Area distortion of quasiconformal mappings

K. Astala

1 Introduction

A homeomorphism f : Ω → Ω′ between planar domains Ω and Ω′ is called K-quasicon-formal if it is contained in the Sobolev class W 1

2,loc(Ω) and its directional derivativessatisfy

maxα|∂αf(x)| ≤ Kminα|∂αf(x)| a.e. x ∈ Ω.

In recent years quasiconformal mappings have been an efficient tool in the study ofdynamical systems of the complex plane. We show here that, in turn, methods or ideasfrom dynamical systems can be used to solve a number of open questions in the theoryof planar quasiconformal mappings.

It has been known since the work of Ahlfors [A] and Mori [Mo] that K-quasiconformalmappings are locally Holder continuous with exponent 1/K. The function

f0(z) = z|z|1K−1 (1)

shows that this exponent is the best possible. In addition to distance, quasiconformalmappings distort also the area by a power depending only on K, as shown first byBojarski [Bj]. Since |f0B(r)| = π1− 1

K |B(r)|1K , where B(r) = z ∈ C : |z| < r, it is

natural to expect that the optimal exponent in area distortion is similarly 1/K.In this paper we give a positive answer to this problem and prove the following

result which was conjectured and formulated in this precise form by Gehring and Reich[GR]. We shall denote by ∆ the open unit disk and by |E| the area of the planar setE.

Theorem 1.1 Suppose f : ∆ → ∆ is a K-quasiconformal mapping with f(0) = 0.Then we have

|fE| ≤M |E|1/K (2)

for all Borel measurable sets E ⊂ ∆. Moreover, the constant M = M(K) depends onlyon K with M(K) = 1 +O(K − 1).

For the proof of (2) we consider families Bin1 of disjoint disks Bi = Bi(λ) whichdepend holomorphically on the parameter λ (in a sense to be defined later). After anapproximation (2) now becomes equivalent to

n∑i=1

|Bi(λ)| ≤ C( n∑i=1

|Bi(0)|) 1−|λ|

1+|λ| , (3)

1

where C depends only on λ. Furthermore, iterating the configuration one is led tomeasures on Cantor sets and there we shall apply the Ruelle-Bowen thermodynamicformalism [Bw]; if we write (3) in terms of the topological pressure, then the proofcomes out in a transparent way.

The function f0 is extremal in the distortion of area as well as distance, and thereforeit is natural to ask [I, 9.2] if for quasiconformal mappings the Holder continuity alone,rather than the dilatation, implies the inequality (2). However, this turns out to befalse, as shown recently by P.Koskela [K].

As is well known the optimal control of area distortion answers several questionsin this field. For example, in general domains Ω one can interpret (2) in terms of thelocal integrability of the Jacobian Jf of the quasiconformal mapping f . This leads toa solution of the well known problem [LV], [Ge] on the value of the constant

p(K) = supp : Jf ∈ Lploc(Ω) for each K − quasiconformal f on Ω.

Corollary 1.2 In every planar domain Ω, p(K) =K

K − 1.

In other words, for each K-quasiconformal f : Ω → Ω′,

f ∈W 1p,loc(Ω), p <

2KK − 1

.

The example (1) shows that this is false for p ≥ 2KK−1 .

Theorem 1.1 governs also the distortion of the Hausdorff dimension dim(E) of asubset E.

Corollary 1.3 Let f : Ω → Ω′ be K-quasiconformal and suppose E ⊂ Ω is compact.Then

dim(fE) ≤ 2K dim(E)2 + (K − 1) dim(E)

. (4)

This inequality, as well, is the best possible.

Theorem 1.4 For each 0 < t < 2 and K ≥ 1 there is a set E ⊂ C with dim(E) = tand a K-quasiconformal mapping f of C such that

dim(fE) =2K dim(E)

2 + (K − 1) dim(E).

The estimate (4) was suggested by Gehring and Vaisala [GV]. It can also be for-mulated [IM2] in the symmetric form

1K

( 1dim(E)

− 12

)≤ 1dim(fE)

− 12≤ K

( 1dim(E)

− 12

). (5)

The results 1.3 and 1.4 are closely related to the removability properties of quasireg-ular mappings, since in plane domains these can be represented as compositions ofanalytic functions and quasiconformal mappings. The strongest removability conjec-ture, due to Iwaniec and Martin [IM1], suggest that sets of Hausdorff d−measure zero,d = n

K+1 , are removable for bounded quasiregular mappings in Rn. Here we obtain thefollowing.

2

Corollary 1.5 In planar domains sets E of Hausdorff dimension dim(E) < 2K + 1

are removable for bounded quasiregular mappings.Conversely, for each K ≥ 1 and t > 2/(K + 1) there is a t−dimensional set E ⊂ Cwhich is not removable for some bounded K−quasiregular mappings.

In addition to [IM1] removability questions have recently been studied for instancein [JV], [KM] and [Ri].

Finally, we mention the applications to the regularity results of quasiregular map-pings. Recall that a mapping

f ∈W 1q,loc(Ω), 1 < q < 2,

is said to be weakly quasiregular, if Jf ≥ 0 almost everywhere and

max|h|=1

|Df(x)h| ≤ K min|h|=1

|Df(x)h| a.e. x ∈ Ω.

Then f is K−quasiregular in the usual sense if f ∈ W 12,loc(Ω), i.e. if Jf is locally

integrable.We can now consider the number q(K), the infimum of the q’s such that every

weakly K−quasiregular mapping f ∈W 1q,loc(Ω) is actually K−quasiregular.

Corollary 1.6 q(K) = 2KK + 1 .

Indeed, Lehto and Virtanen [LV] have proven that the precise estimate on the Lp−inte-grability, Corollary 1.2, implies that q(K) ≤ 2K

K+1 . The opposite inequality q(K) ≥ 2KK+1

was shown by Iwaniec and Martin in [IM1].

Quasiconformal mappings are also the homeomorphic solutions of the elliptic dif-ferential equations

∂f(z) = µ(z) ∂f(z);

here µ is the complex dilatation or the Beltrami coefficient of f with ‖µ‖∞ = K−1K+1 < 1.

Hence there are close connections to the singular integral operators and especially tothe Beurling-Ahlfors operator

Sω(z) = − 1π

∫C

ω(ζ) dm(ζ)(ζ − z)2

,

see [I],[IM1],[IK] for example. In fact, this operator was the main tool in the workof Bojarski [Bj] and Gehring-Reich [GR], c.f. also [IM2]. Below we shall use mostlydifferent approaches and the role of the S operator remains implicit. Still the areadistortion inequality has a number of implications on the properties of S. In particular,we have

Corollary 1.7 There is a constant α ≥ 1 such that for any measurable set E ⊂ ∆,∫∆

|SχE | dm ≤ |E| logα

|E|. (6)

It is for this consequence that we must show the asymptotic estimate M(K) = 1 +O(K − 1) and then, actually, Theorem 1.1 is equivalent to 1.7, cf. [GR].1

1David Hamilton and Tadeusz Iwaniec have pointed out that now (6) holds with α = eπ.

