Discrete Holomorphic Local Dynamical Systems

60 Eric Bedford
Fig. 1 Filtration behavior of Hénon maps
3. If (x,y) ∈ V − , then the orbit f n (x,y), n≥ 0, can stay in V − for only finite time.
After f n (x,y) leaves V − , it cannot reenter, which means that it stays in V ∪V + .
In particular, if an orbit is bounded in forward time, then it must enter V and
remain there.
This Theorem is illustrated in Figure 1. The arrows give the possibilities where a
point might map; the dashed circular arrow indicates that an orbit can remain in V −
for only finite time.
K ± = {(x,y) ∈ C 2 : { f ±n (x,y),n ≥ 0} is bounded}
= {(x,y) ∈ C 2 : with bounded f orward/backward orbit}
K := K + ∩ K − , J ± = ∂K ± , J = J + ∩ J − , U + = C 2 − K +
Proposition 1.3. The iterates { f n ,n ≥ 0} are a normal family on the interior of K + ,
and for (x0,y0) ∈ J + , there is no neighborhood U ∋ (x0,y0) on which this family is
normal.
Proof. If p ∈ int(K + ), then the forward orbit cannot enter V + . It can remain in V −
for only finite time, so a neighborhood of p must ultimately be in V. Thus the forward
iterates of a neighborhood of p are ultimately bounded by R in a neighborhood
of p, so the iterates are a normal family in a neighborhood of p. IfU is an open set
that intersects J + ,thenU ∩K + �= /0andU ∩(C 2 − K + ) �= /0. Thus U contains points
where the forward iterates are bounded and points whose orbits tend to ∞. Thus f n
cannot be normal on U. ⊓⊔
Exercise: Show that if p ∈ int(K + ) is fixed, then the eigenvalues of Dfp have
modulus ≤1.
Proposition 1.4. We have K ⊂ V and K + ⊂ V ∪V − .Further,U + = �
n≥0 f −n V + ,
where the union V + ⊂ f −1 V + ⊂··· is increasing.