Discrete Holomorphic Local Dynamical Systems

258 Tien-Cuong Dinh and Nessim Sibony
currents. By definition of R,sinceπ∗ is continuous on currents supported on Σ ×K,
ψni converge to ψ in L1 loc (Σ). On the other hand, 〈Θni ,π,s〉 converge to μs. So,the
function ψ ′ (s) := limψni (s)=〈μs,ϕ〉 is equal to ψ(s) almost everywhere. Since ψni
and ψ are p.s.h., the Hartogs’ lemma implies that ψ ′ ≤ ψ. We show the inequality
ψ ′ (s) ≥ ψ(s).
The function ψ is p.s.h., hence it is strongly upper semi-continuous. Therefore,
there is a sequence (sn) converging to s such that ψ ′ (sn) =ψ(sn) and ψ(sn) converge
to ψ(s). Up to extracting a subsequence, we can assume that μsn converge
to some probability measure μ ′ s. By continuity, μ ′ s is totally invariant under fs. We
deduce from Proposition 2.13 that 〈μ ′ s ,ϕ(s,·)〉 ≤〈μs,ϕ(s,·)〉. The first integral
is equal to ψ(s), the second one is equal to ψ ′ (s). Therefore, ψ(s) ≤ ψ ′ (s). The
identity 〈R,π,s〉 = μs follows.
The second assertion in the proposition is also a consequence of the above
arguments. This is clear when ϕ is smooth. The general case is deduced using an
approximation of ϕ by a decreasing sequence of smooth p.s.h. functions. ⊓⊔
Let Jac(F) denote the Jacobian of F with respect to the standard volume form
on Σ × C k . Its restriction to π −1 (s) is the Jacobian Jac( fs) of fs. SinceJac(F) is a
p.s.h. function, we can apply the previous proposition and deduce that the function
Lk(s) := 1 2 〈μs,logJac( fs)〉 is p.s.h. on Σ. Indeed, by Theorem 2.16, this function is
bounded from below by 1 2 logdt, hence it is not equal to −∞. By Oseledec’s theorem
1.119, Lk(s) is the sum of the Lyapounov exponents of fs. We deduce the following
result of [DS1].
Corollary 2.46. Let ( fs)s∈Σ be as above. Then, the sum of the Lyapounov exponents
associated to the equilibrium measure μs of fs is a p.s.h. function on s. In particular,
it is upper semi-continuous.
Pham defined in [PH] the bifurcation (p, p)-currents by Bp :=(ddcLk) p for
1 ≤ p ≤ dimΣ. The wedge-product is well-defined since Lk is locally bounded: it is
bounded from below by 1
2 logdt. Very likely, these currents play a crucial role in the
study of bifurcation as we see in the following observation. Assume that the critical
set of fs0 does not intersect the filled Julia set Ks0 for some s0 ∈ Σ. Since the filled
Julia sets Ks vary upper semi-continuously in the Hausdorff metric, logJac(F) is
pluriharmonic near {s0}×Ks0 . It follows that Lk is pluriharmonic and Bp = 0ina
neighbourhood of s0 8 . On the other hand, using Kobayashi metric, it is easy to show
that f is uniformly hyperbolic on Ks for s close to s0. It follows that Ks = Js and
s ↦→ Js is continuous near s0, see[FS6].
Note that Lk is equal in the sense of currents to π∗(logJac(F) ∧ R), whereRis the current in Proposition 2.45. Therefore, B can be obtained using the formula
B = dd c π∗(logJac(F) ∧ R)=π∗([CF] ∧ R),
since dd c log|Jac(F)| =[CF ], the current of integration on the critical set CF of F.
We also have the following property of the function Lk.
8 This observation was made by the second author for the family z 2 + c, with c ∈ C. Heshowed
that the bifurcation measure is the harmonic measure associated to the Mandelbrot set [SI].