3

If we consider general functions ω ∈ L∞(∆) then the inequality (6) implies thecorrect exponential decay for |z ∈ ∆ : <Sω > t| when t→∞. As a consequence, foreach δ > 1 there is a constant Mδ <∞ such that∫

∆

|Sv| dm ≤ δ

∫∆

|v| log(1 +Mδ|v||v|∆

) dm, v ∈ L logL(∆). (7)

Here |v|∆ = 1π

∫∆ |v| dm is the integral mean of |v|. It is a natural question whether

(7) holds at δ = 1 as in (6) with characteristic functions. This would also implythe Iwaniec-Martin removability conjecture in the planar case. However, in the lastsection we show, again by considering the inequalities arising from the thermodynamicformalism, that in fact (7) fails when δ = 1.

Further results equivalent to the Gehring-Reich conjecture have been given byIwaniec and Kosecki [IK]. These include applications to the L1−theory of analyticfunctions, quadratic differentials and critical values of harmonic functions. Moreover,by results of Lavrentiev, Bers and others the solutions to the elliptic differential equa-tions ∇A(x)∇u = 0 can be interpreted in terms of quasiregular mappings f . ThereforeCorollary 1.2 yields sharp exponents of integrability on the gradient ∇u; note thatthe dilatation of f and so necessarily the optimal integrability exponent depends in acomplicated manner on all the entries of the matrix A rather than just on its ellipticitycoefficient .

Acknowledgements During the preparation of this manuscript a number of peoplefound simplifications in the first preliminary notes. Especially I would like to thankAlexander Eremenko and Jose Fernandez who both pointed out Corollary 2.4. Also,I would thank Tadeusz Iwaniec and Michel Zinsmeister for important discussions andcomments on the topics in this paper and many people, including Alexander Eremenko,Fred Gehring, Seppo Rickman, Juha Heinonen and Pekka Koskela for reading andmaking corrections to the first draft.

2 Holomorphic deformations of Cantor sets

Let us first consider a family Bini=1 of nonintersecting subdisks of ∆. We shall studythe quasiconformal deformation of such families and, in particular, estimate sums

n∑i=1

r(Bi)t, t ∈ R, (8)

where r(Bi) denotes the radius of Bi. Looking for the extremal phenomena we caniterate the configuration Bini=1 and are thus led to Cantor sets. There one needsmeasures µ which reflect in a natural manner the properties of the sums (8). It turnsout that such measures can, indeed, be found by using the thermodynamic formalismintroduced by Ruelle and Bowen, c.f. [Bw], [W].

To describe this in more detail suppose hence that we are given similarities γi,1 ≤ i ≤ n, for which Bi = γi∆. Since the γi are contractions, there is a unique compact

4

subset J of the unit disk for whichn⋃i=1

γi(J) = J.

Thus J is self-similar in the terminology of Hutchinson [H]. We can also reverse thispicture and define the mapping g : ∪iBi → ∆ by g|Bi = γ−1

i . Now

J =∞⋂k=0

g−k∆,

and g is a n-to-1 expanding mapping with J completely invariant, J = gJ = g−1J .Furthermore, g represents the shift on J ; in a natural manner we can identify the pointx ∈ J with the sequence (jk)∞k=0 ∈ 1, . . . , nN by defining jk = i if gk(x) ∈ Bi. Then,in this identification, g : (jk)∞k=0 7→ (jk+1)∞k=0.

In the sequel we use the notation J = J(g) for our Cantor set and say also that itis generated by the similarities γi.

Next, let s = dim(J(g)), the Hausdorff dimension of J(g). Then the Hausdorffs−measure is nonzero and finite on J(g) and after a normalization it defines a prob-ability measure µs which is invariant under the shift g, i.e. µs(g−1E) = µs(E) for allborelian E ⊂ J(g). A general and systematic way to produce further invariant mea-sures is provided by the Ruelle-Bowen formalism: Given a Holder continuous and realvalued function ψ on J(g) there is a unique shift-invariant probability measure µ = µψ,called the Gibbs measure of ψ, for which the supremum

P (ψ) = sup hµ(g) +∫J(g)

ψ dµ : µ is g − invariant (9)

is attained, see [Bw] or [W]. Here hµ(g) denotes the entropy of µ and the quantity P (ψ)is called the topological pressure of ψ.

Let us then look for the Gibbs measures that are related to the sums (8). Recallthat s = dim(J(g)) is the unique solution of P (−s log |g′|) = 0, and this suggest thechoices ψt = −t log |g′|. It then readily follows from [Bw, Lemma I.1.20], that

P (−t log |g′|) = log( n∑i=1

|γ′i|t)

= log( n∑i=1

r(Bi)t). (10)

5

In fact, the functions ψ = ψt are in our situation locally constant and therefore itcan be shown that the system g : (J(g), µψ) → (J(g), µψ) is Bernoulli. In other words,the numbers pi = µψ(J ∩ Bi) satisfy

∑ni=1 pi = 1 and on J(g) ∼= 1, . . . , nN µψ is

the product measure determined by the probability distribution pini=1 of 1, . . . , n.This enables one to make the dynamical approach more elementary, as pointed to us byAlexander Eremenko. We are grateful to him for letting us to include this simplificationhere.

For the readers convenience let us recall the proof of the variational principle, thecounterpart of (9), in the elementary setting of product measures. Then also theentropy of µ = µψ attains the simple form

hµ(g) = −n∑1

pi log pi.

Lemma 2.1 Let ν be a product measure on J(g) determined by the probability distri-bution qini=1. Then for each t ∈ R,

hν(g)− t

∫J(g)

log |g′| dν ≤ log(n∑i=1

r(Bi)t)

with equality if and only if

qi =r(Bi)t∑ni=1 r(Bi)t

, 1 ≤ i ≤ n.

Proof: Since the logarithm is concave on R+,

hν(g)− t

∫J(g)

log |g′| dν =n∑i=1

qi log|γ′i|t

qi=

n∑i=1

qi logr(Bi)t

qi≤ log(

n∑i=1

r(Bi)t)

where the equality holds if and only if qi r(Bi)−t has the same value for each1 ≤ i ≤ n.

Remark. In the elementary setting, one can use (10) as the definition of thepressure P (−t log |g′|). Note also that if s = dim(J(g)), then

∑ni r(Bi)

s = 1, orP (−s log |g′|) = 0, and the extremal measure in Lemma 2.1 is again the normalizedHausdorff s−measure.

