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

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.