Discrete Holomorphic Local Dynamical Systems

262 Tien-Cuong Dinh and Nessim Sibony
obtained directly using the Green measures of fs. This was done by Bassanelli-
Berteloot in [BB]. They studied some properties of the bifurcation currents and
obtained nice formulas for that currents in terms of the Green functions. We
also refer to DeMarco, Dujardin-Favre, McMullen, Milnor, Sibony and Silverman
[DM, DM1, DF, MM, MI1, SI, SJ] for results in dimension one.
Exercise 2.51. If f is an endomorphism in Hd(P k ), denote by Lk( f ) the sum of the
Lyapounov exponents of the equilibrium measure. Show that f ↦→ Lk( f ) is locally
Hölder continuous on Hd(P k ). Deduce that the bifurcation currents are moderate.
Hint: use that the lift of f to C k+1 has always a Lyapounov exponent equal to logd.
Exercise 2.52. Find a family ( fs)s∈Σ such that Js does not vary continuously.
Exercise 2.53. A family (Xs)s∈Σ of compact subsets in V is lower semi-continuous
at s0 if for every ε > 0, Xs0 is contained in the ε-neighbourhood of Xs when s is close
enough to s0. If(νs)s∈Σ is a continuous family of probability measures on V, show
that s ↦→ supp(νs) is lower semi-continuous. If ( fs)s∈Σ is a holomorphic family of
polynomial-like maps, deduce that s ↦→ Js is lower semi-continuous. Show that if
Js0 = Ks0 ,thens ↦→ Js is continuous at s0 for the Hausdorff metric.
Exercise 2.54. Assume that fs0 is of large topological degree. Let δ > 0 be a constant
small enough. Using the continuity of s ↦→ μs, show that if ps0 is a repelling
fixed point in Js0 for fs0 , there are repelling fixed points ps in Js for fs, with
|s − s0| < δ, such that s ↦→ ps is holomorphic. Suppose s ↦→ Js is continuous with
respect to the Hausdorff metric. Construct a positive closed current R supported
on ∪ |s−s0|