We shall next consider holomorphic families of Cantor sets or pairs (gλ, J(gλ)),λ ∈ ∆. By this we mean that each set J(gλ) is generated as above by similaritiesγi,λ(z) = ai(λ)z + bi(λ), 1 ≤ i ≤ n, where the coefficients ai(λ) 6= 0, bi(λ) now dependholomorphically on the parameter λ. On the other hand, we can also consider theBi(λ) = γi,λ∆ and say that Bi(λ)n1 is a holomorphic family of disjoint disks in ∆.

Both of these configurations can be described as holomorphic motions; recall thata function Φ : ∆×A→ C is called a holomorphic motion of a set A ⊂ C if

6

(i) for any fixed a ∈ A, the map λ 7→ Φ(λ, a) is holomorphic in ∆(ii) for any fixed λ ∈ ∆, the map a 7→ Φλ(a) = Φ(λ, a) is an injection,

and(iii) the mapping Φ0 is the identity on A.In fact, (global) quasiconformal mappings and holomorphic motions are just differ-

ent expressions of the same geometric quantity. For instance, according to Slodkowski’sgeneralized λ−lemma ([Sl], see also [AM], 3.3) the correspondence γi,0(z) 7→ γi,λ(z) forz ∈ ∆ and 1 ≤ i ≤ n, extends to a quasiconformal mapping Φλ : C → C withK(Φλ) ≤ 1+|λ|

1−|λ| . Therefore the estimate

n∑i=1

|Bi(λ)| ≤ C( n∑i=1

|Bi(0)|) 1−|λ|

1+|λ| (11)

is a special case of the Gehring-Reich conjecture. But after simplifying arguments,given later in Lemmas 3.1 and 3.3, we will see that the conjecture is in fact equivalentto (11).

Expressing this inequality now in terms of the topological pressure (10) we end upwith the following formulation.

Theorem 2.2 Suppose that (gλ, J(gλ)) depends holomorphically on the parameter λ ∈∆. Then

1 + |λ|1− |λ|

P (−2 log |g′0|) ≤ P (−2 log |g′λ|) ≤1− |λ|1 + |λ|

P (−2 log |g′0|).

Proof: By the variational inequality 2.1 for each λ there is a unique (product) measureµλ such that

P (−2 log |g′λ|) = hµλ(gλ)− 2

∫J(gλ)

log |g′λ|(z) dµλ(z) (12)

and clearly log |g′λ|(z) is harmonic in λ. To use Harnack’s inequality we ’freeze’ themeasure µλ. In other words, given a probability distribution pin1 on 1, . . . , n, definefor each λ ∈ ∆ a product measure µλ on J(gλ) by the condition µλ(J(gλ)∩Bi(λ)) = pi;this is possible since the disks Bi(λ) remain disjoint. By the construction, hµλ

(gλ) isalso constant in λ.

Moreover, we have that P (−2 log |g′λ|) < 0, since P (−s log |g′λ|) is strictly decreasingin s and it vanishes for s = dim(J(g)) < 2. Alternatively, we may also use here theidentity (10) to P (−2 log |g′λ|) = log(

∑ni=1 r(Bi(λ))2) < 0.

If now the numbers pi are so chosen that µ0 = µ0 (the maximizing measure in(12) when the parameter λ = 0), then Harnack’s inequality with 2.1 implies that

1 + |λ|1− |λ|

P (−2 log |g′0|) =1 + |λ|1− |λ|

(hµ0

(g0)− 2∫

J(g0)

log |g′0| dµ0

)

≤ hµλ(gλ)− 2

∫J(gλ)

log |g′λ| dµλ ≤ P (−2 log |g′λ|)

7

which proves the first of the required inequalities. The second follows similarly bysymmetry in λ and 0.

When t > 2 the same inequalities hold for P (−t log |g′λ|) as well. However, smallerexponents must change with |λ| and we shall later see how this reflects in the precisedistortion of Hausdorff dimension under quasiconformal mappings.

Corollary 2.3 If (gλ, J(gλ)) is as above and 0 < t ≤ 2, set t(λ) = t(1 + |λ|)(1− |λ|+ t|λ|) .

Then1t(λ)

P (−t(λ) log |g′λ|) ≤1− |λ|1 + |λ|

1tP (−t log |g′0|).

Proof: If µλλ∈∆ is a family of product measures on J(gλ), all defined by a fixedprobability distribution pin1 like in the previous theorem, then by 2.1

1t(λ)

hµλ(gλ)−

∫J(gλ)

log |g′λ| dµλ = hµλ(gλ)(

1t(λ)

− 12) +

12hµλ

(gλ)−∫

J(gλ)

log |g′λ| dµλ

≤ 1− |λ|1 + |λ|

(hµ0

(g0)(1t− 1

2) +

12hµ0

(g0)−∫

J(g0)

log |g′0| dµ0

)

≤ 1− |λ|1 + |λ|

1tP (−t log |g′0|)

and taking the supremum over the product measures on J(gλ) proves the claim.

The above estimates for the topological pressure hold actually in a much greatergenerality. We can consider, for instance, polynomial-like mappings of Douady andHubbard [DH]. More precisely, suppose we have a family of holomorphic functions fλdefined on the open sets Uλ, λ ∈ ∆, such that Uλ ⊂ fλUλ. We need to assume that

J(fλ) =∞⋂n=0

fλ−nUλ

is a mixing repeller for fλ. That is, f ′λ 6= 0 for z ∈ J(fλ) and J(fλ) is compact in Cwith no proper fλ−invariant relatively open subsets. Then the fλ are expanding onJ(fλ) and the thermodynamic formalism extends to fλ : J(fλ) → J(fλ), see [Bw] or[Ru].

To consider the dependence on the parameter, let Uλ depend continuously on λand let (λ, a) 7→ fλ(a) be holomorphic whenever defined. Because the functions areexpanding, we have a holomorphic motion of the periodic points ([MSS], p.198). Sincethese are dense in the repeller J(fλ), by the λ−lemma of Mane, Sad and Sullivan weobtain a holomorphic motion Φ of J(f0) such that J(fλ) = ΦλJ(f0) and fλ Φλ =Φλ f0.

