Geometrically finite transcendental entire functions

Abstract

For polynomials, local connectivity of Julia sets is a much-studied and important property. Indeed, when the Julia set of a polynomial of degree (Formula presented.) is locally connected, the topological dynamics can be completely described as a quotient of a much simpler system: angle (Formula presented.) -tupling on the circle. For a transcendental entire function, local connectivity is less significant, but we may still ask for a description of the topological dynamics as the quotient of a simpler system. To this end, we introduce the notion of docile functions (Definition 1.1): a transcendental entire function with bounded postsingular set is docile if it is the quotient of a suitable disjoint-type function. Moreover, we prove docility for the large class of geometrically finite transcendental entire functions with bounded criticality on the Julia set. This is an analogue of the local connectivity of Julia sets for geometrically finite polynomials, first proved by Douady and Hubbard, and extends previous work of the second author and of Mihaljević for more restrictive classes of entire functions. We deduce a number of further results, concerning bounded Fatou components and the local connectivity of Julia sets. In particular, we show that the Julia set of the sine function is locally connected, answering a question raised by Osborne.