Yongcheng Yin, Proof of the Branner-Hubbard
conjecture and applications Abstract By means of a nested sequence of some critical pieces constructed by Kozlovski, Shen, and van Strien, and by using a covering lemma recently proved by
Kahn and Lyubich, we prove that a component of the
filled-in Julia set of any polynomial is a point if and only if its forward
orbit contains no periodic critical components. It follows immediately that
the Julia set of a polynomial is a Cantor set if and only if each critical
component of the filled-in Julia set is aperiodic.
This result was a conjecture raised by Branner and
Hubbard in 1992. Some applications will also be given. |