Combining these facts we conclude that

1t(λ)

P (−t(λ) log |f ′λ|) ≤1− |λ|1 + |λ|

1tP (−t log |f ′0|). (13)

8

Namely, since the variational principle generalizes to this setting, the proof of (13) is asin Lemma 2.3. In this case to show that P (−t(λ) log |f ′λ|) < 0 we may use Manning’sformula [M]

dim(µ) =hµ(fλ)∫

J(fλ)

log |f ′λ| dµ

and the fact [Su] that dim(µ) ≡ infdim(E) : µ(E) = 1 ≤ dim(J(fλ)) < 2. These holdfor any ergodic fλ−invariant measure on J(fλ). Especially, starting from a measure µon J(f0) we can take the images µλ = Φ∗

λµ under the holomorphic motion, and sincethe entropy is an isomorphism invariant, (13) follows.

On the other hand, if one looks for the minimal approach to the quasiconformalarea distortion, then the above leads also to a proof for (11) that avoids the thermo-dynamic formalism. In fact, this was shown to us by A. Eremenko and J.Fernandez,who independently pointed out the following result on the (nonharmonic !) functionlog ‖f(z)‖.

Corollary 2.4 Let Bn = z ∈ Cn : ‖z‖ < 1. If f : ∆ → Bn is a holomorphicmapping such that all of its coordinate functions fi are everywhere nonzero, then

1 + |z|1− |z|

log ‖f(0)‖ ≤ log ‖f(z)‖ ≤ 1− |z|1 + |z|

log ‖f(0)‖.

Proof: If f = (f1, . . . , fn), consider numbers pi > 0 with∑n

1 pi = 1 and set

u(z) =n∑i=1

pi logpi

|fi(z)|2.

Then u(z) is harmonic and by Jensen’s inequality, e−u(z) ≤∑n

1 pi|fi(z)|2pi

< 1, u is alsopositive. Hence using the concavity of the logarithm and the Harnack’s inequality wemay deduce

log ‖f(z)‖2 ≥n∑i=1

pi log|fi(z)|2

pi≥ 1 + |z|

1− |z|

n∑i=1

pi log|fi(0)|2

pi.

Choosing finally pi = |fi(0)|2‖f(0)‖2 proves the first inequality. The second follows by symme-

try.

3 Distortion of area

We shall reduce the proof of the area distortion estimate |fE| ≤ M |E|1/K into twodistinct special cases. In the first, where we use the inequalities of the previous section,let us assume that E is a finite union of nonintersecting disks Bi = B(zi, ri) ⊂ ∆,1 ≤ i ≤ n.

9

Lemma 3.1 Suppose that f : ∆ → ∆ is K−quasiconformal with f(0) = 0. If f isconformal in E = ∪n1Bi, then

n∑i=1

|fBi| ≤ C(K)( n∑i=1

|Bi|) 1

K ,

where the constant C(K) depends only on K. Moreover, C(K) = 1 +O(K − 1).

Proof: Extend f first to C by a reflection across S1 and assume without loss ofgenerality that f(1) = 1. Then we can embed f to a holomorphic family of quasicon-formal mappings of C. However, in order to control the distortion as K →∞ we needto modify f near ∞. Thus, if µ is the Beltrami coefficient of (the extended) f , definenew dilatations by

µλ(z) =λK+1K−1µ(z), |z| ≤ 2,

0, |z| > 2.

By the measurable Riemann mapping theorem there are unique µλ−quasiconformalmappings fλ : C → C normalized by the condition

fλ(z)− z = O(1|z|

) as |z| → ∞.

Then fλ is conformal in E, fλ(z) and its derivatives (when z ∈ E) depend holomorphi-cally on λ [AB, Theorem 3], f0(z) ≡ z and if λ0 = K−1

K+1 , then

fλ0 = Φ f, (14)

where Φ is conformal in fB(0, 2).To apply Theorem 2.2 note that by Koebe’s 1/4−theorem

Di(λ) ≡ B(fλ(zi),

ri4|f ′λ(zi)|

)⊂ fλB(zi, ri)

and similarly fλB(0, 2) ⊂ B(0, 8). Also Di(λ) = ψλ,iDi(0), where

ψi,λ(z) = f ′λ(zi) (z − zi) + fλ(zi),

and thus Di(λ)n1 is a holomorphic family of disjoint disks contained in B(0, 8). There-fore we need only choose extra similarities φi : B(0, 8) → Di(0), 1 ≤ i ≤ n, setγi,λ = ψi,λ φi and note that these generate a holomorphic family of Cantor setsJ(gλ) ⊂ B(0, 8). By Theorem 2.2 P (−2 log |g′λ|) ≤

1−|λ|1+|λ|P (−2 log |g′0|) or, in other

words, by (11)n∑i=1

|f ′λ(zi)|2r2i ≤ 324|λ|

1+|λ|( n∑i=1

r2i

) 1−|λ|1+|λ|

.

The Lemma will then be completed by simple approximation arguments. Since theimages of circles under global quasiconformal mappings have bounded distortion,

|fλBi| ≤ π maxz∈∂Bi

|fλ(z)− fλ(zi)|2 ≤ πC0(|λ|) minz∈∂Bi

|fλ(z)− fλ(zi)|2

10

≤ πC0(|λ|)|f ′λ(zi)|2r2i ,

where the last estimate follows from the Schwarz lemma. Moreover, the correct expres-sion for the constant C0(|λ|), see [L, p.16], shows that C0(|λ|) = 1+O(|λ|). If we chooseλ0 = K−1

K+1 , it then follows that |fλ0E| ≤ C1(K)|E|1K with C1(K) = 1 +O(K − 1).

It remains to show that the function Φ in (14) satisfies |Φ′(z)| ≥ C2(K) = (1 +O(K − 1))−1 for all z ∈ ∆. First, since the diameter of fλ0B(0, 2) is at least four [P,11.1], the basic bounds on the circular distortion, see [L, I.2.5], imply that

B(fλ0(0), ρ(K)) ⊂ fλ0∆ = Φ∆

for a ρ(K) > 0 depending only on K. As above, |Φ′(0)| ≥ ρ(K) by the Schwarz lemma.Yet another application of the Schwarz lemma, this time to the function λ 7→ fλ(z)−z,gives

|fλ(z)− z| ≤ 10|λ|, z ∈ B(0, 2).

This shows that we may choose ρ(K) = (1 +O(K − 1))−1.Furthermore, as fS1 = S1 and f−1 is uniformly Holder continuous with constants

depending only on K, fB(0, 2) ⊃ B(0, R) for an R = R(K) > 1. Then Koebe’sdistortion theorem combined with Lehto’s majorant principle [L, II.3.5] proves that

|Φ′(z)| ≥ |Φ′(0)|((1− | zR |)

3

1 + | zR |

)K−1K+1 , z ∈ ∆,

and the required estimates follow.

Remark 3.2 The above proof gives us the following ”variational principle” for planarquasiconformal mappings:Suppose we are given numbers pi > 0 with

∑ni=1 pi = 1 and disjoint disks Bi ⊂ ∆.

