Discrete Holomorphic Local Dynamical Systems

Dynamics in Several Complex variables 185
Recall that if ν is a positive measure on P k , the pull-back f ∗ (ν) is defined by
the formula 〈 f ∗ (ν),ϕ〉 = 〈ν, f∗(ϕ)〉 for ϕ continuous on P k . Observe that since
f is a ramified covering, f∗(ϕ) is continuous when ϕ is continuous, see Exercise
A.11 in Appendix. So, the above definition makes sense. For ϕ = 1, we obtain that
� f ∗ (ν)� = d k �ν�, since the covering is of degree d k .Ifν is the Dirac mass at a
point a, f ∗ (ν) is the sum of Dirac masses at the points in f −1 (a).
Recall that a measure ν is PB if quasi-p.s.h. are ν-integrable and ν is PC if it
is PB and acts continuously on DSH(P k ) with respect to the weak topology on this
space, see Appendix A.4. We deduce from Proposition 1.34 the following result
where the norm �·�μ on DSH(P k ) is defined by
�ϕ�μ := |〈μ,ϕ〉| + inf�S ± �,
with S ± positive closed such that dd c ϕ = S + − S − . We will see that μ is PB, hence
this norm is equivalent to �·�DSH, see Proposition A.43.
Theorem 1.35. Let f be as above. If ν is a PB probability measure, then
d −kn ( f n ) ∗ (ν) converge to a PC probability measure μ which is independent of
ν and totally invariant under f . Moreover, if ϕ is a d.s.h. function and cϕ := 〈μ,ϕ〉,
then
�Λ n (ϕ) − cϕ�μ ≤ d −n �ϕ�μ and �Λ n (ϕ) − cϕ�DSH ≤ Ad −n �ϕ�DSH,
where A > 0 is a constant independent of ϕ and n. In particular, there is a constant
c > 0 depending on ν such that
|〈d −kn ( f n ) ∗ (ν) − μ,ϕ〉| ≤ cd −n �ϕ�DSH.
Proof. Since ν is PB, d.s.h. functions are ν integrable. It follows that there is a
constant c > 0suchthat|〈ν,ϕ〉| ≤ c�ϕ�DSH. Otherwise, there are d.s.h. functions
ϕn with �ϕn�DSH ≤ 1and〈ν,ϕn〉 ≥2 n , hence the d.s.h. function ∑2 −n ϕn is not
ν-integrable.
It follows from Proposition 1.34 that f ∗ (ν) is PB. So, d −kn ( f n ) ∗ (ν) is PB for
every n.Defineforϕ d.s.h.,
and inductively
c0 := 〈ω k FS,ϕ〉 and ϕ0 := ϕ − c0
cn+1 := 〈ω k FS ,Λ(ϕn)〉 and ϕn+1 := Λ(ϕn) − cn+1 = � Λ(ϕn).
A straighforward computation gives
Therefore,
Λ n (ϕ)=c0 + ···+ cn + ϕn.
〈d −kn ( f n ) ∗ (ν),ϕ〉 = 〈ν,Λ n (ϕ)〉 = c0 + ···+ cn + 〈ν,ϕn〉.