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.