Then for each K−quasiconformal mapping f : ∆ → ∆ for which f(0) = 0 and

f |∪n1Bi is conformal, (15)

we have the inequality

n∑i=1

pi log|fBi|pi

≤ 1K

n∑i=1

pi log|Bi|pi

+ C(K) (16)

where C(K) = O(K − 1) depends only on K.

In fact, choosing pi = |fBi|/(∑n

i=1 |fBi|)

shows that (16) generalizes Lemma 3.1.Somewhat curiously, the variational inequality (16) is not true for general quasi-

conformal mappings, for mappings which do not satisfy (15). We shall return to thisin Section 5 where it will have implications on the estimates of the L logL−norm ofthe Beurling-Ahlfors operator.

To prove the complementary case in the area distortion inequality we use thereforea different method. We shall apply here the approach due to Gehring and Reich [GR]based on a parametric representation.

11

Lemma 3.3 Let f : ∆ → ∆ be K−quasiconformal with f(0) = 0. If E ⊂ ∆ is closedand f is conformal outside E, then

|fE| ≤ b(K)|E|

where b(K) = 1 +O(K − 1) depends only on K.

Proof: As in [GR] define the Beltrami coefficients

νt(z) = sgn(µ(z)) tanh(t

Tarctanh |µ(z)|), t ∈ R+,

where µ is the complex dilatation of f , T = logK and sgn(w) = w|w| if w 6= 0 with

sgn(0) = 0. By the measurable Riemann mapping theorem we can find νt−quasicon-formal ht : ∆ → ∆ with ht(0) = 0.

If A(t) = |htE|, then Gehring and Reich show that

d

dtA(t) =

∫∆

φS(χhtE

) dxdy + c(t)|htE| (17)

where S is the Beurling-Ahlfors operator and |c(t)| is uniformly bounded. The functionφ depends only on the family ht, not on E, and from [GR, (2.6) and (3.6)] we concludethat ‖φ‖∞ ≤ 1 and that φ(w) = 0 whenever µ(h−1

t (w)) = 0.Suppose now that f is conformal outside the compact subset E ⊂ ∆. Then µ ≡ 0

in ∆\E and, in particular, we obtain

|φ(z)| ≤ χhtE

(z).

But S : L2 → L2 is an isometry and therefore for any set F ⊂ C,∫F

|SχF | dxdy ≤ |F |12

(∫C

|SχF |2 dxdy

) 12 = |F |. (18)

Thusd

dtA(t) ≤ C0A(t), 0 < t <∞,

and an integration gives |htE| = A(t) ≤ eC0tA(0) = eC0t|E|. Taking t = logK showsthat |fE| ≤ eC0 logK |E| = b(K)|E|, where b(K) = 1 +O(K − 1).

Remark. On the other hand, as kindly pointed out by the referee, if one considers inC the normal solutions fλ(z) = z+O(|z|−1) of the Beltrami equation fz = µfz with µsupported on E, then in that situation the following argument gives a direct proof fora very precise estimate

|f(E)| ≤ K|E|.

Namely, for ω = fz we have fz = 1 + Sω with ω = µ(1 + Sµ+ SµSµ+ . . .). In view of‖S‖2 = 1 we obtain

|f(E)| =∫E|1 + Sω|2 − |ω|2 ≤ |E|+ 2<

∫ESω,

12

where for the k’th iterate∫E |SµSµ . . . Sµ| ≤ ‖µ‖k∞|E| as in (18). Thus

|f(E)| ≤ |E|+ 2‖µ‖∞|E|+ 2‖µ‖2∞|E|+ . . . = |E|(−1 +2

1− ‖µ‖∞) = K|E|.

The area distortion inequality is now an immediate corollary of the two previouslemmas 3.1 and 3.3.

Proof of Theorem 1.1: Suppose that f : ∆ → ∆ is K−quasiconformal and f(0) = 0.In proving the estimate

|fE| ≤M |E|1/K

it suffices to study sets of the type E = ∪n1Bi, where the Bi are subdisks of ∆ withpairwise disjoint closures. The general case follows then from Vitali’s covering theorem.

To factor f we find by the measurable Riemann mapping theorem a K−quasicon-formal mapping g : ∆ → ∆, g(0) = 0, with complex dilatation µg = χ

∆\Eµf . Then g is

conformal in E and f = h g, where h : ∆ → ∆ is also K−quasiconformal, h(0) = 0,but now h is conformal outside gE. Since quasiconformal mappings preserve sets ofzero area, |h(∂gE)| = |∂gE| = 0, and then Lemmas 3.1 and 3.3 imply

|fE| = |h(gE)| ≤ b(K)|gE| ≤ b(K)C(K)|E|1K ,

where M(K) ≡ b(K)C(K) = 1 +O(K − 1) as required. 2

One of the equivalent formulations of Theorem 1.1 is the statement that for aK−quasiconformal f the Jacobian Jf belongs to the class weak-Lp(∆), p = K

K−1 .

Corollary 3.4 If f : ∆ → ∆ is K−quasiconformal, f(0) = 0, then for all s > 0,

|z ∈ ∆ : Jf (z) ≥ s| ≤(Ms

) KK−1 ,

where M depends only on K. Moreover the exponent p = KK−1 is the best possible.

Proof: If Es = z ∈ ∆ : Jf (z) ≥ s, then by Theorem 1.1

s|Es| ≤∫Es

Jf dm = |fEs| ≤M(K)|Es|1K .

No p larger than KK−1 will do, since |Es| = π(Ks)

−KK−1 for f(z) = z|z|

1K−1.

Proof of Corollary 1.2: If D is a compact disk in the domain Ω and f : Ω → Ω′ isK−quasiconformal, choose conformal ψ, φ which map neigbourhoods of D and fD,respectively, onto the unit disk. As ψ|D and φ|fD are bilipschitz, applying Corollary3.4 to φ f ψ−1 proves that Jf ∈ Lploc(Ω) for all p < K

K−1 .

2David Hamilton has informed us that the same methods can also be used to obtain good boundsfor the constant M in |fE| ≤ M |E|1/K if one considers instead of the case f : ∆ → ∆ those mappingsf which are conformal outside ∆ with f(z)− z = O(1/|z|).

13

4 Distortion of dimension

In the previous section we determined the quasiconformal area distortion from theproperties of the pressure P (−2 log |g′λ|). Similarly Corollary 2.3, or the variationalinequality (16) with a suitable choice of the probabilities pi, also admits a geometricinterpretation:

If f : ∆ → ∆ is K−quasiconformal with f(0) = 0 and if, in addition, f is conformalin the union of the disks Bi ⊂ ∆, 1 ≤ i ≤ n, then

∑i

|fBi|tK

1+t(K−1) ≤ C(K)(∑

i

|Bi|t) 1

1+t(K−1) , 0 < t ≤ 1,

where the constant C(K) depends only on K.Since the complementary lemma 3.3 fails for exponents t < 2, in the general case

we content with slightly weaker inequalities.

