Escape-rate Green function of a polynomial ($G_f$)

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.