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.