Discrete Holomorphic Local Dynamical Systems

204 Tien-Cuong Dinh and Nessim Sibony
Remark 1.71. The above estimate on cap(Λ n (R)) can be seen as a version of
Lojasiewicz’s inequality for currents. It is quite delicate to obtain. We also have an
explicit estimate on the speed of convergence. Indeed, we have for an appropriate
d ′ < d:
� −pn n ∗ p
dist2 d ( f ) (S),T � := sup |〈d −pn ( f n ) ∗ (S) − T p ,Φ〉| � d ′n d −n .
�Φ� C 2≤1
The theory of interpolation between Banach spaces [T1] implies a similar estimate
for Φ Hölder continuous.
Exercise 1.72. If f :=[zd 0 : ···: zd k ], show that {zp 1 = zq
2 }, for arbitrary p,q,isinvariant
under f . Show that a curve invariant under an endomorphism is an image of P1 or a torus, possibly singular.
Exercise 1.73. Let f be as in Example 1.68. LetS be a (p, p)-current with strictly
positive Lelong number at [1:0:··· :0]. Show that any limit of d −pn ( f n ) ∗ (S) has
a strictly positive Lelong number at [1 :0:··· :0] and deduce that d −pn ( f n ) ∗ (S)
do not converge to T p . Note that for f generic, the multiplicity of the set of critical
values of f N at every point is smaller than δ N .
Exercise 1.74. Let f be as in Theorem 1.70 for p = k and Λ the associated Perron-
Frobenius operator. If ϕ is a C 2 function on P k , show that
�Λ n (ϕ) −〈μ,ϕ〉�∞ ≤ cd ′n d −n
for some constant c > 0. Deduce that Λ n (ϕ) converge uniformly to 〈μ,ϕ〉. Givean
estimate of �Λ n (ϕ) −〈μ,ϕ〉�∞ for ϕ Hölder continuous.
Exercise 1.75. Let f be an endomorphism of algebraic degree d ≥ 2ofP k . Assume
that V is a totally invariant hypersurface, i.e. f −1 (V )=V. LetVi denote the irreducible
components of V and hi minimal homogeneous polynomials such that
Vi = {hi = 0}. Defineh = ∏hi. Show that h ◦ f = ch d where c is a constant. If F
is a lift of f to C k+1 , prove that Jac(F) contains (∏hi) d−1 as a factor. Show that
V is contained in the critical set of f and deduce 3 that degV ≤ k + 1. Assume now
that V is reducible. Find a totally invariant positive closed (1,1)-current of mass 1
which is not the Green current nor the current associated to an analytic set.
Exercise 1.76. Let u be a p.s.h. function in C k , such that for λ ∈ C ∗ , u(λ z) =
log|λ | + u(z). If{u < 0} is bounded in C k , show that dd c u + is a positive closed
current on P k which is extremal in the cone of positive closed (1,1)-currents 4 .
Deduce that the Green (1,1)-current of a polynomial map of C k which extends
holomorphically to P k ,isextremal.
Exercise 1.77. Let v be a subharmonic function on C. Suppose v(e iθ z)=v(z) for
every z ∈ C and for every θ ∈ R such that e iθdn
= 1 for some integer n. Prove
3 It is known that in dimension k = 2, V is a union of at most 3 lines, [CL, FS7, SSU].
4 Unpublished result by Berndtsson-Sibony.