## Monday, 2 April 2007

### Essay Update - The Böttcher Isomorphism

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.

#### 1 comment:

Lexi_love08 said...

Ok.
So im a sophomore in highschool in an honors geometry class.
and all this seems like gibberish to me.
My class has an assignment and I got stuck with Bottcher's proof.
I'm suppose to prove it.
and figure out why he used the squares and triangles that he did and why they are that size and everything.
So basically im suppose to put it into detail that everyone can understand.
And so far all i found is that this proff is a "proof without words" And it's not helping me any.
You have any ideas?
And maybe can put it in a very very easy way to understand haha.
Thank you =]