Discrete Holomorphic Local Dynamical Systems

320 Dierk Schleicher
8. if f has a Baker wandering domain, then A( f ) and I( f ) are connected, and each
component of A( f ) ∩ J( f ) is bounded; otherwise, each component of A( f ) ∩
J( f ) is unbounded;
9. every component of A( f ) is unbounded;
10. wandering domains may escape slow enough for L( f ) and fast enough for A( f ).
Item (1) comes out of Eremenko’s proof that I( f ) is non-empty (compare
Theorem 1.8). In (2), invariance is built into the definition, and the inclusion
comes from [BH99]. For Statement (3), note that ∂A( f ), ∂L( f ), and∂Z( f ) are all
closed invariant sets, so they contain J( f ) by Theorem 1.7, and the claim reduces to
the fact that any Fatou component that intersects A( f ), L( f ),orZ( f ) is contained in
these sets. This follows from Lemma 4.5 and also implies (4) (see[RS00]). Statement
(5) is from [RS00, BH99]. Statement (6) is due to Baker (see Lemma 4.5).
Statements (7) and(9) are from [RS05a], while (8) is unpublished recent work by
Rippon and Stallard. Wandering domains in L( f ) were given in Examples 2.9 and
2.10, while Baker wandering domains are in A( f ); thisis(10).
Recent work by Rippon and Stallard is on “spider’s webs”: these are subsets of
I( f ) for which every complementary component is connected (such as the orbit of a
Baker wandering domain, together with certain parts of I( f ) connecting the various
wandering domains). Rippon and Stallard propose the idea that this feature may be
no less common and prototypical for the dynamics of entire functions as Cantor
Bouquets are nowadays often considered to be.
Theorem 4.8 (Slow Escape Possible).
For every real sequence Kn → ∞,thereisaz∈ I( f ) ∩ J( f ) so that | f ◦n (z)| < Kn for
all sufficiently large n.
This result is also due to Rippon and Stallard [RS].
It had been observed by Fatou that I( f ) often contains curves to ∞. Thiswas
shown for exponential maps in [DGH], and for more general entire functions having
logarithmic tracts satisfying certain geometric conditions in [DT86]. This leads to
the following.
Definition 4.9 (Dynamic Ray).
A dynamic ray tail is an injective curve γ : (τ,∞) → I( f ) so that
• f ◦n (γ(t)) → ∞ as t → ∞ for every n ≥ 0, and
• f ◦n (γ(t)) → ∞ as n → ∞ uniformly in t.
A dynamic ray is a maximal injective curve γ : (0,∞) → I( f ) so that for every τ > 0,
the restriction γ| (τ,∞) is a dynamic ray tail. An endpoint of a dynamic ray γ is a point
a with a = limt↘τ γ(t).
Remark 4.10. Dynamic rays are sometimes called “hairs”; we prefer the term “ray”
in order to stress the similarity to the polynomial case.
Eremenko asked whether every point in I( f ) can be connected to ∞ by a curve in
I( f ) (compare Conjecture 4.2). This conjecture has been confirmed for exponential