Discrete Holomorphic Local Dynamical Systems

Dynamics in Several Complex variables 199
then with image contained in one of the B j. In the same way, we show that for
n large enough, each B j admits (1 − 2ε2 )μ(B)d k(n−N) inverse branches for f n−N
with images in B. Therefore, B admits at least (1 − 2ε2 ) 2μ(B)d kn inverse branches
gi : B → Ui for f n with image Ui ⋐ B. Observe that every holomorphic map
g : B → U ⋐ B contracts the Kobayashi metric and then admits an attractive fixed
point z. Moreover, gl converges uniformly to z and ∩lgl (B)={z}. Therefore, each
gi admits a fixed attractive point ai. This point is fixed and repelling for f n .They
are different since the Ui are disjoint. Finally, since μ is totally invariant, its support
is also totally invariant under f . Hence, ai, which is equal to ∩lgl i (supp(μ) ∩ B), is
necessarily in supp(μ). We deduce that
#Pn ∩ B ≥ (1 − 2ε 2 ) 2 μ(B)d kn ≥ (1 − ε)d kn μ(B).
This completes the proof. ⊓⊔
Note that the periodic points ai, constructed above, satisfy �(Df n ) −1 (ai)� �
d −n/2 . Note also that in the previous theorem, one can replace Pn with the set of all
periodic points counting with multiplicity or not.
Exercise 1.58. Let f be an endomorphism of algebraic degree d ≥ 2ofP k as above.
Let K be a compact set such that f −1 (K) ⊂ K. Show that either K contains Jk,the
Julia set of order k, orK is contained in the exceptional set E .Provethata �∈E if
and only if ∪ f −n (a) is Zariski dense.
Exercise 1.59. Let f be as above and U an open set which intersects the Julia set
Jk. Show that ∪n≥0 f n (U) is a Zariski dense open set of P k . Deduce that f is topologically
transitive on Jk, that is, for any given non-empty open sets V,W on Jk,
there is an integer n ≥ 0 such that f n (V)∩W �=∅.IfE = ∅, show that f n (U)=P k
for n large enough. If E ∩ Jk = ∅, show that f n (V )=Jk for n large enough.
Exercise 1.60. Assume that p is a repelling fixed point in Jk. Ifg is another endomorphism
close enough to f in Hd(P k ) such that g(p)=p, show that p belongs
also to the Julia set of order k of g. Hint: use that g ↦→ μg is continuous.
Exercise 1.61. Using Example 1.10, construct a map f in Hd(P k ), d ≥ 2, such that
for n large enough, every fiber of f n contains more than d (k−1/2)n points. Deduce
that there is Zariski dense open set in Hd(P k ) such that if f is in that Zariski open
set, its exceptional set is empty.
Exercise 1.62. Let ε be a fixed constant such that 0 < ε < 1. Let P ′ n the set of
repelling periodic points a of prime period n on the support of μ such that all
the eigenvalues of Df n at a are of modulus ≥ (d − ε) n/2 . Show that d −kn ∑a∈P ′ n δa
converges to μ.
Exercise 1.63. Let g be as in Theorem 1.56. Show that repelling periodic points
on supp(μX ) are equidistributed with respect to μX. In particular, they are Zariski
dense.