Böttcher theorem

Near a superattracting fixed point, a holomorphic map is conformally conjugate to its leading monomial.

For a degree-$d$ polynomial $f(z)=a_d{z}^d+\cdots$, the Böttcher coordinate near infinity satisfiesIts modulus extends naturally throughout the basin of infinity.

The escape-rate Green function isIt vanishes on the filled Julia set, is positive and harmonic on the basin of infinity, satisfies $G_f\circ f=d{G}_f$, and equals $\log |{\varphi }_f|$ near infinity and by continuation throughout the basin.

The Julia set of a polynomial is connected exactly when every finite critical point belongs to the filled Julia set. In this case the Böttcher coordinate extends conformally over the entire basin of infinity; an escaping critical point is precisely an obstruction to that continuation.

For $f_c(z)=z^2+c$ outside the Mandelbrot set, evaluate the dynamical Böttcher coordinate at the critical value:This parameter Böttcher map is a conformal isomorphism from the complement of the Mandelbrot set to the exterior unit disc, proving that the Mandelbrot set is full and connected.