Discrete Holomorphic Local Dynamical Systems

Dynamics of Rational Surface Automorphisms 97
be the associated birational map. The point p∗ =(−b,−a) is the unique point of
indeterminacy for fY ,andΣγ is the exceptional locus. Setting q =(−a,0), letus
define the following subset of parameter space:
{Vn = {(a,b) ∈ C 2 a,b
: f j
Y q �= p∗,0 ≤ j < n, f n Y q = p∗}.
Theorem 2.11. fY is a rational surface automorphism if and only if (a,b) ∈ Vn for
some n.
Proof. If (a,b) /∈ Vn for any n, then by Theorem 2.10 we have that λ ( fY ) is the
largest root of t 3 −t −1. This is not a Salem number, so fa,b is not an automorphism.
Conversely, let us suppose that (a,b) ∈ Vn for some n. LetZ denote the manifold
obtained by blowing up the n + 1 points in the orbit q, fY q,..., f n Y q = p∗. It follows
that the induced map fZ is an automorphism. ⊓⊔
The action of fZ ∗ on cohomology is given by:
P2 → Σ0 = H − P1 − P2 → P1 → ΣB = H − P1 − Q (11)
which is like what we have seen already from the action of fY , except that now the
point q ∈ ΣB has been blown up, so we must subtract Q to obtain the representation
of ΣB = {y = 0} as an element of Pic(Z). The behavior of the new blowup fibers
Q → fQ→···→ f n Q = P∗ → ΣC = p2q = H − P2 − Q (12)
Finally, since a generic line L intersects all three lines Σ0, Σβ ,andΣγwith multiplicity
one, the image fL will be a quadric passing through e2, e1, andq. Thus we
have
H → 2H − P1 − P2 − Q. (13)
Theorem 2.12. If (a,b) ∈ Vn, then the characteristic polynomial of fZ∗ is
χn = x n+1 (x 3 − x − 1)+x 3 + x 2 − 1.
Thus δ( f )=λn, which is the largest root of χn, and λn > 1 if n ≥ 7.
We note that λn increases to the number λ ∼ 1.324... from the previous section.
An interesting consideration is to ask whether fa,b has an invariant curve. The maps
which posses invariant curves have a number of interesting properties; we describe
one of them below.
There are rational functions ϕ j : C → C 2 a,b such that if (a,b) =ϕ j(t) for some
t ∈ C,thenfa,b has an invariant curve S with j irreducible components. The curve S
is a singular cubic. For instance, the first of these functions is
�
t − t3 − t4 1 − t5
ϕ1(t)=
,
1 + 2t + t2 t2 + t3 �
.