Discrete Holomorphic Local Dynamical Systems

94 Eric Bedford
which corresponds to the matrix
⎛ ⎞
2 0 0 1
⎜
⎜−1
10−1 ⎟
⎝−2
01−2⎠
−3 00−2
As expected, the square of this matrix is the identity.
Exercise: Define a new ordered basis 〈H,E1,E2,E3〉 for Pic(X3) by setting E1 =
F1 + F2 + F3, E2 = F2 + F3, E3 = F3. Show that with respect to this basis, L∗ is
represented by the matrix (6).
Exercise: Perform an analysis on the maps L j(x,y)=(x,−y + x j ) for j = 1,2,3,...
2.5 Linear Fractional Recurrences
Here we consider the possibility of producing a space X and an automorphism
f ∈ Aut(X) by the following procedure. That is, we consider a birational map f
of projective space P 2 , and we would like to perform some blowups π : X → P 2
such that the induced map fX will be an automorphism. Now if the original map
f fails to be an automorphism, then there will be an exceptional curve C which is
blown down to a point p1. We can try to fix this by blowing up the point p1 to produce
a space π1 : X1 → P2 . If we are lucky, then the induced map fX1 will not map
C to a point but will map it to the whole curve P1. However, this is only a temporary
success if p1 is not a point of indeterminacy, because fX1 will map the fiber P1 to
the point p2 := fp1. Thus we see that the blowing-up procedure cannot work unless
we have the property that any exceptional curve C in P2 is eventually mapped to a
point of indeterminacy.
We spend this section looking at the example of linear fractional recurrences:
� �
y + a
f (x,y)= y, . (9)
x + b
See [BK3] for further discussion of this family. In projective coordinates this map
takes the form
f [x0 : x1 : x2]=[x0(bx0 + x1) : x2(bx0 + x1) : x0(ax0 + x2)].
We explain that to obtain the homogeneous representation, we write
and so
�
f = 1:y :
[x0 : x1 : x2]=[1:x1/x0 : x2/x0]=[1:x : y]
� �
y + a
= 1:
x + b
x2
:
x0
x2
x0
x1
x0
�
+ a
+ b
�
= 1: x2
:
x0
x2
�
+ ax0
.
x1 + bx0
(8)