Discrete Holomorphic Local Dynamical Systems

62 Eric Bedford
Proposition 1.8. W s (K)=K + .
Proof. Since K is bounded, it is clear from the definition that W s (K) ⊂ K + .On
the other hand, let p ∈ K + be given. Then by the filtration properties, we have
ω(p) ⊂ V. Sinceω(p) is invariant under f −1 , is is contained in K − . We conclude
that ω(p) ⊂ K,sodist( f n p,K) → 0asn → ∞. ⊓⊔
1.3 Fatou Components
We define the Fatou set to be the points p for which there is a neighborhood U such
that the restrictions of the forward iterates { f n |U : n ≥ 0} are equicontinuous. If
we consider C 2 as imbedded in the projective plane P 2 (this is defined in 2nd part),
then the iterates f n : C 2 → P 2 are locally equicontinuous on C 2 − K + .Thatis,they
converge locally uniformly to the point [0:0:1] ∈ P 2 ,therethey-axis intersects the
line at infinity. By the previous section on Filtration, then, we have:
Proposition 1.9. The Fatou set of a complex Hénon map is C 2 − J + .
A domain U ⊂ C 2 is said to be a Runge domain if for each compact subset X ⊂ U,
the polynomial hull
is also contained in U.
ˆX := {q ∈ C 2 : |P(q)|≤max|P(x)|
for all polynomials P}
x∈X
Proposition 1.10. int(K + ) is Runge.
Proof. Suppose that X is a compact subset of int(K + ). By the filtration properties of
the previous Section, we know that sup n≥0 sup x∈X || f n (x)|| = C < ∞. Since the components
of f n are polynomials, it follows from the definition of the polynomial hull
that sup x∈X || f n (x)|| ≤C. Thus ˆX is contained in K + . Further, since X is compact, we
know that the translates X + v are contained in int(K + ) if ||v|| is sufficiently small.
Thus we have �X + v = ˆX + v ⊂ K + . It follows that dist(X,∂K + )=dist( ˆX,∂K + )
and thus ˆX is in the interior of K + . ⊓⊔
If Ω is a Fatou component (i.e., a connected component of the Fatou set), then f Ω
is again a Fatou component. Let δ denote the (constant) Jacobian of f .Thereare
two distinct cases: |δ| = 1and|δ| �= 1. We will first consider the case |δ| �= 1. In
this case, after passing to f −1 if necessary, we may suppose that |δ| < 1. If |δ| < 1,
then the Fatou components consist of C2 − K + and the connected components of
int(K + ).
Problem: Is every Fatou component periodic, i.e., is f nΩ = Ω for some n > 0? A
non-periodic Fatou component (if it existed) would be said to be wandering.
A Fatou component Ω is said to be recurrent if there is a point p ∈ Ω such that
Ω ∩ ω(p) �= /0. Equivalently, Ω is recurrent if there are a compact set C ⊂ Ω and a