1549055259-Ubiquitous_Quasidisk__The__Gehring_

14 1. PRELIMINARIES

FIGURE 1.7

for each choice of x in (0, oo). On the other hand,

x < If (zi) - f (z2) I :::; c If (z1) -f (z3) I = 2 c

by Theorem 1.3.4 where c = c(K) and we have a contradiction.

If Dis a K-quasidisk, then oD is the image of a circle under a self-homeomor-

phism f of R
2
which is differentiable a .e. Thus oD is a Jordan curve which is a
circle or line when K = 1. Hence it is natural to ask if oD has any nice analytic
properties when 1 < K < oo. For example, is oD locally rectifiable?
Our third example shows that the answer is no and that, from the standpoint
of Euclidean geometry, the boundary of a quasidisk can be quite wild. See Gehring-
Viiisiilii [70].

f

FIGURE 1.8

EXAMPLE 1.4.6. For each 1 <a< 2 there exists a quasidisk D such that

dim(8D) 2:: a

where dim denotes Hausdorff dimension.

We will sketch a proof of this. We say that a square is oriented if its sides are

parallel to the coordinate axes and we let Q and Q' denote t he open squares

Q = Q' = {z = x + iy: lxl < 1, lvl < 1}.