1. INTRODUCTIO N 7

x2+x

= l = 0-x = - i W 3

x6

+ x +

!

=

o _ x =??

X3

y3

+

xy

2

_2„.

+ 5 =

i 1 _

T J - -

= 0 an d

_ A » « .

— U r X ,

y

— ??

Newton'

\p^ni Un)

in C 2 o r

s Method :

- ( x

p2

y)

FIGURE

1.7. Root s o f equation s

F({- • • ,p_2,P-l}) = ({•• • ,P-

2

,P-l,F(P-l)}).

Similarly, F ' lift s t o a ma p F

f

o n T^ .

We say that F i s hyperbolic o n K i f there exists a continuous splittin g E

u

0 E

s

of the tangen t bundl e o f K suc h tha t th e subbundle s E

u,

E

s

ar e preserve d b y F

;

,

and fo r som e constant s C , c 0, A 1, /x 1,

\{Fn)'{0\

c\

n\£\,

£eE\

n = l,2,.-- .

One o f the mai n ope n question s fo r rationa l map s o n P 1 i s whether th e map s

which ar e hyperboli c o n their Juli a se t ar e dens e i n the rationa l maps .

A weake r propert y tha n hyperbolicit y i s that o f stability. A family A o f map s

{F}FEA

i s stable a t Fo if there exist s a n ope n neighborhoo d U of Fo suc h tha t al l

maps G G U ar e topologicall y conjugat e t o FQ , i.e . ther e exist s a homeomor -

phism h : M - * M s o that G o h = h o F0.

It i s known that th e space of rational maps on P

1

i s stable on an open dense set

([MSS]) whe n restricte d t o th e Juli a set , wit h th e obviou s definition o f topologica l

conjugacy.

See figures 1.7, 1.8 fo r a n outlin e o f thes e notes . A s i s see n fro m th e figures,

we do not discus s work on Local Dynamics around on e point, Fatou , [Fa2] , Hakim ,

[H], Ueda , [U3 ] an d Weickert , [W] .