## Wednesday, 28 March 2007

### Essay Update - My 3rd Year Essay

So, this is the first of my essay updates, aimed towards mathematics students. It is partly a selfish plan, I have to admit, as it will allow me to keep myself focussed on how I want the essay to progress.

The outline of this essay is the study of external rays of Julia Sets of polynomial maps, and then later to extend this on to the Mandelbrot Set. An external ray is, in laymen's terms, the result of drawing a radial line out from the origin and morphing it in such a way that the outside of the unit disc is transformed to the outside of the Julia Set.

Now, I'll explain what a Julia Set is. Consider a function
$f:\mathbb{C}\rightarrow\mathbb{C}$. Now we can study the effect of the functions $f^n(z)$ as $z\rightarrow\infty$. Any point which is bounded upon repeated iteration of $f$ is said to be in the Filled Julia Set $\mathcal{K}$ of $f$. That is, $\mathcal{K}=\mathbb{C}\setminus\{z\,:\,f^n(z)\rightarrow\infty\text{ as }n\rightarrow\infty\}$. The Julia Set is then said to be $\mathcal{J}=\partial\mathcal{K}$.

The next idea, which will be one of the first of my essay, is the idea of the Böttcher Isomorphism. This is an isomorphism $\varphi:\mathbb{C}\setminus\mathbb{D}_1\rightarrow\mathbb{C}\setminus\mathcal{K}$ (Well, in fact the Böttcher isomorphism is in fact $\varphi^{-1}$).

We can now study external rays. We can take the lines $\mathcal{l}_{\theta}=\{re^{i\theta}:r>1\}$ and perform the isomorphism $\varphi$ on them, giving external rays of the Julia Set.

Keep watching my blog for further updates to how the essay is going!