Discrete Holomorphic Local Dynamical Systems

Dynamics of Rational Surface Automorphisms 59
One advantage of this reduction is that the action of f may be replaced by the shift.
That is, if the f -orbit of the point (x ′ ,y ′ )= f (x,y) corresponds to the sequence
then y ′ n = yn+1.
1.2 Filtration
...,y ′ −2,y ′ −1,y ′ 0,y ′ 1,y ′ 2,...,
WewillfinditusefultolookatHénon maps in terms of their escape to infinity. Let
us define
V = {|x|,|y|≤R}, V + = {|y|≥max{|x|,R}}, V − = {|x|≥max{|y|,R}}.
Thus V is a bi-disk, and the sets V ± are (topologically) the product of a disk and an
annulus. Let us choose R to be sufficiently large that
�
1
|p(t)|−|δt|≥max
2 |td �
|,2|t| , for all |t|≥R. (1)
It follows that:
and thus
(x,y) ∈ V + ⇒ |y1|≥2|y0| = 2|x1| ⇒ (x1,y1) ∈ V +
(x,y) ∈ V + ⇒ |yn|≥2 n |y|≥2 n R (2)
for n = 1,2,3... In fact, it is possible to show that for any ε > 0 we may choose R
sufficiently large that for (x0,y0) ∈ V + we have
(1 − ε)|y0| dn
≤|yn|≤(1 + ε)|y0| dn
. (3)
Theorem 1.1. If (x,y) ∈ V − , then there is a number N such that f N (x,y) ∈ V ∪V + .
Proof. Suppose not. Then we have (xn,yn) ∈ V − for all n ≥ 0. Thus |yn|≤|xn|.Since
xn = yn−1,wehave|x1|≥|x2|≥···≥R,and|y1|≥|y2|≥···, and the both approach
the same limit M. Thus for N sufficiently large, we will have |xN|∼|yN|∼M ≥ R.
Thus by (*), we will have |yN+1|≥2M, which is a contradiction. ⊓⊔
We summarize this discussion in the following Theorem:
Theorem 1.2. 1. fV + ⊂ V + , and for each (x,y) ∈ V + , the forward orbit escapes to
infinity.
2. f (V ∪V + ) ⊂ V ∪V + .