Discrete Holomorphic Local Dynamical Systems

Dynamics of Entire Functions 325
Theorem 5.9 (Hausdorff Dimension 2).
For every entire map of bounded type and finite order, the Julia set has Hausdorff
dimension 2. More generally, if f has bounded type and for every ε > 0 there is a
rε > 0 so that
loglog| f (z)|≤(log|z|) q+ε
for |z| > rε, then the Julia set has Hausdorff dimension at least 1 + 1/q.
(Note that functions of finite order satisfy the condition for q = 1, so the second
statement generalizes the first.)
The following result is due to Stallard [St97].
Theorem 5.10 (Explicit Values of Hausdorff Dimension).
For every p ∈ (1,2], there is an explicit example of an entire function for which the
Julia set has Hausdorff dimension p.
Finally we would like to mention the recent survey article on Hausdorff dimension
of entire functions by Stallard [St08].
6 Parameter Spaces
In addition to the study of individual complex dynamical systems, a substantial
amount of attention is given to spaces (or families) of maps. This work comes
in at least two flavors: in most of the early work, specific usually complex onedimensional
families of maps are considered; in transcendental dynamics, this is
most often the family of exponential maps (parametrized as z ↦→ λ e z with λ ∈ D ∗
or z ↦→ e z + c with c ∈ C). Pioneering work in this direction was by Baker and
Rippon [BR84], Devaney, Goldberg, and Hubbard [DGH],andbyEremenkoand
Lyubich [EL92]. Another flavor is to consider larger “natural” parameter spaces. In
rational dynamics, such natural parameter spaces are finite-dimensional and easy
to describe explicitly (such as the family of polynomials or rational maps of given
degree d ≥ 2). In transcendental dynamics, reasonable notions of natural parameter
spaces are less obvious. Early work in this direction was done for instance by
Eremenko and Lyubich [EL92].
We start with a few remarks on general parameter spaces of entire functions and
especially a recent theorem by Rempe. We then discuss the family of exponential
maps as the space of prototypical entire functions and compare it with the Mandelbrot
set as the space of prototypical polynomials. We conclude this section with a
question of Euler.
The following definition is often seen as the natural parameter space of transcendental
entire functions of bounded type.
Definition 6.1 (Quasiconformally Equivalent Entire Functions).
Two functions f ,g of bounded type are called quasiconformally equivalent near ∞
if there are quasiconformal homeomorphisms ϕ,ψ : C → C such that ϕ ◦ f = g ◦ ψ
near ∞.