Yongcheng Yin

Rational maps whose Julia sets are Cantor circles

In this paper, we give a family of rational maps whose Julia sets are Cantor circles and show that every rational map whose Julia set is a Cantor set of circles must be topologically conjugate to one map in this family on their corresponding Julia sets. In particular, we give the specific expressions of some rational maps whose Julia sets are Cantor circles, but they are not topologically conjugate to any McMullen maps on their Julia sets. Moreover, some non-hyperbolic rational maps whose Julia sets are Cantor circles are also constructed.

A tableau approach of the KSS nest

The KSS nest is a sophisticated choice of puzzle pieces given in [Ann. of Math. 165 (2007), 749–841]. This nest, once combined with the KLLemma, has proven to be a powerful machinery, leading to several important advancements in the field of holomorphic dynamics. We give here a presentation of the KSS nest in terms of tableau. This is an effective language invented by Branner and Hubbard to deal with the complexity of the dynamics of puzzle pieces. We show, in a typical situation, how to make the combination between the KSS nest and the KL-Lemma. One consequence of this is the recently proved Branner–Hubbard conjecture. Our estimates here can be used to give an alternative proof of the rigidity property.