1. INTRODUCTIO N 3
A-\ ^ ? 1
if we assume tha t e is small. W e se e tha t
F(z, w) = (z, w) - A~
XR
= (z
2
+ ew, ez)
Such map s ar e usuall y calle d comple x Heno n maps . Newton' s ma p F ha s
a fixed poin t a t th e root , (0,0) . Thi s poin t i s a n attractin g fixed poin t o f F. Thi s
means tha t th e eigenvalue s o f the 2 x 2 matri x F
f(0,0)
ar e smaller tha n one , which
implies tha t al l point s p i i n a smal l neighborhoo d o f (0,0 ) ar e attracte d t o (0,0) ,
Fn(Pi) -^ (0,0) . Th e larges t suc h se t i s a n ope n subse t o f C 2 an d i s calle d th e
basin o f attractio n o f (0,0) . Thi s se t consist s o f al l initia l guesse s givin g tha t
root.
In thi s cas e th e basi n o f attractio n i s quit e large , holomorphicall y equivalen t
to th e whol e spac e C
2,
nevertheles s i t i s also quit e small . S o both th e se t o f initia l
guesses givin g th e roo t an d initia l guesse s no t givin g th e roo t ar e quit e large . A
more precis e descriptio n o f thes e set s wa s onl y obtaine d i n th e las t fe w years . Se e
figure 1.2. Nevertheless , this kind of sets was studied alread y in the 20' s and 30's by
Fatou ( [Fal] ) an d Bieberbac h ([Bi]) . The y ar e calle d Fatou-Bieberbac h domains .
See Rosa y an d Rudin , [RR] , fo r man y result s o n Fatou - Bieberbac h domain s an d
see Stensones , [S] , for a Fatou - Bieberbac h domai n wit h C°° boundary .
In th e cas e o f Henon map s R, th e iterate s {F
n}
o f Newton's metho d i s a nor -
mal famil y o n th e basi n o f attractio n o f (0,0) . I n general , whe n w e have a ma p
F, w e cal l th e larges t ope n se t wher e {F n} i s a norma l famil y th e Fato u set .
Moreover, w e call the complemen t o f the Fato u se t th e Juli a set . S o the Juli a se t
is always closed .
In orde r t o understan d Newton' s method , a mai n proble m the n i s to describ e
the Fato u se t an d th e Juli a se t o f a ma p F . Sinc e points o n the Juli a se t ar e neve r
roots o f F, w e would lik e to kno w whether th e Juli a se t i s small. I n addition , som e
components o f the Fatou set migh t no t consis t o f initial guesses of any root. Henc e
we ar e le d t o as k whethe r th e Juli a se t ha s zer o volume . Als o w e ar e le d
to as k wha t ar e th e possibl e kind s o f Fato u components .
Next le t u s discuss th e behaviou r o f Newton's Metho d o n the Juli a se t o f F. I t
is known that polynomia l map s o n C are always chaotic o n their Juli a sets . Henc e
we are le d t o as k whethe r F i s chaotic o n it s Juli a se t a s well.
A continuou s ma p F : K K o n a compac t metri c spac e (K,d) i s chaoti c
([De]) i f
i.- F i s sensitiv e t o initia l conditions . Thi s mean s tha t ther e exist s a positiv e
6 so that fo r an y x E K an d an y positiv e e , ther e i s a point y G K close r t o x tha n
e an d a n intege r n 1 so that d(F n(x), F n(y)) 6. Se e Figure 1.3.
ii.- F i s topologicall y transitive . Thi s mean s tha t wheneve r x,y G K an d 6 is
a positiv e number , the n ther e i s a poin t z G K an d a n intege r n 1 so that bot h
d(x,z) an d d(F
n(z),y)
ar e smalle r tha n 6. See figure 1.4.
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