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] .
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