Geometrically finite transcendental entire functions

Abstract

For polynomials, local connectivity of Julia sets is an important and muchstudied property, as it leads to a complete description of the topological dynamics as a quotient of a much simpler system, namely angle d-tupling on the circle, where d ≥ 2 is the degree.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 do so, we introduce the notion of docile functions: a transcendental entire function with bounded postsingular set is docile if its dynamics can be described as 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 can be seen as 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-Brandt, for more restrictive classes of entire functions.Our techniques allow us deduce a number of other results for geometrically finite functions with bounded criticality, 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.

Local connectivity of Julia sets for rational maps with Siegel disks

“…Note that in Theorems A and B, the maps cannot have parabolic basins. The local connectivity of the Julia sets of geometrically finite maps has been proved by constructing expanding orbifold metrics (see [TY96] and [ARS20]). One may weaken the conditions in Theorems A and B such that the parabolic basins are allowed.…”

“…For transcendental entire functions, the local connectivity of the Julia sets is also significant for understanding the dynamics and this property was studied for a lot of classes. For examples, the cases of hyperbolic [Mor99], [BFR15], semi-hyperbolic [BM02], strongly geometrically finite [ARS20] and strongly postcritically separated [Par21]. See also [BD00], [Mih12] and [Osb13] for related results.…”