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 with and , then there exists a local holomorphic change of coordinate with which conjugates to through a neighbourhood of . Furthermore, is unique up to multiplication by a root of unity.
So, what does this mean? Essentially, near any critical point of , we find that it is conjugate to a map with . In the case we are looking at, that is polynomial maps of degree taking , we know that it extends to a rational map of which has a superattracting fixed point at with local degree .
Next up is the proof of this theorem. In my essay, I am looking at the specific case of . 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 with a superattracting fixed point at , and our function has a superattracting fixed point at . To get around this, we simply use a conformal isomorphism - that is a function which essentially drags around the Riemann Sphere to a new function . Since and we want , we can simply define . Expanding this, we find that . We can now find the power series around the point of interest - - and find that if is equal to . This was found using a custom MATLAB script using the symbolic math toolbox. We can see now that is of the correct form for Böttcher's Theorem. The Böttcher Theorem will be proven for the case in the next Essay Update.