ABSTRACT

The Branner-Hubbard conjecture has recently been proved in its full generality (see [KS,QY]):

Theorem (Qiu-Yin, Kozlovski-van Strien). Let f : C → C be a complex polynomial. Denote by Kf its filled Julia set. Assume that every component of Kf containing a critical point is aperiodic. Then Kf is a Cantor set.