Discrete Holomorphic Local Dynamical Systems

176 Tien-Cuong Dinh and Nessim Sibony
Applying the above computation to T instead of S, we obtain that
d −1 f ∗ (T )=ωFS + dd c v + dd c� d −1 g ◦ f � = ωFS + dd c g.
Hence, f ∗ (T )=dT. On smooth forms f∗ ◦ f ∗ is equal to d k times the identity; this
holds by continuity for positive closed currents. Therefore,
f∗(T )= f∗( f ∗ (d −1 T )) = d k−1 T.
It remains to prove the estimate in the theorem. Recall that we can write dd c Φ =
R + − R − where R ± are positive measures such that �R ± �≤�Φ�DSH. Wehave
�
�〈d −n ( f n ) ∗ (S) − T,Φ〉 � � = � �〈dd c (v + ···+ d −n+1 v ◦ f n−1 + d −n u ◦ f n − g),Φ〉 � �
= � � 〈v + ···+ d −n+1 v ◦ f n−1 + d −n u ◦ f n − g,dd c Φ〉 � �
= � � � d −n u ◦ f n − ∑ d
i≥n
−i v ◦ f i ,R + − R −���. Since u and v are bounded, the mass estimate for R ± implies that the last integral is
� d −n �Φ�DSH. The result follows. ⊓⊔
Theorem 1.16 gives a convergence result for S quite diffuse (with bounded potentials).
It is like the first main theorem in value distribution theory. The question
that we will address is the convergence for singular S, e.g. hypersurfaces.
Definition 1.17. We call T the Green (1,1)-current and g the Green function of f .
The power T p := T ∧ ...∧ T , p factors, is the Green (p, p)-current of f , and its
support Jp is called the Julia set of order p.
Note that the Green function is defined up to an additive constant and since
T has a continuous quasi-potential, T p is well-defined. Green currents are totally
invariant: we have f ∗ (T p )=d p T p and f∗(T p )=d k−p T p . The Green (k,k)-current
μ := T k is also called the Green measure, the equilibrium measure or the measure
of maximal entropy. We will give in the next paragraphs results which justify the
terminologies. The iterates f n , n ≥ 1, have the same Green currents and Green
function. We have the following result.
Proposition 1.18. The local potentials of the Green current T are γ-Hölder continuous
for every γ such that 0 < γ < min(1,logd/logd∞),whered∞ := lim�Dfn� 1/n
∞ .
In particular, the Hausdorff dimension of T p is strictly larger than 2(k − p) and T p
has no mass on pluripolar sets and on proper analytic sets of Pk .
Since Dfn+m (x) =Dfm ( f n (x)) ◦ Dfn (x), it is not difficult to check that the
sequence �Dfn� 1/n
∞ is decreasing. So, d∞ = inf�Dfn� 1/n
∞ . The last assertion of the
proposition is deduced from Corollary A.32 and Proposition A.33 in Appendix.
The first assertion is equivalent to the Hölder continuity of the Green function g, it
was obtained by Sibony [SI] for one variable polynomials and by Briend [BJ] and
Kosek [KO] in higher dimension.