Discrete Holomorphic Local Dynamical Systems

Dynamics in Several Complex variables 229
Theorem 1.118. Let f be an endomorphism of algebraic degree d ≥ 2 of P k .Then
the equilibrium measure μ of f is the unique invariant measure of maximal entropy
k logd.
Proof. We have seen in Corollary 1.117 that hμ( f ) ≤ k logd. Moreover, μ has no
mass on analytic sets, in particular on the critical set of f . Therefore, if f is injective
on a Borel set K, then f∗(1K) =1 f (K) and the total invariance of μ implies that
μ( f (K)) = d k μ(K). So,μ is a measure of constant Jacobian d k . It follows from
Theorem 1.116 that its entropy is at least equal to k logd. So,hμ( f )=k logd.
Assume now that there is another invariant probability measure ν of entropy
k logd. We are looking for a contradiction. Since entropy is an affine function on ν,
we can assume that ν is ergodic. This measure has no mass on proper analytic
sets of P k since otherwise its entropy is at most equal to (k − 1)logd, see Exercise
1.122 below. By Theorem 1.45, ν is not totally invariant, so it is not of constant
Jacobian. Since μ has no mass on critical values of f , there is a simply connected
open set U, not necessarily connected, such that f −1 (U) is a union U1 ∪ ...∪U d k of
disjoint open sets such that f : Ui → U is bi-holomorphic. One can choose U and Ui
such that the Ui do not have the same ν-measure, otherwise μ = ν. So, we can
assume that ν(U1) > d −k . This is possible since two ergodic measures are mutually
singular. Here, it is necessarily to chose U so that μ(P k \U) is small.
Choose an open set W ⋐ U1 such that ν(W ) > σ for some constant σ > d −k .
Let m be a fixed integer and let Y be the set of points x such that for every n ≥ m,
there are at least nσ points f i (x) with 0 ≤ i ≤ n − 1 which belong to W.Ifm is large
enough, Birkhoff’s theorem implies that Y has positive ν-measure. By Brin-Katok’s
theorem 1.114,wehaveht( f ,Y ) ≥ hν( f )=k logd.
Consider the open sets Uα := Uα0 ×···×Uαn−1 in (Pk ) n such that there are at
least nσ indices αi equal to 1. A straighforward computation shows that the number
of such open sets is ≤ d kρn for some constant ρ < 1. Let Vn denote the union of
these Uα. Using the same arguments as in Theorem 1.108, we get that
and
1
k logd ≤ ht( f ,Y ) ≤ lim
n→∞ n logvolume(Γn ∩ Vn)
�
k!volume(Γn ∩ Vn)= ∑ ∑ Π
0≤is≤n−1 α Γn∩Uα
∗
i1 (ωFS) ∧ ...∧ Π ∗
i (ωFS).
k
Fix a constant λ such that ρ < λ < 1. Let I denote the set of multi-indices
i =(i1,...,ik) in {0,...,n − 1} k such that is ≥ nλ for every s. We distinguish two
cases where i �∈I or i ∈ I. In the first case, we have
∑ α
�
Π
Γn∩Uα
∗
i1 (ωFS) ∧ ...∧ Π ∗
i (ωFS) ≤
k
since i1 + ···+ ik ≤ (k − 1 + λ )n.
�
Γn
Π ∗
i1 (ωFS) ∧ ...∧ Π ∗
i k (ωFS)
�
=
Pk( f i1 ∗
) (ωFS) ∧ ...∧ ( f ik ∗
) (ωFS)
= d i1+···+ik (k−1+λ )n
≤ d ,