Discrete Holomorphic Local Dynamical Systems

242 Tien-Cuong Dinh and Nessim Sibony
Proof of Theorem 2.11. We can replace ν with d −1
t f ∗ (ν) in order to assume that
ν is supported on U. The measure ν can be written as a finite or countable sum of
bounded positive forms, we can assume that ν is a bounded form.
Consider a smooth p.s.h. function ϕ on a neighbourhood of K . It is clear that
Λ n (ϕ) are uniformly bounded for n large enough. Therefore, the constant cϕ is finite.
We deduce from Lemma 2.12 that Λ n (ϕ) converge in L 1 loc (V) to cϕ. It follows
that
〈d −n
t ( f n ) ∗ (ν),ϕ〉 = 〈ν,Λ n (ϕ)〉→cϕ.
Let φ be a general smooth function on V. We can always write φ as a differ-
ence of p.s.h. functions on U. Therefore, 〈d −n
t ( f n ) ∗ (ν),φ〉 converge. It follows
that the sequence of probability measures d −n
t ( f n ) ∗ (ν) converges to some proba-
bility measure μ. Sincecϕ does not depend on ν, the measure μ does not depend
on ν. Consider a measure ν supported on U \ K . So, the limit μ of d −n
t ( f n ) ∗ (ν)
is supported on ∂K . The total invariance is a direct consequence of the above
convergence.
For the rest of the theorem, assume that ϕ is a general p.s.h. function on a
neighbourhood of K . Since limsupΛ n (ϕ) ≤ cϕ, Fatou’s lemma implies that
〈μ,ϕ〉 = 〈d −n
t ( f n ) ∗ (μ),ϕ〉 = 〈μ,Λ n (ϕ)〉≤〈μ,limsupΛ
n→∞
n (ϕ)〉 = cϕ.
On the other hand, for ν smooth on U,wehavesinceϕ is upper semi-continuous
cϕ = lim 〈ν,Λ
n→∞ n (ϕ)〉 = lim 〈d
n→∞ −n
t ( f n ) ∗ (ν),ϕ〉≤〈lim d
n→∞ −n
t ( f n ) ∗ (ν),ϕ〉 = 〈μ,ϕ〉.
Therefore, cϕ = 〈μ,ϕ〉. Hence, Λ n (ϕ) converge to 〈μ,ϕ〉 for an arbitrary p.s.h.
function ϕ. This also implies that 〈d −n
t ( f n ) ∗ (ν),ϕ〉→〈μ,ϕ〉. �
The measure μ is called the equilibrium measure of f . We deduce from the
above arguments the following result.
Proposition 2.13. Let ν be a totally invariant probability measure. Then ν is supported
on K . Moreover, 〈ν,ϕ〉 ≤〈μ,ϕ〉 for every function ϕ which is p.s.h. in a
neighbourhood of K and 〈ν,ϕ〉 = 〈μ,ϕ〉 if ϕ is pluriharmonic in a neighbourhood
of K .
Proof. Since ν = d −n
t ( f n ) ∗ (ν), it is supported on f −n (V) for every n ≥ 0. So, ν is
supported on K . We know that limsupΛ n (ϕ) ≤ cϕ, then Fatou’s lemma implies
that
〈ν,ϕ〉 = lim 〈d
n→∞ −n
t ( f n ) ∗ (ν),ϕ〉 = lim 〈ν,Λ
n→∞ n (ϕ)〉≤cϕ.
When ϕ is pluriharmonic, the inequality holds for −ϕ; we then deduce that
〈ν,ϕ〉≥cϕ. The proposition follows. ⊓⊔
Corollary 2.14. Let X1,X2 be two analytic subsets of V such that f −1 (X1) ⊂ X1 and
f −1 (X2) ⊂ X2. ThenX1 ∩ X2 �=∅. In particular, f admits at most one point a such
that f −1 (a)={a}.