Lemma 4.1 If 0 < t < 1, f : ∆ → ∆ is K−quasiconformal, f(0) = 0 and Bin1 arepairwise disjoint sets in ∆, then

∑i

|fBi|p ≤ CK(t, p)(∑

i

|Bi|t) 1

1+t(K−1)

whenever (1 + t(K − 1))−1tK < p ≤ 1.

Proof: We use the integrability of the Jacobian Jf as in [GV]. Sincep(1 + t(K − 1)) > tK we can choose an exponent 1 < p0 <

KK−1 such that

1Kq0 < p

1 + t(K − 1)tK

, (19)

where q0 =p0

p0 − 1is the conjugate exponent. Then using Holder’s inequality twice one

obtains ∑i

|fBi|p =∑i

(∫Bi

Jf dm)p≤

∑i

(∫Bi

Jfp0 dm

) pp0 |Bi|

pq0 ≤

(∑i

∫Bi

Jfp0 dm

) pp0

(∑i

|Bi|pq0

p0p0−p

) p0−p

p0 ≤(∫∆

Jfp0 dm

) pp0

(∑i

|Bi|pq0

p0p0−p

) p0−p

p0 .

On the other hand, as p0K−1K < 1 < p1+t(K−1)

tK , it follows thatp0

p0 − p> 1 + t(K − 1). Combining this with Corollary 3.4 (or 1.2) yields

∑i

|fBi|p ≤M(∑

i

|Bi|pq0

(1+t(K−1))) 11+t(K−1) ,

where M depends only on p0 and K. Since by (19) also t <p

q0(1+ t(K−1)), the claim

follows.

14

Proof of Corollary 1.3: If f : Ω → Ω′ is K−quasiconformal, let E ⊂ Ω be a compactsubset with dim(E) < 2. Choose also a number 1

2dim(E) < t ≤ 1 and cover E bysquares Bi with pairwise disjoint interiors.

According to [LV], Theorems III.8.1 and III.9.1, dia(fBi)2 ≤ C0|fBi|, where theconstant C0 depends only on K,E and Ω. Hence we conclude from Lemma 4.1 that

∑i

dia(fBi)δ ≤ C1

(∑i

dia(Bi)2t) 1

1+t(K−1) , δ >2tK

1 + t(K − 1).

With a proper choice of the covering Bi the sum on the right hand side can be madearbitrarily small and thus dim(fE) ≤ δ. Consequently,

dim(fE) ≤ 2K dim(E)2 + (K − 1) dim(E)

(20)

which proves the corollary.

In the special case of K−quasicircles Γ, the images of S1 under global K−quasi-conformal mappings, Corollary 1.3 reads as

dim(Γ) ≤ 1 +(K − 1K + 1

)= 2− 2

K + 1.

This sharpens recent results due to Jones-Makarov [JM] and Becker-Pommerenke [BP].On the other hand, Becker and Pommerenke showed that if the dilatation K ∼ 1, then

1 + 0.09(K−1K+1

)2≤ dim(Γ) ≤ 1 + 37

(K−1K+1

)2. These results suggest the following

Question 4.2 If Γ is a K−quasicircle, is it true that

dim(Γ) ≤ 1 +(K − 1K + 1

)2.

In the positive case, is the bound sharp ?

Let us next show that the equality can occur in (20) for any value of K and dim(E).Note first that in terms of the holomorphic motions Corollary 1.3 obtains the followingform.

Corollary 4.3 . Let Φ : ∆ × E → C be a holomorphic motion of a set E ⊂ C andwrite d(λ) = dim(Φλ(E)). Then

d(λ) ≤ 2d(0)

(2− d(0))1−|λ|1+|λ| + d(0)

. (21)

Proof: By Slodkowski’s extended λ−lemma Φλ is a restriction of a K−quasiconformalmapping of C, K ≤ 1+|λ|

1−|λ| , and hence the claim follows from 1.3.

For the converse, we start by constructing holomorphic motions of Cantor sets suchthat the equality holds in (21) up to a given ε > 0. Thus for each, say, n ≥ 10 find

15

disjoint disks B(zi, r) ⊂ ∆ all of the same radius r = rn, such that 12 ≤ nr2 ≤ 1. If

0 < t < 2, let alsoβ(t) = log(n1/tr). (22)

For n large enough, β(t) > 0 and

t

2≤ log r

log r − β(t)≤ t

2+ ε. (23)

Set then at(λ) = exp(−β(t)1−λ1+λ). Clearly at is holomorphic in ∆ with at(∆) =

∆\0. Therefore we can consider the holomorphic family of similarities

γi,λ(z) = rat(λ)z + zi.

Since the disks γi,λ∆ ⊂ B(zi, r) are disjoint, the similarities γi,λ generate Cantor sets(gλ, J(gλ)) as in Section 2. Furthermore, the derivatives |γ′i,λ| do not depend on i and sothe dimension d(λ) = dim(J(gλ)) is determined from the equation n(r|at(λ)|)d(λ) = 1.By (22) n(r|at(0)|)t = 1 and therefore

d(0) = t.

Similarly, if 0 < λ < 1, it follows from (23) that

d(0)d(λ)

=log(r|at(λ)|)log(r|at(0)|)

=log r − β(t)1−λ

1+λ

log r − β(t)

≤ d(0)2

+(1− d(0)

2

)1− λ

1 + λ+ ε. (24)

Proof of Theorem 1.4: Choose a countable collection Bk∞1 of pairwise disjointsubdisks of ∆ and define, using the argument above, in each disk Bk a holomorphicmotion Φ of a Cantor set Jk with Φλ(Jk) ⊂ Bk. If d(λ) = dim(Φλ(Jk)), we may assumethat d(0) = t and that for each k (24) holds with ε = 1

k .Clearly this construction determines a holomorphic motion Ψ of the union J =⋃

k Jk. Writing still d(λ) = dim(Ψλ(J)) we have

d(λ) =2d(0)

(2− d(0))1−λ1+λ + d(0)

, 0 ≤ λ < 1.

Now Slodkowski’s generalized λ−lemma applies and Ψ extends to a K−quasiconformalmapping f of C, where K = 1+λ

1−λ , 0 ≤ λ < 1. In other words, if E = J , then dim(E) = tand dim(fE) = (2Kdim(E))/(2 + (K − 1)dim(E)).

