Discrete Holomorphic Local Dynamical Systems

70 Eric Bedford
Definition 1.24. A function ψ is upper-semicontinuous if ψ(z0) ≥ limsup z→z0 ψ(z)
at all points z0. We say that a function ψ is pluri-subharmonic,orsimplypsh on a
domain Ω ⊂ C 2 if ψ is upper-semicontinuous, and if for all α,β ∈ C 2 , the function
ζ ↦→ ψ(αζ + β ) is subharmonic in ζ, wherever it is defined. ψ is said to be pluriharmonic
if both ψ and −ψ are psh. If ψ is pluri-harmonic, then it is locally the
real part of a holomorphic function.
Recall that log + |t| := max{log|t|,0} = log(max{|t|,1}) is continuous on all of C.
Proposition 1.25. G + (x,y) := limn→∞ G + n (x,y) exists uniformly on V + .Further
G + > 0 on V + and is pluri-harmonic there.
Proof. We have yn �= 0onV + ,solog + |yn| = log|yn| is pluriharmonic there. It suffices
to show uniform convergence of the limit. For this we rewrite it as a telescoping
sum
G + N
N−1
(x,y) =G0(x,y)+ (Gn+1(x,y) − Gn(x,y))
Now on V + we have � ���
yn+1
y d n
∑
n=1
N−1 1
= log|y| + ∑
n=1 dn �
1
d log|yn+1|−log|yn|
�
= log|y| +∑ 1
� �
� �
log�
�
dn � � .
�
�
− 1�
�
yn+1
y d n
κ κ
< ≤
|yn| 2 R2 so the series converges uniformly. ⊓⊔
The formula G + ◦ f n = d n G + holds on V + , and this formula may be used to
extend G + to U + = �
n≥0 f −n V + = C 2 − K + . Recall that H1(V + ;Z) ∼ = Z. Since
f (x,y)=(y,y d )+···, we see that the action of f∗ on H1(V + ;Z) to itself is multipli-
cation by d. WemayuseG + to describe the homology of U + .
Theorem 1.26. The map defined by γ ↦→ ω(γ)= 1 πi
ω : H1(U + ;Z) → Z[ 1 d ].
�
γ ∂G+ yields an isomorphism
Proof. Let τ denote the circle inside V + which is given by t ↦→ (0,ρe 2πit ).First
we show that τ is nonzero in H1(U + ;Z). For this we recall that on V + we have
G + (x,y)=log|y| + O(y −1 ). Thus we have
�
∂G
τ
+ =
�
∂ log|y| + O(y
τ
−2 )=
� dy
2y + O(y−2 )=iπ + O(ρ −1 ).
Letting ρ → ∞, we see that the integral is iπ �= 0, so τ is nonzero in H1(U + ).Further,
τ generates H1(V + ;Z).
Now if γ ∈ H1(U + ;Z) is arbitrary, there exists m ≥ 0 such that f m ∗ γ is supported
in V + . Thus f ∗ mγ ∼ kτ for some k ∈ Z. Thus γ ∼ kd −m τ. ⊓⊔