6 1. INTRODUCTIO N
FIGURE 1.6. Stabl e an d unstabl e manifold s
z fo r whic h F n(z)— p, a s w e hav e see n above . I f al l th e eigenvalue s ar e strictl y
larger tha n one , th e poin t p i s said t o b e repelling . I n thi s cas e ther e i s an ope n
neighborhood U of p suc h tha t i f q G U \ {p}, the n ther e exist s an intege r n 1 so
that F
n(q)
#U.
Another importan t cas e i n highe r dimensiona l dynamic s whic h doe s no t occu r
in dimension 1 is when som e eigenvalue s {A f }JL1? but no t all , ar e strictl y les s tha n
one an d al l the others , {AJ}^
=1
ar e strictl y large r tha n one . Suc h point s ar e calle d
saddle points . Then , ther e i s a neighborhoo d U(p) o f p containin g repellin g an d
attracting comple x submanifold s Wy, , ^"uipV t n e loca l stabl e an d unstabl e
manifolds o f complex dimensio n k an d I respectively.
The local stable manifold Wy, N consist s of the points q G Up so that F
n(q)—
p
and {F
n(q)}^L1C
U(p). Th e loca l unstabl e manifol d W^ , N consist s o f the point s
q G U(p) wit h a sequenc e {q n}%Li C U(p), q
n
- p , F(q n+l) = q n, q
x
= q. Se e
figure 1.6.
The global stabl e set , W*, o f p consists of all points q G M s o that F n(q)— p.
The points considered above , attracting fixes points, repellings points, and sad -
dle point s ar e hyperboli c fixe d points . I n additio n ther e ar e fixed point s wher e
some eigenvalu e o f th e derivativ e i s o n th e uni t circle . Th e dynamic s nea r thes e
points i s much mor e difficul t t o describe .
The sam e terminolog y applie s t o periodi c orbit , {zk}k=oi zk+i F{zk),zi
ZQ. The y ar e hyperboli c i f (F l)'(zo) ha s n o eigenvalu e o f modulus 1.
The loca l dynamic s nea r hyperboli c point s i s stable, i.e . map s clos e to F als o
have hyperboli c periodi c point s clos e t o p .
The concep t o f hyperbolicit y generalize s t o compac t subsets , no t onl y singl e
points o r periodi c orbits , ([Ru]) . Le t F b e a holomorphi c self-ma p o n a comple x
manifold M and le t K b e a compac t subse t o f M. Assum e tha t K i s surjectivel y
forward invariant , i.e . F(K) = K. W e don't assum e that F i s a homeomorphism ,
so a poin t ma y hav e severa l preimages . Th e se t o f invers e orbits , K C i^ N, K : =
{{pn}n=-oc 'iF(Pn) = Pn+i}
5
i s compac t i n th e produc t topology . W e defin e th e
tangent bundle , T
K
, o f K a s th e se t o f (p,£ ) wher e p = {p
n
} G K an d wher e
£
TM(P-I)
i s a tangen t vector . The n F lift s t o a homeomorphis m F : K - K,
given b y
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