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 .
2.9 . We say that a Fatou componen t ft c P
is a Siege l domai n
if there exist s a subsequence f
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
is hyperbolically embedded. Then
f l U f~
= : J(C).
Hence all periodic points with one eigenvalue of modulus strictly larger than 1 are
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.