Discrete Holomorphic Local Dynamical Systems

330 Dierk Schleicher
Definition 7.1 (Basin and Immediate Basin).
For a root α of f , we define its basin as Uα := {z ∈ C: N◦n f (z) → α} as n → ∞.The
immediate basin is the connected component of Uα containing α.
Theorem 7.2 (Immediate Basins Simply Connected).
Every root of f has simply connected immediate basin.
This was shown in [MS06]. It is an open question whether every Fatou component
of Nf is simply connected. After work by Taixes, the last open case is whether Baker
domains are always simply connected (compare [FJT08]).
Theorem 7.3 (Wandering Newton Domains Simply Connected).
If a Newton map has a wandering domain, then it is simply connected.
This follows from work of Bergweiler and Terglane [BT96]: in analogy with
classical work of Shishikura [Sh09], they prove that a multiply connected wandering
domain of a transcendental meromorphic map g (such as a Newton map) would
require that g has a weakly repelling fixed point; but this is not the case for Newton
maps.
Definition 7.4 (Virtual Immediate Basin).
A virtual immediate basin is a maximal subset of C (with respect to inclusion)
among all connected open subsets of C in which the dynamics converges to ∞ locally
uniformly and which have an absorbing set. (An absorbing set in a domain V
is a subset A such that Nf (A) ⊂ A and every compact K ⊂ V has a n ≥ 0sothat
(K) ⊂ A.)
N ◦n
f
Theorem 7.5 (Virtual Immediate Basins Simply Connected).
Every virtual immediate basin is simply connected.
This was also shown in [MS06]. Every virtual immediate basin is contained in a
Baker domain; it is an open question whether this basin equals a Baker domain (this
is true for simply connected Baker domains). The principal difficulty is the question
whether every Baker domain has an absorbing set as in Definition 7.4; this would
also imply that every Fatou component of a Newton map is simply connected.
Theorem 7.6 (Two Accesses Enclose Basin).
Let f be an entire function (polynomial or transcendental) and let Uα be the immediate
basin of α for Nα. LetΓ1,Γ2 ⊂ Uα represent two curves representing different
invariant accesses to ∞, and let W be an unbounded component of C \ (Γ1 ∪ Γ2).
Then W contains the immediate basin of a root of f or a virtual immediate basin,
(z) ∩W is finite for all z ∈ C.
provided the following finiteness condition holds: N −1
f
Remark 7.7. In the case of a polynomial f , the finiteness condition always holds,
and there is no virtual immediate basin. The result thus says that any two accesses
of any immediate basins enclose another immediate basin. Theorem 7.6 is due to
Rückert and Schleicher [RüS07].