1549055259-Ubiquitous_Quasidisk__The__Gehring_

7.4. HOMOGENEITY 91

A set E C R

2

is homogeneous with respect to a family F of mappings if for

each z 1 , z2 E E there exists a mapping f E F such that

f(E) = E,

Every simply connected domain D C R

2

is homogeneous with respect to con-

formal, and hence quasiconformal, mappings of D onto itself. The situation changes

if we consider self-mappings of R

2

. For example, disks and half-planes are the only
homogeneous Jordan domains with respect to conformal mappings of R
2
. This
result has a counterpart for quasidisks and quasiconformal mappings.
We let QC(K) denote the family of all K-quasiconformal self-mappings of R
2
.
Then QC(l) is simply the family of mappings generated by the family M and the
reflection r(z) = z.
If D is a K-quasidisk, then D = f(H) and 8D = f(8H) where f is a K-

quasiconformal self-mapping of R

2

. Hence D and 8D are both homogeneous with
respect to the family of mappings

F = f o Mo f-^1 c QC(K^2 ).

Moreover a Jordan domain D is a quasidisk if either D or its boundary 8D is
homogeneous with respect to the family QC(K) for some K. More precisely we
have the following results.

THEOREM 7.4.1 (Brechner, Erkama [27], [37]). A simply connected domain D
is a quasidisk if and only if 8D is homogeneous with respect to the family QC(K)
for some fixed K.

THEOREM 7.4.2 (Sarvas [153]). A Jordan domain D is a quasidisk if and only

if it is homogeneous with respect to the family QC(K) for some fixed K.

The following example shows that the hypothesis that D be a Jordan domain
is necessary in Theorem 7.4.2.

EXAMPLE 7.4.3 (Hjelle [86]). There exists a simply connected domain D which
is not a quasidisk but which is homogeneous with respect to the family QC(K) for
a fixed K > 1.

We will sketch a proof of this. Let G denote the group of 1-quasiconformal
mappings generated by the reflections gl, g2 in the circles lz + 21 = 1, lz - 21 = 1
and the translation
h(z)=z+i

and let Go denote the family consisting of the identity and the sixteen transforma-
tions

h, h-1 ' gj 0 h ' gjoh-^1 , h ogj,

h-^1 0 gj 0 h, hogjoh-^1

where j = 1, 2. Next let

E = { z = x + iy: lxl:::; 2 - Jl=Y2, fyf:::; 1/2},

and let
D = LJ g(E), Do= int( LJ g(E)).
gEG gEGo