1549055259-Ubiquitous_Quasidisk__The__Gehring_

30 2. GEOMETRIC PROPERTIES

E'

FIGURE 2.5

THEOREM 2.4.7 (Gehring [ 48 ], Walker [165]). The image of a c-lin early lo-

cally connected set E C R

2

under a Mobi1•s transformation f is c' -linearly locally

connected where c' = g-^1 (c) and

g(t) = ti;2 + c112 _ l.

PROOF. By R emark 2.4.2 it is enough to show that E' = f(E) satisfies condi-
tion 1 °. So assume that this is not true, i.e. that there exist wo E R^2 and r > 0 so
that Ei = E' n B(w 0 , r) does not lie in a component of E' n B(w 0 , c'r). Let C' be
a n arbitrary connected subset of E' containing Ei and set

E~ = C' \ B(wo, c'r) =f. 0.

Then C = f-^1 (C') is an arbitrary connected set in E containing E 1 = f-^1 (Ei)
and E 2 = f-^1 (E~) CC. By Lemma 2.4.5, E 1 and E 2 are separat ed by a n annulus

{ z : a < I z - zo I < ca}

in contradiction to the hypothesis that E is c-linearly locally connected. 0

2.5. Decomposition
The following observation is the b asis for another property of quasidisks.
EXAMPLE 2.5.l. A doma in D ~ R^2 is a disk or ha lf-plane if and only if for
each pair of points z 1 , z 2 E D there exists a disk D' with

Z1 ' Z2 E D' c D.
To see this, we need only prove the sufficiency, and we co nsider two cases. If
D is bounded, we can choose z 11 and z 21 in D so that

lz1 1 - z21 I --+ diam(D) = 2 r

as j --+ oo. By hypothesis, there exists for each j a disk B 1 = B(w 1 , r 1 ) with

z 11 , z21 E B 1 C D and, by passing to a subsequence, we m ay assume that w 1 --+z 0 E