Discrete Holomorphic Local Dynamical Systems

Dynamics of Rational Surface Automorphisms 101
We will describe a number of properties of the maps fa; the details are given in
[BK4], and we do not repeat them here. Each fa maps the line {y = 0} to the point
[0 :1:0] ∈ P 2 ,and f −1
a maps {x = 0} to [1 :0:0]. The line at infinity {z = 0} is
invariant and is mapped according to f [1:w :0]=[1:c−1/w :0].For1≤ s we set
f s [0:1:0]=[1:ws :0]. We note that if ( j,n)=1andifwesetc = ±2cos( jπ/n),
then f n−1 [0:1:0]=[1:0:0], and the restriction of f to {z = 0} has period n.
We define Cn = {±2cos( jπ/n) : ( j,n)=1}, where we choose “+”, “−”, or both,
according to the condition that w1 ···wn−2 = 1. With c ∈ Cn, we obtain the surface
Xa by performing 2k + 1 iterated blowups over each point [1:ws :0],0≤ s ≤ n −1.
The fibers over [1:ws :0] are denoted F j
s ,1≤ j ≤ 2k + 1. From this construction,
it follows that the fibers map as follows:
F 1 0 →···→F 1 s → F 1 s+1 →···→F1 n−1 → F 1 0
F j
j
0 →···→Fs → F j
j
s+1 →···→Fn−1 {y = 0}→F 2k+1
0
→ F 2k+2− j
0
→···→F
2k+1
→···→F n−1 →{x = 0}
2k+2− j 2k+2− j
n−1 → F0 A further observation is that these maps give rational surface automorphisms:
Theorem 2.17. Let 1 ≤ j < n satisfy ( j,n)=1. There is a nonempty set Cn ⊂ R such
that for even k ≥ 2 and for all choices of c ∈ Cn and aℓ ∈ C, the map f in (14) is an
automorphism.
Now we let S denote the subgroup of Pic(Xa) spanned by {z = 0} and the fibers
F j
s ,0≤ s ≤ n − 1, 1 ≤ j ≤ 2k + 1. From [BK4] wealsohave:
Proposition 2.18. The intersection form of Xa, when restricted to S, is negative
definite.
We let T := S⊥ denote the vectors of Pic(Xa) which are orthogonal to S. By
the Proposition, we see that S ∩ T = 0, so we have Pic(Xa) =S ⊕ T. SinceSis invariant, T is also invariant. For each 0 ≤ s ≤ n − 1, we let γs denote the projection
of the class {F 2k+1
s }∈Pic(X ) to T. Thus the γs, 0≤ s ≤ n − 1giveabasisofT .
Following {x = 0} through the various blowups in the construction of X ,wemay
show that {x = 0} = −γ0 + kγ1 + ···kγn−1. And from the mapping of the fibers, we
see that f∗ maps according to
γ0 → γ1 →···→γn−1 →{x = 0} = −γ0 + kγ1 + ···+ kγn−1. (15)
Computing the characteristic polynomial for the transformation (14), we obtain:
Theorem 2.19. The dynamical degree δ( fa) is the largest root of the polynomial
n−1
χn,k(x)=1 − k x ℓ + x n . (16)
∑
ℓ=1