1549055259-Ubiquitous_Quasidisk__The__Gehring_

90 7. MISCELLANEOUS PROPERTIES

DEFINITION 7.2.6. A Jordan domain D has the comparable Dirichlet integral

property if there exists a constant c ;::: 1 such that

~ r Jgrad uj^2 dm ::; f, Jgrad vj^2 dm ::; cf, Jgrad uj^2 dm

C lv D• D

for each pair of functions u and v which are harmonic in D and D*, respectively,

with continuous and equal boundary values.

THEOREM 7.2. 7 (Ahlfors [4], Springer [154]). A Jordan domain D is a K -

quasidisk if and only if D has the comparable Dirichlet integral property with con-

stant c, where K and c depend only on each other.

7.3. Quasiconformal groups

Suppose that G is a group of self-homeomorphisms g of R

2

, i.e., a family which

is closed under composition and taking inverses. We say that w 0 is in the limit set

L(G) of G if there exist distinct gj E G and a point z 0 E R

2

such that

(7.3.1) wo = _lim gj(zo).

J-+00

Suppose next that D is a K-quasidisk. Then there exists a K-quasiconformal

self-mapping f of R^2 that maps the upper half-plane H onto D. Let g 1 and g2 be

the Mobius transformations

g1(z) = -1/z, g2(z) = z + 1,

and let G = (g1,g2), the group generated by g1 and g2. Then G has 8H as its limit

set (Beardon [18]). Next let

F = {! o go r^1 : g E C}.

Then Fis a group of K^2 -quasiconformal self-mappings of R

2

.

If W1 E 8D, then wo = f-^1 (w 1 ) E 8H and there exists a sequence of distinct

gj E G and a point zo E R
2
for which (7.3.1) holds. Thus

W1 = hm f 0 gj 0 r^1 (z1) E L(F)

J-+00

where z1 = f(zo). Reversing the above argument shows that each w 1 in L(F) is a

point of 8D.

We conclude that the boundary of each quasidisk is the limit set of a finitely

generated group of self-mappings of R
2
that are K~quasiconformal for some fixed
K. The following result shows that this property actually characterizes the family
of quasidisks.

THEOREM 7.3.2 (Maskit [125], Sullivan [156], Tukia [161]). A Jordan domain

D is a quasidisk if and only if 8D is the limit set of a finitely generated group of

K -quasiconformal self-mappings of R

2

for some fixed K.

7 .4. Homogeneity

We recall the standing assumption that Dis a subdomain of R

2

of hyperbolic

type.