Discrete Holomorphic Local Dynamical Systems

318 Dierk Schleicher
Examples of Fatou components in I( f ) have been given in Examples 2.8 (a Baker
domain), 2.9,and2.10 (wandering domains).
We would like to review especially Fatou’s Example 2.9 with f (z)=z+1+e −z .
There is a single Fatou component, it contains the right half plane, and in it the orbits
converge quite slowly to ∞. However, for integers k, points on i(2k + 1)π + R −
converge very quickly to ∞ (at the speed of iterated exponentials); these are countably
many curves in I( f ) (dynamic rays), and the backwards orbits of these curves
form many more curves in I( f ). In fact, I( f ) contains uncountably many curves to
∞ (a Cantor bouquet).
Eremenko [Er89] raised some fundamental questions on I( f ) that inspired further
research.
Question 4.2 (Eremenko’s Questions on I( f )).
Weak version Is every component of I( f ) unbounded?
Strong version Can every z ∈ I( f ) be connected to ∞ within I( f )?
These questions have inspired a substantial amount of work, and are often referred
to as Eremenko’s conjecture (in its weak and strong form). There are various
partial results on them, positive and negative. Rippon and Stallard [RS05a, RS05b]
showed the following.
Theorem 4.3 (Baker Wandering Domains and I( f )).
Let f be an entire function. Then I( f ) always has an unbounded component. If f
has a Baker wandering domain, then I( f ) is connected.
These results are based on a study of the set A( f ) as described below.
Rempe [Re07] established sufficient conditions for the weak version of Eremenko’s
question, as follows.
Theorem 4.4 (Unbounded Components of I( f )).
If f is an entire function of bounded type for which all singular orbits are bounded,
then every component of I( f ) is unbounded.
Substantial attention has been given to the speed of escape for points in I( f ). A
classical lemma, due to Baker [Ba81], is the following.
Lemma 4.5 (Homogeneous Speed of Escape in Fatou Set).
If z,w ∈ I( f ) are two escaping points from the same Fatou component of an entire
function f , then log| f ◦n (z)|/log| f ◦n (w)| is bounded as n → ∞. If z is in a (periodic)
Baker domain, then log| f ◦n (z)| = O(n).
This result follows from the observation that the hyperbolic distance between w
and z in the hyperbolic metric of F( f ) cannot increase, and that this distance is
essentially bounded below by the hyperbolic distance of C \ R − 0 .
Unlike other types of periodic Fatou components, Baker domains need not contain
singular values. However, if a Baker domain exists, the set of singular values
must be unbounded by Theorem 3.4; there is a stronger result by Bargmann [Ba01]