Discrete Holomorphic Local Dynamical Systems

Dynamics in Several Complex variables 205
that v(z) =v(|z|) for z ∈ C. Hint: use the Laplacian of v. Let f be as in Example
1.68, R a current in C k+1 ,andv a p.s.h. function on C k+1 such that R = dd c v and
v(F(z)) = dv(z), whereF(z) := (z d 0 ,...,zd k ) is a lift of f to Ck+1 . Show that v is
invariant under the action of the unit torus T k+1 in C k+1 . Determine such functions
v. Recall that T is the unit circle in C and T k+1 acts on C k+1 by multiplication.
Exercise 1.78. Define the Desboves map f0 in M4(P 2 ) as
f0[z0 : z1 : z2] :=[z0(z 3 1 − z3 2 ) : z1(z 3 2 − z3 0 ) : z2(z 3 0 − z3 1 )].
Prove that f0 has 12 indeterminacy points. If σ is a permutation of coordinates,
compare f0 ◦ σ and σ ◦ f0. Define
and
Φ λ (z0,z1,z2) := z 3 0 + z3 1 + z3 2 − 3λ z0z1z2, λ ∈ C
L[z0 : z1 : z2] :=[az0 : bz1 : cz2], a,b,c ∈ C.
Show that for Zariski generic L, fL := f0 + Φ λ L is in H4(P 2 ). Show that on the
curve {Φ λ = 0} in P 2 , fL coincides with f0, andthat f0 maps the cubic {Φ λ = 0}
onto itself. 5
Exercise 1.79. Let f be an endomorphism of algebraic degree d ≥ 2ofP k .Show
that there is a finite invariant set E, possibly empty, such that if H is a hypersurface
such that H ∩ E = ∅, thend −n deg(H) −1 ( f n ) ∗ [H] converge to the Green
(1,1)-current T of f .
1.6 Stochastic Properties of the Green Measure
In this paragraph, we are concerned with the stochastic properties of the equilibrium
measure μ associated to an endomorphism f .Ifϕ is an observable, (ϕ ◦ f n )n≥0
can be seen as a sequence of dependent random variables. Since the measure is
invariant, these variables are identically distributed, i.e. the Borel sets {ϕ ◦ f n < t}
have the same μ measure for any fixed constant t. The idea is to show that the
dependence is weak and then to extend classical results in probability theory to our
setting. One of the key point is the spectral study of the Perron-Frobenius operator
Λ := d −k f∗. It allows to prove the exponential decay of correlations for d.s.h. and
Hölder continuous observables, the central limit theorem, the large deviation theorem,
etc. An important point is to use the space of d.s.h. functions as a space of
observables. For the reader’s convenience, we recall few general facts from ergodic
theory and probability theory. We refer to [KH, W] for the general theory.
5 This example was considered in [BO]. It gives maps in H4(P 2 ) which preserves a cubic. The
cubic is singular if λ = 1, non singular if λ �=1. In higher dimension, Beauville proved that a
smooth hypersurface of P k , k ≥ 3, of degree > 1 does not have an endomorphism with dt > 1,
unless the degree is 2, k = 3 and the hypersurface is isomorphic to P 1 × P 1 [BV].