Discrete Holomorphic Local Dynamical Systems

240 Tien-Cuong Dinh and Nessim Sibony
Proof. The first assertion is clear, using the characterization of polynomial-maps
by their graphs. We prove the second one. Fix a constant δ with d ∗ p < δ < dt and
an open set W such that U ⋐ W ⋐ V. Fix an integer N large enough such that
�( f N )∗(S)�W ≤ δ N for any positive closed (k − p,k − p)-current S of mass 1 on U.
If g is close enough to f ,wehaveg −N (U) ⋐ f −N (W ) and
with ε > 0 a small constant. We have
�(g N �
)∗(S)�U =
≤
�(g N ) ∗ (ω p ) − ( f N ) ∗ (ω p )� L ∞ (g −N (U)) ≤ ε
g−N S ∧ (g
(U)
N ) ∗ (ω p )
�
f −N S ∧ ( f
(W )
N ) ∗ (ω p �
)+
g−N (U)
≤�( f N )∗(S)�W + ε ≤ δ N + ε < d N t .
S ∧ � (g N ) ∗ (ω p ) − ( f N ) ∗ (ω p ) �
Therefore, the dynamical ∗-degree d ∗ p(g N ) of g N is strictly smaller than d N t . Lemma
2.6 implies that d ∗ p(g) < dt. ⊓⊔
Remark 2.8. The proof gives that g ↦→ d ∗ p(g) is upper semi-continuous on g.
Consider a simple example. Let f : C 2 → C 2 be the polynomial map f (z1,z2)=
(2z1,z 2 2 ). The restriction of f to V := {|z1| < 2,|z2| < 2} is polynomial-like and
using the current S =[z1 = 0], it is not difficult to check that d1 = d∗ 1 = dt = 2. The
example shows that in general one may have d∗ k−1 = dt.
Exercise 2.9. Let f : C 2 → C 2 be the polynomial map defined by f (z1,z2) :=
(3z2,z 2 1 + z2). Show that the hyperplane at infinity is attracting. Compute the topological
degree of f . Compute the topological degree of the map in Example 2.1.
Exercise 2.10. Let f be a polynomial map on C k of algebraic degree d ≥ 2, which
extends to a holomorphic endomorphism of P k .LetV be a ball large enough centered
at 0 and U := f −1 (V). Prove that the polynomial-like map f : U → V satisfies
d ∗ p = d p and dt = d k . Hint: use the Green function and Green currents.
2.2 Construction of the Green Measure
In this paragraph, we introduce the first version of the dd c -method. It allows to construct
for a polynomial-like map f a canonical measure which is totally invariant. As
we have seen in the case of endomorphisms of P k , the method gives good estimates
and allows to obtain precise stochastic properties. Here, we will see that it applies
under a very weak hypothesis. The construction of the measure does not require any
hypothesis on the dynamical degrees and give useful convergence results.
Consider a polynomial-like map f : U → V of topological degree dt > 1as
above. Define the Perron-Frobenius operator Λ acting on test functions ϕ by
Λ(ϕ)(z) := d −1
t f∗(ϕ)(z) := d −1
t ∑
w∈ f −1 (z)
ϕ(w),