Discrete Holomorphic Local Dynamical Systems

Dynamics in Several Complex variables 245
Defineforagivenδ > 0 and for any integer j,
Wj := � exp(− jδ) < J ≤ exp(−( j − 1)δ) �
and
τ j(x) := 1
N #�q, f q (x) ∈ Wj and 0 ≤ q ≤ N − 1 � .
We have ∑τj = 1and
νN(U α �
N ) ≤
Uα �
1
exp
dt N
�
�
∑−( j − 1)δτj
�N dν.
Using the inequality gm ≤ logM/J,wehaveonU α N
Therefore,
α < sN < ∑τj(logM + jδ)=∑ jδτj + logM.
−∑( j − 1)δτj < −α +(logM + δ).
We deduce from the above estimate on νN(U α N ) that
νN(U α �
N ) ≤
Uα N
� exp(−α)M exp(δ)
dt
� N
dν.
So, for every α > log(M/dt)+δ,wehaveνN(Uα N ) → 0.
Choosing δ arbitrarily small, we deduce from the above discussion that
lim
N→∞ 〈μN,gm〉≤log(M/dt).
Since gm is continuous and μN converge to μ,wehave〈μ,gm〉≤log(M/dt). Letting
m go to infinity gives 〈μ,logJ〉≥logdt. ⊓⊔
Exercise 2.17. Let ϕ be a strictly p.s.h. function on a neighbourhood of K ,i.e.a
p.s.h. function satisfying ddcϕ ≥ cddc�z�2 , with c > 0, in a neighbourhood of K .
Let ν be a probability measure such that 〈d −n
t ( f n ) ∗ (ν),ϕ〉 converge to 〈μ,ϕ〉 and
that 〈μ,ϕ〉 is finite. Show that d−n t ( f n ) ∗ (ν) converge to μ.
Exercise 2.18. Using the test function ϕ = �z� 2 , show that
when n goes to infinity.
�
f −n ( f
(U)
n ) ∗ (ω k−1 ) ∧ ω = o(d n t ),
Exercise 2.19. Let Y denote the set of critical values of f . Show that the volume of
f n (Y ) in U satisfies volume( f n (Y) ∩U)=o(dn t ) when n goes to infinity.