Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 157 / 3

For $f(z)=a_d{z}^d+\cdots$, the Böttcher coordinate at infinity is the conformal coordinate ${\varphi }_f$ defined near infinity byIts modulus has a dynamically natural extension to the entire basin of infinity

The escape-rate Green function of a polynomial isIt is zero on the filled Julia set $K(f)$, positive and harmonic on $A_{\infty }(f)$, and satisfies $G_f(f(z))=d{G}_f(z)$. Near infinity,so $e^{G_f}$ is precisely the extension of $|{\varphi }_f|$ throughout the basin.

Assume $J(f)$ is connected. Then the filled Julia set $K(f)$ is connected and full, so its complement $A_{\infty }(f)$ is simply connected. The local Bottcher coordinate therefore extends by the functional equation to a conformal isomorphismIf a finite critical point $c$ lay in $A_{\infty }(f)$, differentiatingat $c$ would giveNeither factor on the right vanishes on the exterior disc under a conformal coordinate, a contradiction. Hence every finite critical point lies in $K(f)$. This is one direction of the connected Julia set criterion for a polynomial.

For $f_c(z)=z^2+c$, the Mandelbrot set is the set of parameters for which the critical orbit of zero is bounded. If $c$ lies outside it, the critical value $c$ is in the basin of infinity and one defines the parameter Bottcher mapThe dynamical functional equation, holomorphic dependence on $c$, and the normalization at infinity show that $\Phi$ is a proper degree-one holomorphic mapIt is therefore a conformal isomorphism. Since the exterior disc is connected and simply connected, the complement of $M$ has no bounded component; equivalently, the Mandelbrot set connectedness from the parameter Böttcher coordinate proves that $M$ is connected.