Random Iteration of Rational Functions Page: 91

                                
Thus it suffices to show that T-'(Q) C B, or T- (Q) n B = 0. To see this, note
that
Q C Un AT()
CUnT(B)
C U \T(8B)
T41(Q) C T-1(U \ T(OB2))
C XK\&BX
CE \Bx U (evn Bx)
Since Tv (Q) is connected, this completes the proof.
The next logical step would be to apply Lemma 5.8 to get ((i) . However, we will
instead delay this step and instead perform a somewhat more complicated logical maneuver:
We will first prove the existence of a number 6 > 0 such that when Lemma 5.8 is applied,
the resulting sequence ( 1( satisfies (5.1.3). To this end we will prove the following:
CLAIM 5.11.
mult(Vi)1x < 3r
PROOF. Fix x e C. By Lemma 2.13, there exists a neighborhood Be of x such that
TBx := T1 Bx: Be- T(Bx)
is proper of degree k := multr(x). Since U is nice, there exists Ar(s) e B(< with T(x) E
Ar(s) C T(Be).
As above, let
As = T- (Ar(g)) N B = T-(AT());

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