Discrete Holomorphic Local Dynamical Systems

314 Dierk Schleicher
The following fundamental estimate of Eremenko and Lyubich [EL92, Lemma 1]
is very useful. Write H := {z ∈ C: Rez > 0} for the right half plane.
Lemma 3.12 (Expansion on Logarithmic Tracts).
If T ′ is a logarithmic tract and F : T ′ → H is a conformal isomorphism, then
|F ′ (z)|≥ 1
ReF(z) . (1)
4π
Remark 3.13. All we are using about T ′ is that it is simply connected and disjoint
from T ′ + 2πik for k ∈ Z \{0}.
Proof. Since F : T ′ → H is a conformal isomorphism, it has a conformal inverse
G: H → T ′ .LetD be the open disk around F(z) with radius R = ReF(z). Then
by the Koebe 1/4-theorem, G(D) contains a disk around G(F(z)) = z of radius
R|G ′ (F(z))|/4 = ReF(z)/4|F ′ (z)|; but by periodicity of the tracts in the vertical
direction, the image radius must be at most π,and(1) follows. ⊓⊔
Sketch of proof of Theorem 3.4. For a function F : � T ′ j,k → H in logarithmic coordinates,
we claim that the set of points z with ReF ◦n (z) → ∞ has no interior (here
we assume for simplicity that, possibly by translating coordinates, F(T ′ j,k )=H for
all tracts T ′ j,k ). The key to this is Lemma 3.12. Suppose that F : � T ′ j,k → H has an
open set in the escaping set, say the round disk Dε(z) around some point z with
ReF ◦n (z) → ∞. Then(1) implies |F ′ (F◦n (z))| →∞, hence |(F◦n ) ′ (z)| →∞. Using
the Koebe 1/4 theorem it follows that I(F) contains disks of radii ε|(F ◦n ) ′ (z)|/4.
Since these radii must be bounded by π, it follows that ε = 0. This proves the claim,
and this implies that for functions f ∈ B, the escaping set I( f ) has empty interior.
This proves Theorem 3.4, except for the fact that functions F ∈ S do not have wandering
domains. We do not describe this proof here. It uses the essential ideas of
Sullivan’s proof in the rational case (see [Mi06]), in particular quasiconformal deformations,
and depends on the fact that small perturbations of functions in class S
live in a finite-dimensional space. ⊓⊔
Corollary 3.14 (Entire Functions with Empty Fatou Set).
If an entire function f in class B has the property that all its singular values are
strictly preperiodic or escape to ∞, then f has empty Fatou set.
Indeed, any Fatou component of f would require an orbit of a singular value that
converges to an attracting or parabolic periodic orbit, or accumulates on the boundary
of a Siegel disk, by Theorems 2.3 and 3.4.
Functions f ∈ B have a most useful built-in partition with respect to which
symbolic dynamics can be defined for those orbits that stay sufficiently far away
from the origin.
Definition 3.15 (Symbolic Dynamics and External Address).
Consider a function f ∈ B and let R > 0 be large enough so that DR(0) contains all
singular values as well as f (0),andletF : T ′ j,k → HlogR be a function in logarithmic
coordinates corresponding to f .Herekranges through the integers so that T ′ j,k+1 =
T ′ j,k + 2πi,andj ranges through some (finite or countable) index set J,say.