Discrete Holomorphic Local Dynamical Systems

100 Eric Bedford
corresponds to the Cremona inversion. That is, if we write the action of Re0 on the
subspace with ordered basis 〈e0,e1,e2,e3〉, then the restriction of Re0 is represented
by the matrix:
⎛
⎞
2 1 1 1
⎜
⎜−1
0 −1 −1 ⎟
⎝−1
−1 0 −1⎠
−1 −1 −1 0
In §2.3, we saw that after the involutions J, K, andLhave been regularized to become
automorphisms, the actions of J∗ , K∗ and L∗ can all be written in the form of
this matrix.
Let WN denote the group generated by the reflections Rα j ,0≤ j ≤ N −1. ThusWN
is generated by the permutations of the e j’s, 1 ≤ j ≤ N, together with the Cremona
inversion. The following result is classical (see [Do, Theorem 5.2]):
Theorem 2.16. Let X be a rational surface, and let e0,...,eN be a geometric basis
for H 2 (X). If f∈ Aut(X),then f∗ ∈ WN.
Problem: It remains unknown which elements of WN can arise from rational surface
automorphisms.
Let us return to the linear fractional automorphisms and see how f ∗ is related to
WN. Since the space Z was constructed by simple blowups, we have a geometric
basis for Pic(Z) by letting e0 be the class of a general line and then setting e1 = P,
e2 = E2, e3 = E1, e4 = Q, ..., e j = f j−4Q,4≤ j ≤ N − 4. We see that cyclic permutation
σ =(123...N) is equal to the composition Rα1 ◦ Rα2 ◦···◦RαN−1 ,andthat
f∗ itself (cf. equation (11-13)) is equal to Rα0 ◦ σ = Rα0 ◦···RαN−1 is a product of
the reflections that generate WN.
2.8 More Automorphisms
We give some more examples of rational surface automorphisms of positive entropy.
Our goal here is to give families of maps which illustrate that such automorphisms
can occur in continuous families of arbitrarily high dimension. Namely, for each
even k we give a family { fa : a ∈ C k 2 −1 } of birational maps of P 2 , and for each map
fa there is a rational surface π : Xa → P 2 ,and fa lifts to an automorphism of Xa.We
note that the complex structures of the surfaces Xa are allowed to vary with a, but
the smooth structures are locally constant. The smooth dynamical systems ( fa,Xa),
however, may be shown to vary nontrivially with a. The maps we discuss are
⎛
⎜
f (x,y)= ⎝y,−x + cy +
aℓ 1
+
yℓ yk ⎞
⎟
⎠ (14)
k−2
∑
ℓ=2
ℓ even
where the sum is taken only over even values of ℓ.