1549055259-Ubiquitous_Quasidisk__The__Gehring_

1.4. QUASIDISKS 15

Next set

3 1 1

Z1 = 4' Z2 = 4' Z3 = -4,

and

l+i -l+i 1-i -1-i

W1 = -

2

-, W2 = -

2

-, W3 = -

2

-, W4=-

2

-,

and fix 0 < r < 1/2. Then choose 0 < s < 1 so that

log 4

---=a

log(2/ s) ·

Finally for j = 1, 2, 3, 4 let Q1 denote the open oriented square with center z 1 and
side length r and let Qj be the open oriented square with center w 1 and side length
s. Then we can choose a piecewise linear homeomorphism
4 4
Jo: Q \ LJ Q1 -t Q' \ LJ Qj
j=l j=l

such that Jo is the identity on fJQ and is of the form a 1 z + b 1 , a 1 > 0, on 8Q 1 with
Jo(8Q1) = 8Qj. Then Jo is K-quasiconformal in Q \ LJ 1 Q 1 where K = K(r, s).
Next for each j choose oriented squares Qj,k in Qj and Qj ,k in Qj in the same
way as the squares Q j and Qj were chosen in Q and Q', respectively. By scaling
we can extend Jo to obtain a piecewise linear homeomorphism

4 4

Ji : Q \ u Qj,k -t Q' \ u Qj,k

j,k=l j,k=l

which is K-quasiconformal in Q \ Uj,k Qj ,k·
Continuing in this way, we obtain a homeomorphism

J : Q \ E -t Q' \ E'

where E and E' are Cantor sets. Then J can be extended by continuity to give a
K-quasiconformal mapping which maps Q onto Q' and is the identity on fJQ.

Set J( z ) = z in R

2

\ Q. Then J is a K-quasiconformal self-mapping of R

2

which maps the upper half-plane H onto a quasidisk D with Hausdorff dimension

log 4

dim( fJD) ::'.". log( 2 / 8 ) = a.

See Beardon [17] or page 67 in Mattila [127].

Although the Hausdorff dimension of the boundary fJD of a quasidisk D can
be arbitrarily close to 2, it always satisfies m(fJD) = 0 where mis planar Lebesgue
measure. This follows from Lusin's property (N) of quasiconformal mappings in
Theorem 1.1.8. On the other hand, a result due to Astala [13] gives the estimate

2K

dim(fJD) < --

• K+l

for the Hausdorff dimension of the boundary fJD of a K-quasidisk D.
Our final example, or rather class of examples, in this section illustrates how
quasidisks arise naturally in complex dynamics.