Discrete Holomorphic Local Dynamical Systems

Dynamics of Rational Surface Automorphisms 67
Proof. By the stable manifold theorem, W s (p) is homeomorphic to a disk, so as a
Riemann surface it is conformally equivalent either to the disk or to C. Inorderto
show it is equivalent to C, it suffices to show that W s (p) contains an infinite increasing
family of disjoint annuli A(k) so that A(1) surrounds p, A(k+1) surrounds A(k),
and the moduli of the A(k) are bounded below.
Let Bp(r)={q : ||p − q|| < r} be the ball with radius r and center p. Then for r2
small, the connected component of Bp(r2) ∩W s (p) containing p will be a disk, and
for 0 < r1 < r2, Ap :=(Bp(r2)−Bp(r1))∩W s (p) will be an annulus in W s (p) which
goes around p. By compactness, there is a lower bound on the moduli of the annuli
Ap for p ∈ J. By the uniform contraction of hyperbolicity, there is an n such that for
every p, Ap(1) := f −n A f n p contains Ap in its “hole”. Thus A(k) := f −nk A f nk p is the
desired increasing sequence of annuli in W s (p). ⊓⊔
Theorem 1.19. If f is a complex Hénon map with |δ| �= 1, and if J is a hyperbolic
set, then the interior of K + consists of the basins of finitely may hyperbolic sink
orbits.
Proof. We may assume that |δ|≤1, for if |δ| > 1, then int(K + ) is empty, and there
is nothing to prove. The theorem will be a consequence of the following assertions:
(1) Each component is periodic and recurent.
(2) Each component is the basin of a sink.
(3) There are finitely many sink orbits.
Proof of (1). So assume that |δ| < 1. Let C be a wandering component of int(K + ),
and let p be a point of C.Fornsufficiently large, f n p will be a point of V.SinceVis compact, then, the ω-limit set ω(p) is a nonempty compact subset of V.Sinceω(p)
is invariant under f −1 and bounded, it follows that ω(p) ⊂ K ⊂ K− .Since|δ| < 1,
we have J− = K− . Thus ω(p) ⊂ J− .
Now we claim that ω(p) intersects int(K + ). For otherwise, ω(p) ⊂ J + = ∂K + ,
which means that ω(p) ⊂ J. Thus p ∈ W s (J), and by the Proposition above, we
conclude that p ∈ J + , which is not the case.
Now choose q0 ∈ ω(p) ∩ int(K + ),andletC0denote the component of int(K + )
containing q0. Therearen1and n2 so that f n1 p and f n2 p are arbitrarily close to q0,
and so they both belong to C0. We conclude, then, that f n1C = f n2C, from which
we conclude that C must be periodic, and since it contains a point of ω(p) it is
recurrent.
Proof of (2) Assume |δ| < 1. We Know By Theorem 1.12 that each recurrent component
is the basin of either a sink or a Siegel disk or Herman ring. Suppose that is
the basin of Σ, which is a Siegel disk or a Herman ring. By Theorem 1.12 again, Σ
is a subvariety of C. SinceΣisbounded and invariant under f −1 ,wehaveΣ⊂K− .
Finally, since ∂Σ ⊂ ∂C ⊂ ∂K + ,wehave∂Σ ⊂ J. By the rotation property of f on
Σ, the closure of the orbit of a point of Σ is a compact curve. If we take q ∈ Σ close
to J, then the closure of the orbit will be contained in a small neighborhood of J.
However, this contradicts the maximality property of J. We conclude that the Siegel
disk Herman ring cases cannot occur, so C must be the basin of a sink.
Proof of 3. The sink orbits are all contained in V, which is compact. Define the
set L to be the limit points of sequences of (distinct) sinks. If there are infinitely