Discrete Holomorphic Local Dynamical Systems

Dynamics in Several Complex variables 253
We deduce that for any neighbourhood W of K , there is a constant c > 0 such that
|〈μ,ϕ〉| ≤ c�ϕ� L 1 (W) for ϕ p.s.h. on W.
We now show that there is a constant 0 < λ < 1 such that sup V Λ(ϕ) ≤ λ if
ϕ is a p.s.h. function on V, bounded from above by 1, such that 〈μ,ϕ〉 = 0. This
property implies the last assertion in the proposition. Assume that the property is
not satisfied. Then, there are functions ϕn such that sup V ϕn = 1, 〈μ,ϕn〉 = 0and
sup V Λ(ϕn) ≥ 1 − 1/n 2 . By definition of Λ,wehave
sup
U
ϕn ≥ supΛ(ϕn)
≥ 1 − 1/n
V
2 .
The submean value inequality for p.s.h. functions implies that ϕn converge to 1 in
L1 loc (V ). On the other hand, we have
1 = |〈μ,ϕn − 1〉| ≤ c�ϕn − 1� L 1 (W ) .
This is a contradiction.
Finally, consider a positive closed (1,1)-current S of mass 1 on W. By
Proposition A.16, there is a p.s.h. function ϕ on a neighbourhood of U with
bounded L 1 norm such that dd c ϕ = S. The submean inequality for p.s.h functions
implies that ϕ is bounded from above by a constant independent of S. We can
after subtracting from ϕ a constant, assume that 〈μ,ϕ〉 = 0. The p.s.h. functions
λ −n Λ n (ϕ) are bounded above and satisfy 〈μ,λ −n Λ n (ϕ)〉 = 〈μ,ϕ〉 = 0. Hence,
they belong to a compact subset of PSH(U) which is independent of S. IfW ′ is
a neighbourhood of K such that W ′ ⋐ U, the mass of dd c� λ −n Λ n (ϕ) � on W ′ is
bounded uniformly on n and on S. Therefore,
�dd c ( f n )∗(S)� W ′ ≤ cλ n d n t
for some constant c > 0 independent of n and of S. It follows that d ∗ k−1 ≤ λ dt. This
implies property 1). ⊓⊔
Theorem 2.34. Let f : U → V be a polynomial-like map with large topological
degree. Let P be a bounded family of p.s.h. functions on V. Let K be a compact
subset of V such that f −1 (K) is contained in the interior of K. Then, the equilibrium
measure μ of f is Hölder continuous on P with respect to dist L 1 (K) . In particular,
this measure is moderate.
Let DSH(V ) denote the space of d.s.h. functions on V, i.e. functions which are
differences of p.s.h. functions. They are in particular in L p
loc (V) for every 1 ≤ p <
+∞. Consider on DSH(V) the following topology: a sequence (ϕn) converges to ϕ
in DSH(V) if ϕn converge weakly to ϕ and if we can write ϕn = ϕ + n − ϕ− n with ϕ± n
in a compact subset of PSH(V), independent of n. We deduce from the compactness
of bounded sets of p.s.h. functions that ϕn → ϕ in all L p
loc (V) with 1 ≤ p < +∞.
Since μ is PC, it extends by linearity to a continuous functional on DSH(V ).
Proof of Theorem 2.34. Let P be a compact family of p.s.h. functions on V .We
show that μ is Hölder continuous on P with respect to dist L 1 (K) . We claim that Λ