Discrete Holomorphic Local Dynamical Systems

Dynamics in Several Complex variables 261
We now estimate the integral on K ∩ S. Its absolute value is bounded by
〈ΩK∩S,|logJac( fs)|〉 + 〈ΩK∩S,|logJac( ft)|〉.
We deduce from the estimate 〈ΩK,e λ |logJac( fs)| 〉≤A that volume(K ∩ S) ≤ Ar 2λα .
Therefore, by Cauchy-Schwarz’s inequality, we have
〈ΩK∩S,|logJac( fs)|〉 � volume(K ∩ S) 1/2� ΩK,|logJac( fs)| 2� 1/2
� r λα 〈ΩK,e λ |logJac( fs)| 〉 1/2 � r λα .
The estimate holds for ft instead of fs. Hence, ψ is Hölder continuous. The fact that
B p are moderate follows from Theorem A.34. �
The following result of Pham generalizes Corollary 2.46 and allows to define
other bifurcation currents by considering dd c Lp or their wedge-products [PH].
Theorem 2.50. Let ( fs)s∈Σ be a holomorphic family of polynomial-like maps as
above. Let χ1(s) ≥ ··· ≥ χk(s) be the Lyapounov exponents of the equilibrium
measure μs of fs. Then, for 1 ≤ p ≤ k, the function
Lp(s) := χ1(s)+···+ χp(s)
is p.s.h. on Σ. In particular, Lp is upper semi-continuous.
Proof. Observe that Lp(s) ≥ p
k Lk(s) ≥ p
2k logdt. We identify the tangent space of
V at any point with Ck . So, the differential Dfs(z) of fs at a point z ∈ Us is a
linear self-map on Ck which depends holomorphically on (s,z). It induces a linear
self-map on the exterior product �p k p C that we denote by D fs(z). This map depends
holomorphically on (s,z). In the standard coordinate system on �p Ck ,the
function (s,z) ↦→ log�D p fs(z)� is p.s.h. on UΣ . By Proposition 2.45, the function
ψ1(s) := 〈μs,log�D p fs�〉 is p.s.h. or equal to −∞ on Σ. Define in the same way the
functions ψn(s) := 〈μs,log�D p f n s �〉 associated to the iterate f n s of fs.Wehave
Hence,
D p f n+m
s (z)=D p f m s ( f n s (z)) ◦ D p f n s (z).
�D p f n+m
s (z)�≤�D p f m s ( f n s (z))� �D p f n s (z)�.
We deduce using the invariance of μs that
ψm+n(s) ≤ ψm(s)+ψn(s).
Therefore, the sequence n −1 ψn decreases to infn n −1 ψn. So, the limit is p.s.h. or
equal to −∞. On the other hand, Oseledec’s theorem 1.119 implies that the limit is
equal to Lp(s) which is a positive function. It follows that Lp(s) is p.s.h. ⊓⊔
Consider now the family fs of endomorphisms of algebraic degree d ≥ 2of
P k with s ∈ Hd(P k ). We can lift fs to polynomial-like maps on C k+1 and apply
the above results. The construction of the bifurcation currents B p can be