Discrete Holomorphic Local Dynamical Systems

234 Tien-Cuong Dinh and Nessim Sibony
In dimension 1, Fatou and Julia considered their theory as an investigation to solve some
functional equations. In particular, they found all the commuting pairs of polynomials [F, JU], see
also Ritt [R] and Eremenko [E] for the case of rational maps. Commuting endomorphisms of P k
were studied by the authors in [DS0]. A large family of solutions are Lattès maps. We refer to
Berteloot, Dinh, Dupont, Loeb, Molino [BR, BDM, BL, D4, DP1] for a study of this class of maps,
see also Milnor [MIL] for the dimension 1 case.
We do not consider here bifurcation problems for families of maps and refer to Bassanelli-
Berteloot [BB]andPham[PH] for this subject. Some results will be presented in the next chapter.
In [Z], Zhang considers some links between complex dynamics and arithmetic questions. He is
interested in polarized holomorphic maps on Kähler varieties, i.e. maps which multiply a Kähler
class by an integer. If the Kähler class is integral, the variety can be embedded into a projective
space P k and the maps extend to endomorphisms of P k . So, several results stated above can be
directely applied to that situation. In general, most of the results for endomorphisms in P k can
be easily extended to general polarized maps. In the unpublished preprint [DS14], the authors
considered the situation of smooth compact Kähler manifolds. We recall here the main result.
Let (X,ω) be an arbitrary compact Kähler manifold of dimension k. Let f be a holomorphic
endomorphism of X. We assume that f is open. The spectral radius of f ∗ acting on H p,p (X,C) is
called the dynamical degree of order p of f . It can be computed by the formula
��
dp := lim ( f
n→∞ X
n ) ∗ (ω p ) ∧ ω k−p� 1/n
.
The last degree dk is the topological degree of f , i.e. equal to the number of points in a generic
fiber of f . We also denote it by dt.
Assume that dt > dp for 1 ≤ p ≤ k − 1. Then, there is a maximal proper analytic subset E
of X which is totally invariant by f ,i.e. f −1 (E )= f (E )=E .Ifδa is a Dirac mass at a �∈E ,
then d −n
t ( f n ) ∗ (δa) converge to a probability measure μ, which does not depend on a. Thisisthe
equilibrium measure of f . It satisfies f ∗ (μ)=dt μ and f∗(μ)=μ. IfJ is the Jacobian of f with
respect to ω k then 〈μ,logJ〉≥logdt. The measure μ is K-mixing and hyperbolic with Lyapounov
exponents larger or equal to 1 2 log(dt/dk−1). Moreover, there are sets Pn of repelling periodic
points of order n, on supp(μ) such that the probability measures equidistributed on Pn converge to
μ,asn goes to infinity. If the periodic points of period n are isolated for every n, an estimate on the
norm of ( f n ) ∗ on H p,q (X,C) obtained in [D22], implies that the number of these periodic points is
dn t + o(dn t ). Therefore, periodic points are equidistributed with respect to μ. We can prove without
difficulty that μ is the unique invariant measure of maximal entropy logdt and is moderate. Then,
we can extend the stochastic properties obtained for Pk to this more general setting.
When f is polarized by the cohomology class [ω] of a Kähler form ω, there is a constant λ ≥ 1
such that f ∗ [ω]=λ [ω]. It is not difficult to check that dp = λ p . The above results can be applied
for such a map when λ > 1. In which case, periodic points of a given period are isolated. Note
also that Theorem 1.108 can be extended to this case.
2 Polynomial-like Maps in Higher Dimension
In this section we consider a large family of holomorphic maps in a semi-local
setting: the polynomial-like maps. They can appear as a basic block in the study
of some meromorphic maps on compact manifolds. The main reference for this
section is our article [DS1] wherethedd c -method in dynamics was introduced.
Endomorphisms of P k can be considered as a special case of polynomial-like maps.
However, in general, there is no Green (1,1)-current for such maps. The notion of
dynamical degrees for polynomial-like maps replaces the algebraic degree. Under