concept can give some surprising results right from the offset.

Generally speaking, a good point to start at in the study of Julia Sets is looking at the Julia Set of a Polynomial Map - $f_c(z)=z^2+c$ (note that it is a simple exercise to notice that in fact any complex polynomial map of the form $ax^2+bx+c$ can be reduced to one of the form $z^2+c$ ). So, let us look at the most simple of these maps, that is $f_0(z)=z^2$ . Now, let us try to find which points are in the set $\{z:\left\vert f^n(z)\right\vert<M\text{ as }n\rightarrow\infty\}$ . In this particular case, it is simple. As we know, if we take a point $re^{i\theta}\in\mathbb{C}$ , then $(re^{i\theta})^n=r^ne^{in\theta}$ , and so all points such that $r<1$ will move towards $0$ , all points with $r>1$ will move toward infinity and all points with $r=0$ will stay on the unit circle. We can now say that the filled julia set $\mathcal{K}=\{re^{i\theta}:r\leq1\}$ . Furthermore, the Julia set is $\mathcal{J}=\partial\mathcal{K}=\{re^{i\theta}:r=1\}$ . Finally, in terms of definitions, the Fatou Set of this function is defined to be $\mathcal{F}=\mathbb{C}\setminus\mathcal{K}$ , that is all the points which are not in the filled Julia Set.

So, having seen a very simple example, we may wonder what will the effect be when we change our value of $c$ ? Well, this is where we get our surprising result. Running a simple script through MATLAB (which I will write about in the future), we find that our Filled Julia set (with $c=0.52+0.25i$ ) now looks like this:

This is a set of fractal nature, that is to say that it has an infinitely fine structure (so if you zoom in on the edge of it, it remains detailed at arbitrary levels) and small parts of it bear a resemblance to the entire set.

Julia Sets were some of the first things I came across when writing my second year essay, and my interest in them has really pushed me towards learning about complex dynamics in general, thus my third year essay!

If you want to find out more about Julia Sets, and in fact fractals in general, I would suggest that the book Fractal Geometry: Mathematical Foundations and Applications by Kenneth Falconer is brilliant, and well worth a read for anyone with interest in this area.

Right, I'm now at the point in my essay that I need to explore a theorem known as "Böttcher's Theorem." What Böttcher's Theorem states is that given a function of the form $f(z)=a_nz^n+a_{n+1}z^{n+1}+\ldots$ with $n\geq 2$ and $a_n\neq 0$ , then there exists a local holomorphic change of coordinate $w=\varphi(z)$ with $\varphi(0)=0$ which conjugates $f$ to $w\mapsto w^n$ through a neighbourhood of $0$ . Furthermore, $\varphi$ is unique up to multiplication by a $(n-1)\text{th}$ root of unity.

So, what does this mean? Essentially, near any critical point of $f$ , we find that it is conjugate to a map $\varphi\circ f\circ\varphi^{-1}:w\mapsto w^n$ with $n\geq 2$ . In the case we are looking at, that is polynomial maps of degree $d\geq 2$ taking $\mathbb{C}\rightarrow\mathbb{C}$ , we know that it extends to a rational map of $\widehat{\mathbb{C}}=\mathbb{C}\cup\infty$ which has a superattracting fixed point at $\infty$ with local degree $d=n$ .

Next up is the proof of this theorem. In my essay, I am looking at the specific case of $f_c(z)=z^2+c$ . Of course, our issue with this function is that it is not in a form that the Böttcher Theorem can deal with - it needs a function of the form $a_nz^n+a_{n+1}z^{n+1}+\ldots$ with a superattracting fixed point at $0$ , and our function has a superattracting fixed point at $\infty$ . To get around this, we simply use a conformal isomorphism - that is a function which essentially drags $f_c$ around the Riemann Sphere $\widehat{\mathbb{C}}$ to a new function $F_c$ . Since $f_c(\infty)=\infty$ and we want $F_c(0)=0$ , we can simply define $F_c(\zeta)=\frac{1}{f_c(\frac{1}{\zeta})}$ . Expanding this, we find that $F_c(\zeta)=\frac{1}{\frac{1}{\zeta^2}+c}=\frac{\zeta^2}{1+\zeta^2c}$ . We can now find the power series around the point of interest - $0$ - and find that if is equal to $F_c(\zeta)=\zeta^2-c\zeta^4+c^2\zeta^6-c^3\zeta^8+\ldots$ . This was found using a custom MATLAB script using the symbolic math toolbox. We can see now that $F_c$ is of the correct form for Böttcher's Theorem. The Böttcher Theorem will be proven for the case $F_c$ in the next Essay Update.

Ravings of a Mathematician

Friday, 6 April 2007

Essay Backtrack - Julia Sets

Monday, 2 April 2007

Essay Update - The Böttcher Isomorphism

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