3 - Meromorphic functions with two completely invariant domains

Summary

Abstract. We show that if a meromorphic function has two completely invariant Fatou components and only finitely many critical and asymptotic values, then its Julia set is a Jordan curve. However, even if both domains are attracting basins, the Julia set need not be a quasicircle. We also show that all critical and asymptotic values are contained in the two completely invariant components. This need not be the case for functions with infinitely many critical and asymptotic values.

INTRODUCTION AND MAIN RESULT

Let f be a meromorphic function in the complex plane ℂ. We always assume that f is not fractional linear or constant. For the definiitions and main facts of the theory of iteration of meromorphic functions we refer to a series of papers by Baker, Kotus and Lü [2, 3, 4, 5], who started the subject, and to the survey article [8]. For the dynamics of rational functions we refer to the books [7, 11, 21, 25].

A completely invariant domain is a component D of the set of normality such that f⁻¹(D) = D. There is an unproved conjecture (see [4, p. 608], [8, Question 6]) that a meromorphic function can have at most two completely invariant domains. For rational functions this fact easily follows from Fatou's investigations [15], and it was first explicitly stated by Brolin [10, §8].