4
24
G. Levin, S. van Strien
Sm+1
Sm+1−1
g
≤
2|U | = 2 · |g
(U )|, and since g
: U → U extends to
m
m+1
∗
a diffeomorphism onto U
, we get by the Koebe Principle that the sum
n−k−1
j
of the lengths of g (U), j = 0, ..., S
− 1, is close to zero when n and
m+1
m are large. Therefore, the preimage of the disc D (U ) by the central
∗
∗
g
branch to the critical value c is contained in a set D(U , θ), where the
1
angle θ is close to π/2 for n large. Since U is well inside U ⊂ U , we
m
∗
get that the central domain is contained well inside Ω. Let us now consider
a non-central domain J of the return map to U with return time s. Let
m
s
ˆ
ˆ
J ⊃ J be so that g : J → U is a homeomorphism. The problem is that the
∗
pullback of a Euclidean disc is merely inside a Poinca re´ disc of angle close
ˆ
to π/2. Therefore we need that J is well inside U . This holds for all non-
∗
central domains with return time s > S , and so for these the corresponding
m
domains in the complex plane fit inside Ω. However, unfortunately, for the
Sm
ˆ
non-central branch g : I → U this may not hold. Indeed, let I ⊃ I
m
m
m
m
Sm
ˆ
be so that g : I → U is a homeomorphism. It need not be true that
m
∗
ˆ
I is well inside U . Therefore, let us change the definition of U slightly.
m
∗
∗
Define U (τ) to be the interval with boundary points ± (1 + τ)∂ U . For
∗
r
∗
all τ ∈ [0, τ ] (with τ ∈ (0, 1) some fixed small number), the interval
0
0
ˆ
U (τ) is still well inside U
and there is an interval I (τ) ꢅ c such
∗
n−k−1
m
S−1
g
ˆ
that g : I (τ) → U (τ) is a diffeomorphism. Since g has no periodic
m
∗
attractors or neutral periodic orbits near c (see see Proposition 2.1), we may
S
ˆ
assume that n is so large that g : I (τ) → U (τ) has only one fixed point.
m
∗
This fixed point is repelling and lies in U .
m
ˆ
Claim: There exists τ ∈ [0, τ ] so that I (τ) is well inside U (τ). To prove
0
m
∗
ˆ
this claim, consider τ = (i/8)τ with i = 1, . . . , 8 and let [l , r ] = I (τ ).
i
0
i
i
m
i
If the claim is false for τ then either
i
S
S
|
g (r ) − r | ≤ o(n)|U | or |g (l ) − l | ≤ o(n)|U |
i
i
∗
i
i
∗
where o(n) are functions which tend to zero as n → ∞. So if the claim is
false for all i, then for at least four of the points l , . . . , l we have
1
8
S
|
g (l ) − l | ≤ o(n)|U | as n → ∞
(5.1)
i
i
∗
(
or the same holds for four of the points r). Choose intervals T ⊃ J so that
i
C(T, J) is the cross-ratio determined by these four points (from l , . . . , l ).
From (5.1) it follows that C(g (T), g (J))/C(T, J) is close to one when
n is large. But since U (τ) is well-inside U we get from the (real)
1
8
S
S
∗
n−k−1
g
S−1
ˆ
Koebe principle that the non-linearity of g : Im(τ ) → U (τ ) is uni-
0
∗
0
S−1
ˆ
versally bounded. Hence I (τ ) ⊂ U
and since U , . . . , g (U
)
m
0
n−k
n−k
S
n−k
S
C(g T,g J)
C(gT,gJ)
are disjoint and g has no wandering intervals,
≥ 1 − o(n) where
o(n) → 0 as n → ∞, see [MS]. By the definition of τ, |J|, |l(T \ J)|,
i
|
r(T \ J)| and |U | are all of the same order. But an explicit calculation for
∗
ꢁ
the map z
ꢂ→ z shows that this implies that C(gT, gJ)/C(T, J) > 1 + κ
for some universal constant κ > 0 (the intervals J ⊂ T do not contain the