2. KOBAYASH I HYPERBOLICIT Y 11
F
(pfc+1 /-^/c+ 1
_ P k
where n : C fc+1 » P k i s th e canonica l projection . Thi s F ha s th e impor -
tant propert y tha t i t onl y vanishe s a t th e origin . Thi s fac t distinguishe s
holomorphic map s from rationa l maps . Conversely , i t i s clea r tha t an y ho -
mogeneous F whic h vanishe s onl y a t th e origi n give s rise to a holomorphi c ma p o n
P
k
.
In th e investigatio n o f dynamics, on e alway s need s a way to coun t suc h thing s
as the numbe r o f fixed points . Fo r this , the Bezou t Theore m i s a usefu l tool .
THEOREM 2.4 . Suppose that Pi{z
u
...2 n+i), •,
Pn(zi,..., 2 n+i) o,re homogeneous holomorphic polynomials of degree
d\,..., d
n
respectively. Their zero sets determine complex hypersurfaces
Xi,.. . ,X
n
in P
n
. IfXiC\X2r\' - -nXn is finite, then the number of points, counted
with multiplicity, is d\d
2
. . .d n.
As a consequence , w e obtain :
THEOREM 2.5 . Let F : P k - P k be a holomorphic map of degree d at least
two. Then F has (d fc+1 l)/( d 1) fixed points counted with multiplicity.
We sho w nex t tha t hyperbolicit y occur s generically . Le t Ha denot e th e holo -
morphic map s o n P
2
o f degree d.
THEOREM 2.6 . [FS3 ] Fix an integer d2. Then there exists a dense open
set H f C Hd with the following properties. If f GW ' and C denotes it's critical set.
Then
i) No point of P
2
lies in f
n(C)
for three different n , 0 n 4.
is Kobayashi hyperbolic.
The proo f o f th e theore m use s analyticit y an d a calculatio n nea r a simpl e ex -
plicit map . Th e calculatio n i s used t o prov e th e followin g Lemma .
LEMMA 2.7 . Let f = [z
d
:
wd
:
td].
There exists an arbitrarily small pertur-
bation g of f such that the five (reducible) varieties g
n(C),
n 0, ,4 have no
triple intersections.
The theore m follow s fro m th e Lemm a usin g Greene' s Theore m 2.3 .
The nex t resul t show s tha t periodi c orbit s o f holomorphic sel f map s o f P k ar e
non attractin g i n th e complemen t o f th e critica l orbit s unde r th e hypothesi s o f
Kobayashi hyperbolicity . W e sa y tha t a n ope n se t Q C P 2 i s hyperbolicall y em -
bedded i f Ct is Kobayashi hyperboli c an d i f in addition th e Kobayash i pseudometri c
of ft i s bounded belo w b y a constan t multipl e o f a smooth metri c o n P 2.
THEOREM 2.8 . Let f : P k - P k be a holomorphic map with critical set C.
LetC be the closure ofU*L 0p(C). Assume that P
k\C
is Kobayashi hyperbolic and
hyperbolically embedded. If p is a periodic point for / , f £ (p) = p, with eigenvalues
Ai, A2, , Afc and p $. C, then | A; | 1, 1 i k. Also
I Ai Afc ] 1 or f is an automorphism of the component of P \C containing p.
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