CHAPTER 2
Kobayashi Hyperbolicit y
As mentione d i n th e introduction , som e Fato u component s arisin g i n Newton s
method ar e initia l guesse s leadin g t o roots , bu t other s migh t no t be . Hence , a
main goa l i s t o characteriz e th e natur e o f th e Fato u components . On e
of th e basi c tool s i n on e comple x dimensio n i s the Monte l Theorem , tha t is , tha t
the spac e o f holomorphic map s fro m th e uni t dis c into P
1
= Cu{oo } minu s thre e
points i s a norma l family . Onc e thi s too l wa s discovered, on e dimensiona l comple x
dynamics go t a bi g pus h forwar d becaus e peopl e wer e the n abl e t o prov e globa l
results. Before , th e theor y wa s mostl y local . I n severa l variables , th e Kobayash i
metric serve s thi s purpose . I n thi s sectio n w e firs t giv e som e basi c propertie s o f
the Kobayash i metri c an d the n sho w wher e th e Kobayash i hyperbolicit y enter s i n
investigations o f iterations o f maps .
Let M b e a comple x manifold ; w e will mainl y dea l wit h ope n subset s o f C n
and P n . Pic k a poin t p G M an d a tangen t vecto r £ to M a t p . Le t A denot e th e
unit dis c i n the comple x plane .
Pick an y holomorphi c dis c throug h p tangentia l t o £ , i. e a holomorphi c ma p
/ : A M, /(0 ) = p , fl(d/dz) c£. Th e infinitesima l Kobayash i pseudo-metri c
is obtained b y maximizin g th e discs , ds(p,£) : = inf/{4r} . Se e figure 2.1.
As a n example , i f M = C n o r P n , c can b e arbitraril y large , s o ds = 0 i n eithe r
case. Also , if M = A , ds i s the Poincar e metric .
We sa y tha t M i s Kobayashi hyperboli c i f ds CK\€\ fo r som e CK 0, fo r
any compac t subse t K G M.
Brody ([La] ) prove d a ver y interestin g fac t abou t hyperbolicit y o n compac t
complex manifolds .
M
^ X ^ ^
/(A)
FIGURE 2.1. Th e Kobayash i metri c
9
http://dx.doi.org/10.1090/cbms/087/02
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