Discrete Holomorphic Local Dynamical Systems

Dynamics of Entire Functions 311
points in U0 at arbitrarily short hyperbolic distance within F( f ) would have orbits at
least 2π apart, in contradiction to the fact that hyperbolic distances never increase.
Therefore, F( f )=F(N). Therefore, all basins of roots for the Newton map N move
a distance 2πi in each iteration of f , so they turn into wandering domains for f .
Example 2.10 (A Simply Connected Wandering Domain).
The map f (z)=z + sin z + 2π has a bounded simply connected wandering domain.
This example is due to [Ba84] (more generally, he considers functions of the type
z ↦→ z + h(z), whereh is a periodic function). In order to describe this example, first
observe that for g(z)=z+sinz, all points nπ (for odd integers n) are superattracting
fixed points, so their immediate basins are disjoint. Moreover, g(z+2π)=g(z)+2π,
so the Julia set is 2π-periodic, and as above J( f )=J(g). Therefore, f maps the
immediate basin of nπ to the immediate basin of (n + 2)π for odd n, and this is
a wandering domain for f . All critical points of g are superattracting fixed points,
so it follows easily that these basins are simply connected. It is not hard to see
that these basins of g are bounded (the imaginary axis is preserved under g and all
points other than zero converge to ∞,andthesameistruefor2π-translates, so every
Fatou component has bounded real parts, and points with large imaginary parts have
images with much larger absolute values).
Example 2.11 (A Baker Wandering Domain).
For appropriate values of c > 0andrn → ∞,themap
f (z)=cz 2 ∏(1 + z/rn)
n≥1
has a multiply connected wandering domain.
This is the original example from Baker [Ba63, Ba76]. The idea is the following:
the radii rn grow to ∞ very fast. There are annuli An with large moduli containing
points z with rn ≪|z|≪rn+1. On them, the factors (1 + z/rm) with m > n are very
close to 1 so that even their infinite product can be ignored, while the finitely many
bounded factors essentially equal z/rn, so on the annuli An the map f (z) takes the
form cnz n+2 . The factors are arranged so that f sends An into An+1 with degree n+2.
Therefore all points in An escape to ∞. The point z = 0 is a superattracting fixed
point. The fact that the An are contained in a (Baker) wandering domain, rather than
a (periodic) Baker domain (basin at ∞) follows from simple connectivity of Baker
domains (Theorem 2.5).
Remark 2.12. Baker wandering domains may have any connectivity, finite or infinite
[KS08] (the eventual connectivity will be 2 or ∞). These may coexist with simply
connected wandering domains [Be]. These examples use a modified construction
from Example 2.11: essentially, there is a sequence of radii 0 < ··· < rn < Rn <
rn+1