Discrete Holomorphic Local Dynamical Systems

Dynamics in Several Complex variables 241
where the points in f −1 (z) are counted with multiplicity. Since f is a ramified
covering, Λ(ϕ) is continuous when ϕ is continuous. If ν is a probability measure
on V, define the measure f ∗ (ν) by
〈 f ∗ (ν),ϕ〉 := 〈ν, f∗(ϕ)〉.
This is a positive measure of mass dt supported on f −1 (supp(ν)). Observe that the
operator ν ↦→ d −1
t f ∗ (ν) is continuous on positive measures, see Exercise A.11.
Theorem 2.11. Let f : U → V be a polynomial-like map as above. Let ν be a prob-
ability measure supported on V which is defined by an L 1 form. Then d −n
t ( f n ) ∗ (ν)
converge to a probability measure μ which does not depend on ν. Forϕ p.s.h. on a
neighbourhood of the filled Julia set K , we have 〈d −n
t ( f n ) ∗ (ν),ϕ〉→〈μ,ϕ〉. The
measure μ is supported on the boundary of K and is totally invariant: d −1
t f ∗ (μ)=
f∗(μ)=μ. Moreover, if Λ is the Perron-Frobenius operator associated to f and ϕ
is a p.s.h. function on a neighbourhood of K ,thenΛn (ϕ) converge to 〈μ,ϕ〉.
Note that in general 〈μ,ϕ〉 may be −∞. If〈μ,ϕ〉 = −∞, the above convergence
means that Λ n (ϕ) tend locally uniformly to −∞; otherwise, the convergence is in
L p
loc for 1 ≤ p < +∞, see Appendix A.2. The above result still holds for measures ν
which have no mass on pluripolar sets. The proof in that case is more delicate. We
have the following lemma.
Lemma 2.12. If ϕ is p.s.h. on a neighbourhood of K ,thenΛ n (ϕ) converge to a
constant cϕ in R ∪{−∞}.
Proof. Observe that Λ n (ϕ) is defined on V for n large enough. It is not difficult
to check that these functions are p.s.h. Indeed, when ϕ is a continu-
ous p.s.h. function, Λ n (ϕ) is a continuous function, see Exercise A.11, and
dd c Λ n (ϕ)=d −n
t ( f n )∗(dd c ϕ) ≥ 0. So, Λ n (ϕ) is p.s.h. The general case is obtained
using an approximation of ϕ by a decreasing sequence of smooth p.s.h. functions.
Consider ψ the upper semi-continuous regularization of limsupΛ n (ϕ). Wededuce
from Proposition A.20 that ψ is a p.s.h. function. We first prove that ψ is constant.
Assume not. By maximum principle, there is a constant δ such that sup U ψ <
δ < sup V ψ. By Hartogs’ lemma A.20,forn large enough, we have Λ n (ϕ) < δ on U.
Since the fibers of f are contained in U, we deduce from the definition of Λ that
sup
V
Λ n+1 (ϕ)=sup
V
Λ(Λ n (ϕ)) ≤ supΛ
U
n (ϕ) < δ.
This implies that ψ ≤ δ which contradicts the choice of δ. Soψ is constant.
Denote by cϕ this constant. If cϕ = −∞, it is clear that Λ n (ϕ) converge to −∞
uniformly on compact sets. Assume that cϕ is finite and Λ ni(ϕ) does not converge to
cϕ for some sequence (ni). By Hartogs’ lemma, we have Λ ni(ϕ) ≤ cϕ − ε for some
constant ε > 0andfori large enough. We deduce as above that Λ n (ϕ) ≤ cϕ − ε for
n ≥ ni. This contradicts the definition of cϕ. ⊓⊔