The Dynamics Of The Rational Functions With Cantor Julia Sets

In this dissertation, we study a special class of rational functions with degree no less than 2 whose Julia sets are Cantor sets and obtain a series of results.First, we give a sufficient and necessary condition when the Julia set of a rational function with an invariant Fatou domain is a Cantor set, and this result generalizes the Branner-Hubbard conjecture.Secondly, we discuss whether rational functions carry invariant line fields on their Julia sets and prove that the Cantor Julia sets of rational functions carry no invariant line fields. Next, we solve the quasi-conformal rigidity for this class of rational functions, namely, topologically conjugate implies quasi-conformally conjugate, moreover, we obtain that each Cantor Julia set is dynamically removable.We also study the conformal measures of rational functions and show that the number of ergodic components of a conformal measure on a Cantor Julia set is controlled by the total number of the critical points and the parabolic fixed point belonging to the Julia set.Finally, taking advantage of quasi-conformal surgery and the previous consequences, we prove a theorem associated with the hyperbolic conjecture, which says that each non-hyperbolic rational function with Cantor Julia set can be approximated by hyperbolic rational functions with Cantor Julia sets.Most of the achievements in this dissertation are mainly based on the work provided by O. Kozlovski, W. Shen, and S. van Strien about studying the quasi-conformal rigidity for real polynomials, and the work about resolving the Branner-Hubbard conjecture owed to Prof. Qiu, Weiyuan and Prof. Yin. Yongcheng.