12 2. KOBAYASH I HYPERBOLICIT Y
The ke y idea i n the proof i s that covering s ar e isometries i n the Kobayash i
metric an d inclusions ar e distance decreasing .
DEFINITION
2.9 . We say that a Fatou componen t ft c P
k
is a Siege l domai n
if there exist s a subsequence f
ni
convergin g to the identity ma p on Q.
Using norma l familie s argument s w e obtain:
PROPOSITION 2.10. [FS3] Let C denote the critical set of a holomorphic
map / : P P of degree at least 2. Assume that the complement of the closure
0f[X?=of~n{C)
is hyperbolically embedded. Then
J C
f l U f~
n{C)
= : J(C).
N0nN
Hence all periodic points with one eigenvalue of modulus strictly larger than 1 are
in J{C).
THEOREM 2.11 . [FS3 ] Under the assumptions of Theorem 2.8 we
have : If there is a component U of the Fatou set of f such that f n(U) does not
converge uniformly on compact sets to C, then U is preperiodic to a Siegel domain
Q with dQ c C.
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