Discrete Holomorphic Local Dynamical Systems

64 Eric Bedford
Thus, setting M := {w ∈ Ω : hw = w}, we see that Σ ⊂ M. Conversely,ifw ∈ M,
then gw ∈ M because h(gw)=g(hw)=g(w). Thus Σ = M.
Again, since f ◦ g = g ◦ f ,wehave f Σ = Σ. Further, since Σ consists of infinite
forward limits of points of K + ,wehaveΣ ⊂ K. Thus Σ is a bounded Riemann surface,
so it is covered by the disk. Finally, since the fixed points of f N are isolated, the
restriction f N |Σ is not the identity. Since there is a point of ω(p) ∩ Σ, we know that
the iterates f n |Σ cannot diverge to the boundary of Σ as n → ∞. We conclude that
Σ must be uniformized by either the disk or an annulus, and f must be an irrational
rotation. Thus Σ must be either a Siegel disk or a Herman ring. ⊓⊔
It is clear that Hénon maps can have sinks. For a Siegel disk, let us consider the
diagonal linear map L of Δ × C to itself which is given by (ζ,η) ↦→ (αζ,δη/α),
and α = e πia with a irrational. If Φ : Δ × C → C 2 is a holomorphic imbedding such
that f ◦Φ = Φ ◦L,thenD = Φ(Δ ×{0}) will be contained in a Siegel disk. Further,
if |δ| < 1, we will have that Ω := Φ(Δ × C 2 ) is equal to W s (D).
Such a map occurs for Hénon maps which have the form
f (x,y)=(αx + ···,δy/α + ···)
where the ··· indicate arbitrary terms of higher order. (Or rather we conjugate a
Hénon map by an affine map so that the origin is fixed, and the linear part is diagonal.)
If |δ| < 1andifα satisfies a suitable Diophantine condition (see [Z]), then
there will exist a linearizing map Φ as in the previous paragraph.
Problem: Is it possible for a Fatou component to be the basin of a Herman ring?
Another sort of Fatou component can arise as follows. Suppose that f is a Hénon
map fixing (0,0) and taking the form
f (x,y)=(x + x 2 + ···,by + ···),
where 0 < |b| < 1, and the terms ··· involve both x and y, but they are “smaller.”
This is like a parabolic fixed point in the x-direction and an attracting fixed point
in the y-direction. We may choose r small so that D := {|x + r| < r}×{|y| < r}
is mapped inside itself. In fact, for each p ∈ D there is a neighborhood U such that
f n → (0,0) uniformly on U as n → +∞. The forward basin B = {(x,y) ∈ C 2 : f n →
(0,0) uniformly in a neighborhood of (x,y)} is a non-recurrent Fatou component.
Problem: Is such a semi-attracting basin the ony possible sort of non-recurrent Fatou
component? More precisely, suppose that |δ| �= 1andthatΩ is a periodic Fatou
component which is not recurrent. Is there necessarily a point p ∈ ∂Ω with f n p = p
and such that the multipliers of Df n p are α and δ n /α, whereα is a root of unity?
The volume preserving case: |δ| = 1.
Except for C 2 − K + , all Fatou components are contained in the bounded set
int(K + )=int(K − )=int(K). IfΩ is such a bounded component, then there are
only finitely many other components with the same volume, so Ω is necessarily
periodic.