Finally, Corollary 1.5 is an immediate consequence of 1.3 and 1.4 since K−quasireg-ular mappings f can be factored as f = φ g, where φ is holomorphic and g K−quasi-conformal; for holomorphic φ sets E with dim(E) < 1 are removable by Painleve’stheorem while those with dim(E) > 1 are never removable [Ga, III. 4.5].

16

Therefore in considering the removability questions for K−quasiregular mappings,the dimension dK = 2

K+1 is the border-line case and there we have the Iwaniec-Martinconjecture that all sets of zero Hausdorff dK−measure are removable. More generally, itis natural to ask whether the precise bound on the dimension dim(fE) ≤ 2K dim(E)

2+(K−1) dim(E)given by Corollary 1.3 is still correct on the level of measures.

Question 4.4 Let 0 < τ < 2 and δ = δK(τ) = 2Kτ2+τ(K−1) . If f is a planar K−quasi-

conformal mapping, is it true that

Hτ (E) = 0 ⇒ Hδ(fE) = 0.

If not, what is the optimal Hausdorff measure Hh or measure function h such thatf∗Hh Hτ ?

5 Estimates for the Beurling-Ahlfors operator

As we saw earlier quasiconformal mappings have important connections to the singu-lar integrals and in particular to the Beurling-Ahlfors operator, the complex Hilberttransform

Sω(z) = − 1π

∫C

ω(ζ) dm(ζ)(ζ − z)2

.

There are even higher dimensional counterparts, see [IM1] and the references there.In fact, many properties of the S operator can be reduced to the distortion results

of quasiconformal mappings. We shall here consider only the operation of S on thefunction space L logL and refer to the work of Iwaniec and Kosecki [IK] for furtherresults.

In case of the characteristic functions ω = χE we have then by Corollary 1.7 that∫B

|SχE | dm ≤ |E| logα|B||E|

(25)

for all Borel subsets E of a disk B ⊂ C; the constant α does not depend on E or B.This translates also to the L∞ setting:

Corollary 5.1 Let B ⊂ C be a disk. If ω is a measurable function such that |ω(z)| ≤χB (z) a.e. then

|z ∈ B : |<Sω(z)| > t| ≤ 2α|B|e−t. (26)

Proof: Let E+ = z ∈ B : <Sω > t. Since S has a symmetric kernel,

t|E+| ≤ <∫E+

Sω dm = <∫B

ωSχE+dm ≤ |E+| log

α|B||E+|

by (25). Thus |E+| ≤ α|B|e−t and since by the same argument E− = z ∈ B : <Sω <−t satisfies |E−| ≤ α|B|e−t, the inequality (26) follows.

17

The estimate (26) is sharp since for ω = (z/z)χ∆(z) we have

Sω = (1 + 2 log |z|)χ∆(z).

For the modulus |Sω| Iwaniec and Kosecki [IK, proposition 12] have shown that (25)implies

|z ∈ B : |Sω(z)| > t| ≤ α(1 + 19t)|B|e−t. (27)

It remains open if the linear term 19t can be replaced by a constant.

Corollary 5.2 For each δ > 1 there is a constant M(δ) <∞ such that∫B

|Sv| dm ≤ δ

∫B

|v(z)| log(1 +M(δ)

|v(z)||v|B

)dm(z)

whenever the right hand side is finite.

Proof: Let ω be a function, unimodular in B and vanishing in C\B, such that∫B

|Sv| dm =∫B

ωSv dm = |v|B∫B

Sωv

|v|Bdm.

We apply then the elementary inequality ab ≤ a log(1+a)+exp(b)−1. Since accordingto (27) ∫

B

e|Sω|/δ − 1 dm ≤ |B| α

δ − 1(1 +

19δδ − 1

) = M1(δ)|B|,

it follows that ∫B

|Sv| dm ≤M1(δ)∫B

|v| dm + δ

∫B

|v| log(1 + δ

|v||v|B

)dm.

Define now E0 = z ∈ B : |v(z)| < 1e |v|B. As t 7→ t log 1

t is increasing on (0, 1e ),

−e∫E0|v| log( |v|

|v|B ) dm ≤ |v|B|E0| ≤∫B |v| dm, where we use the convention 0 log 0 = 0.

Thus∫B

|v| log(1 + δ

|v||v|B

)dm ≤

∫B\E0

|v| log( |v||v|B

(e+ δ))dm +

∫E0

|v| log(1 +δ

e) dm

≤∫B

|v| log(M2

|v||v|B

)dm, (28)

where M2 = e2 + eδ. In conclusion, if M = M2 exp(M1(δ)),∫B

|Sv| dm ≤ δ

∫B

|v| log(M

|v||v|B

)dm ≤ δ

∫B

|v| log(1 +M

|v||v|B

)dm,

which completes the estimation.

18

Since the variational inequality (16),

n∑i=1

pi log|fBi|pi

≤ 1K

n∑i=1

pi log|Bi|pi

+ C(K)

with C(K) = O(K−1) and f |∪Bi conformal, was the key in the area distortion Theorem1.1 it is of interest to know whether the inequality is valid without any conformalityassumptions. Another natural question is whether Corollary 5.2 still holds at δ = 1;for characteristic functions this is true and (25) with [IK, proposition 19] implies thatfor nonnegative functions v,∫

B\E

|Sv| dm ≤∫E

|v(z)| log(1 + α

|v(z)||v|B

)dm(z), if supp(v) ⊂ E.

Indeed, it can be shown that these two questions are equivalent (if v ≥ 0 in Corollary5.2). However, it turns out that the answer to them is the negative. We omit here theproof of the equivalence; instead we give first a simple counterexample to the generalvariational inequality and then show how this reflects in the L logL estimates of thecomplex Hilbert transform.

Example 5.3 Choose 0 < ρ < 1 and for 1 ≤ i ≤ n consider the disjoint disks Bi =B(ρi, aρi) where 0 < a < 1−ρ

1+ρ . Let also pi = 1n and f0(z) = z|z|

1K−1. Then

n∑i=1

pi log|Bi|pi

= (n+ 1) log ρ+ log n+ log πa2

whilen∑i=1

pi log|f0Bi|pi

≥ n+ 1K

log ρ+ log n+ C0 =1K

n∑i=1

pi log|Bi|pi

+K − 1K

log n+ C1,

where C0, C1 depend only on K and a. Letting n → ∞ shows that the variationalinequality fails for f0.

Proposition 5.4 For each M < ∞ there is an ε > 0 and a nonnegative functionv ∈ L logL(∆) such that∫

∆

|Sv| dm > (1 + ε)∫∆

|v(z)| log(1 +M

|v(z)||v|∆

)dm(z).

