Discrete Holomorphic Local Dynamical Systems

Dynamics in Several Complex variables 249
Remark 2.26. Assume that f is not with large topological degree but that the series
which defines the ramification current R converges. We can then construct inverse
branches as in Proposition 2.21. To obtain the same exponential estimates on the
diameter of Wi, it is enough to assume that dk−1 < dt. In general, we only have that
these diameters tend uniformly to 0 when n goes to infinity. Indeed, we can use the
estimate in Exercise 2.18. The equidistribution of periodic points and of negative
orbits still holds in this case.
Exercise 2.27. Let f be a polynomial-like map with large topological degree. Show
that there is a small perturbation of f , arbitrarily close to f , whose exceptional set
is empty.
2.4 Properties of the Green Measure
Several properties of the equilibrium measure of polynomial-like maps can be
proved using the arguments that we introduced in the case of endomorphisms of P k .
We have the following result for general polynomial-like maps.
Theorem 2.28. Let f : U → V be a polynomial-like map of topological degree
dt > 1. Then its equilibrium measure μ is an invariant measure of maximal entropy
logdt. Moreover, μ is K-mixing.
Proof. By Theorem 2.16, μ has no mass on the critical set of f . Therefore, it is
an invariant measure of constant Jacobian dt in the sense that μ( f (A)) = dt μ(A)
when f is injective on a Borel set A. We deduce from Parry’s theorem 1.116 that
hμ( f ) ≥ logdt. The variational principle and Theorem 2.4 imply that hμ( f )=logdt.
We prove the K-mixing property. As in the case of endomorphisms of Pk ,
it is enough to show for ϕ in L2 (μ) that Λ n (ϕ) →〈μ,ϕ〉 in L2 (μ). Since
Λ : L2 (μ) → L2 (μ) is of norm 1, it is enough to check the convergence for a
dense family of ϕ. So, we only have to consider ϕ smooth. We can also assume
that ϕ is p.s.h. because smooth functions can be written as a difference of p.s.h.
functions. Assume also for simplicity that 〈μ,ϕ〉 = 0.
So, the p.s.h. functions Λ n (ϕ) converge to 0 in L p
loc
(V). By Hartogs’ lemma
A.20, sup U Λ n (ϕ) converge to 0. This and the identity 〈μ,Λ n (ϕ)〉 = 〈μ,ϕ〉 = 0
imply that μ{Λ n (ϕ) < −δ} →0foreveryfixedδ > 0. On the other hand, by
definition of Λ, |Λ n (ϕ)| is bounded by �ϕ�∞ which is a constant independent of n.
Therefore, Λ n (ϕ) → 0inL 2 (μ) and K-mixing follows. ⊓⊔
The following result, due to Saleur [S], generalizes Theorem 1.118.
Theorem 2.29. Let f : U → V be a polynomial-like map with large topological
degree. Then its equilibrium measure is the unique invariant probability measure of
maximal entropy.