Proof: By inequality (28) it suffices to show that for no M <∞ does∫∆

|Sv| dm ≤∫∆

|v(z)| log(M|v(z)||v|∆

)dm(z) (29)

hold for all nonnegative functions v ∈ L logL(∆).

19

We argue by contradiction. Hence consider first the mapping f(z) = z|z|K−1 andimbedd it to a one parameter family of quasiconformal mappings ht : ∆ → ∆, as inthe proof of Lemma 3.3. Thus for t = logK, ht = f.

Suppose next that we have disjoint open sets Din1 ⊂ ∆ and numbers pi > 0 with∑n1 pi = 1. Set then

vt(z) =n∑i=1

pi|htDi|

χhtDi

(z).

Clearly∫∆ vt dm = 1 and if ψ(t) =

∫∆ vt log(Mvt) dm then by Jensen’s inequality

ψ(t) ≥ 0 for M ≥ π. Furthermore, we can deduce from the Gehring-Reich identity (17)that

ψ′(t) = −n∑i=1

pi|htDi|

d

dt|htDi| = −

∫∆

φSv dm− c(t)

where |c(t)| is uniformly bounded. Thus if (29) holds, then ψ′(t) ≤ ψ(t) − c(t) andafter integration ψ(t) ≤ etψ(0)− et

∫ t0 c(s)e

−sds = etψ(0) + c1(t). Taking t = logK weobtain

n∑i=1

pi log( pi|fDi|

)≤ K

n∑i=1

pi log( pi|Di|

)+ C(K),

where C(K) = (K − 1) logM + c1(logK).Finally, if Bi, pi are as in the previous example with f−1(z) = f0(z) = z|z|

1K−1, we

can choose Di = f0Bi. But this would mean that

n∑i=1

pi log|f0Bi|pi

≤ 1K

n∑i=1

pi log|Bi|pi

− C(K)K

,

contradicting Example 5.3. Therefore (29) cannot hold and so the estimate of Corollary5.2 is sharp.

REFERENCES

[A] Ahlfors, L., On quasiconformal mappings. J. Analyse Math., 3 (1954), 1-58.[AB] Ahlfors, L. & Bers L., Riemann’s mapping theorem for variable metrics. Ann.

Math., 72 (1960), 385-404.[AM] Astala, K. & Martin, G., Holomorphic Motions. Preprint, 1992.[BP] Becker, J. & Pommerenke Chr., On the Hausdorff Dimension of Quasicircles.

Ann. Acad. Sci. Fenn. Ser. A I Math., 12 (1987), 329-333.[Bj] Bojarski, B., Generalized solutions of a system of differential equations of first

order and elliptic type with discontinuous coefficients. Math. Sb., 85 (1957), 451-503.[Bw] Bowen, R., Equilibrium States and the Ergodic Theory of Anosov Diffeomor-

phisms. Lecture Notes in Math., 470. Springer-Verlag, New York-Heidelberg, 1975.[DH] Douady, A. & Hubbard J., On the dynamics of polynomial-like mappings.

Ann. Sci. Ec. Norm. Sup., 18 (1985), 287-345.

20

[Ga] Garnett, J. Analytic Capacity and Measure. Lecture Notes in Math., 297.Springer-Verlag, New York-Heidelberg, 1972.

[Ge] Gehring, F.W. Open problems. Proc.Romanian-Finnish Seminar on Te-ichmuller Spaces and quasiconformal Mappings, Romania (1969), p. 306.

[GR] Gehring, F.W. & Reich E., Area distortion under quasiconformal mappings.Ann. Acad. Sci. Fenn. Ser. A. I.,388 (1966), 1-14.

[GV] Gehring, F.W. & Vaisala J., Hausdorff dimension and quasiconformal map-pings. J. London Math. Soc., 6 (2) (1975), 504-512.

[H] Hutchinson, J.E., Fractals and self-similarity. Indiana Univ. Math. J., 30(1981), 713-747.

[I] Iwaniec, T., Lp−theory of quasiregular mappings. Lecture Notes in Math., 1508.Springer-Verlag, New York-Heidelberg, 1992, pp. 39-64.

[IK] Iwaniec, T. & Kosecki R., Sharp estimates for complex potentials and quasi-conformal mappings. Preprint of Syracuse University.

[IM1] Iwaniec, T. & Martin G., Quasiregular mappings in even dimensions. ActaMath., 170 (1992), 29-81.

[IM2] Iwaniec, T. & Martin G., Quasiconformal mappings and capacity. IndianaUniv. Math. J., 40 (1991), 101-122.

[JM] Jones, P. & Makarov, N., Manuscript.[JV] Jarvi, P. & Vuorinen M., Self-similar Cantor sets and quasiregular mappings.

J. reine angew. Math., 424 (1992), 31-45.[K] Koskela, P., The degree of regularity of a quasiconformal mapping. To appear

in Proc. Amer. Math. Soc.[KM] Koskela, P. & Martio O., Removability theorems for quasiregular mappings.

To appear in Ann. Acad. Sci. Fenn. Ser. A I Math.[LV] Lehto O. & Virtanen K., Quasiconformal mappings in the plane. Second

edition. Springer-Verlag, New York-Heidelberg, 1973.[L] Lehto O., Univalent functions and Teichmuller spaces. Springer–Verlag, New

York-Heidelberg, 1987.[MSS] Mane R., Sad P. & Sullivan D., On the dynamics of rational maps. Ann.

Sci. Ecole Norm. Sup., 16 (1983), 193-217.[Ma] Manning, A., The dimension of the maximal measure for a polynomial map.

Ann. Math., 119 (1984), 425-430.[Mo] Mori, A., On an absolute constant in the theory of quasiconformal mappings.

J. Math. Soc. Japan, 8 (1956), 156-166.[P] Pommerenke, Chr., Univalent Functions. Vandenhoeck & Ruprecht, Gottingen,

1975.[Ri] Rickman, S., Nonremovable Cantor sets for bounded quasiregular mappings.

To appear in Ann. Acad. Sci. Fenn. Ser. A I Math.[Ru] Ruelle, D., Repellers for real analytic maps. Erg. Th. and Dynam. Sys., 2

(1982), 99-107.[Sl] Slodkowski Z., Holomorphic motions and polynomial hulls. Proc. Amer. Math.

Soc., 111 (1991), 347–355.

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[Su] Sullivan D., Quasiconformal homeomorphisms and dynamics I, II, Annals ofMath., 122 (1985), 401–418, II Acta Math., 155 (1985), 243–260.

[W] Walters, P., An Introduction to Ergodic Theory. Springer-Verlag, New York-Heidelberg, 1982.

The University of HelsinkiP.O. Box 4 (Hallituskatu 15)

SF-00014 HelsinkiFinland

Astala@cc.helsinki.fi

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