Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 29, 2004, 121–138

DAVID MAPS AND HAUSDORFF DIMENSION

Saeed Zakeri

Stony Brook University, Institute for Mathematical Sciences Stony Brook, NY 11794-3651, U.S.A.; zakeri@math.sunysb.edu

Abstract. David maps are generalizations of classical planar quasiconformal maps for which the dilatation is allowed to tend to infinity in a controlled fashion. In this note we examine how these maps distort Hausdorff dimension. We show:

– Given α and β in [0, 2], there exists a David map φ: C → C and a compact set Λ such that dim_{H} Λ = α and dim_{H} φ(Λ) = β.

– There exists a David map φ: C → C such that the Jordan curve Γ = φ(S^{1}) satisfies dim_{H} Γ = 2. One should contrast the first statement with the fact that quasiconformal maps preserve sets

of Hausdorff dimension 0 and 2. The second statement provides an example of a Jordan curve with Hausdorff dimension 2 which is (quasi)conformally removable.

1. Introduction

An orientation-preserving homeomorphism φ: U → V between planar domains is called quasiconformal if it belongs to the Sobolev class W_{loc}^{1,1}(U) (i.e., has locally integrable distributional partial derivatives in U) and its complex dilatation μ_{φ} := ¯∂φ/∂φ satisfies

∥μ_{φ}∥_{∞} < 1.

In terms of the real dilatation defined by

K_{φ} := \frac{1 + |μ_{φ}|}{1 − |μ_{φ}|} = \frac{|∂φ| + |¯∂φ|}{|∂φ| − |¯∂φ|},

the above condition can be expressed as

∥K_{φ}∥_{∞} < +∞.

The quantity ∥K_{φ}∥_{∞} is called the maximal dilatation of φ. We say that φ is K -quasiconformal if its maximal dilatation does not exceed K.

For later comparison with the properties of David maps defined below, we recall some basic properties of quasiconformal maps (see [A] or [LV]):

– If φ is K -quasiconformal for some K ≥ 1, so is the inverse map φ^{−1} .

2000 Mathematics Subject Classification: Primary 30C62.

– A K -quasiconformal map φ: U → V is locally Hölder continuous of exponent 1/K. In other words, for every compact set E ⊂ U and every z, w ∈ E,

|φ(z) − φ(w)| ≤ C|z − w|^{1/K}

where C > 0 only depends on E and K.

– A quasiconformal map φ: U → V is absolutely continuous; in fact, the Jacobian J_{φ} = |∂φ|^{2} − |¯∂φ|^{2} is locally integrable in U and

(1.1)  area φ(E) = ∫_{E} J_{φ} dx dy,

for every measurable set E ⊂ U.

– More precisely, the Jacobian J_{φ} of a quasiconformal map φ: U → V is in L_{loc}^{p}(U) for some p > 1. If we define

(1.2) p(K) := sup{p : J_{φ} ∈ L_{loc}^{p}(U) for every K-quasiconformal map φ in U},

then

(1.3)  p(K) = \frac{K}{K − 1}.

(In particular, p(K) is independent of the domain U.) This was conjectured by Gehring and Väisälä in 1971 [GV] and proved by Astala in 1994 [As].

– Let {φ_{n}} be a sequence of K -quasiconformal maps in a planar domain U which fix two given points of U. Then {φ_{n}} has a subsequence which converges locally uniformly to a K -quasiconformal map in U. The measurable Riemann mapping theorem of Morrey–Ahlfors–Bers [AB] as-

serts that every measurable function μ in a domain U which satisfies ∥μ∥_{∞} < 1 is the complex dilatation of some quasiconformal map φ in U, which means φ satisfies the Beltrami equation ¯∂φ = μ · ∂φ almost everywhere in U. Recent progress in holomorphic dynamics has made it abundantly clear that one must also study this equation in the case ∥μ∥_{∞} = 1. With some restrictions on the asymptotic growth of |μ|, the solvability of the Beltrami equation can still be guaranteed. One such condition is given by David in [D]. Let σ denote the spherical area in Ĉ and μ be a measurable function in U which satisfies

(1.4)  σ\{z ∈ U : |μ(z)| > 1 − ε\} ≤ C exp(−\frac{t}{ε})  for all ε < ε_{0}

for some positive constants C, t, ε_{0} . Then David showed that the Beltrami equation ¯∂φ = μ · ∂φ has a homeomorphic solution φ ∈ W_{loc}^{1,1}(U) which is unique up to postcomposition with a conformal map (see also [BJ] for a more geometric approach which gives a stronger theorem). Motivated by this result, we call a homeomorphism φ: U → V a David map if φ ∈ W_{loc}^{1,1}(U) and the complex dilatation μ_{φ} satisfies a condition of the form (1.4). Equivalently, φ is a David map if there are positive constants C, t, K_{0} such that its real dilatation satisfies

(1.5)  σ{z ∈ U : K_{φ}(z) > K} ≤ Ce^{−tK}  for all K > K_{0}.

To emphasize the values of these constants, sometimes we say that φ is a (C, t, K_{0})- David map. Note that when U is a bounded domain in C, the spherical metric in (1.4) or (1.5) can be replaced with the Euclidean area.

David maps enjoy some of the useful properties of quasiconformal maps, but the two classes differ in many respects. As indications of their similarity, let us mention the following two facts:

– Every David map is absolutely continuous; the Jacobian formula (1.1) still holds.

– Tukia’s theorem [T]. “Let C, t, K_{0} be positive and suppose {φ_{n}} is a sequence of (C, t, K_{0})-David maps in a domain U which fix two given points of U. Then {φ_{n}} has a subsequence which converges locally uniformly to a David map in U.” It is rather easy to show that some subsequence of {φ_{n}} converges locally uniformly to a homeomorphism, but the fact that this homeomorphism must be David is quite non-trivial. We remark that the parameters of the limit map may a priori be different from C, t, K_{0} . Here are further properties of David maps which indicate their difference with

quasiconformal maps:

– The inverse of a David map may not be David.

– A David map may not be locally Hölder.

– The Jacobian of a David map may not be in L_{loc}^{p}(U) for any p > 1. As an example, the homeomorphism φ: D(0, 1/e) → D defined by

φ(re^{iθ}) := −\frac{1}{log r} e^{iθ}

is a David map but φ^{−1} is not. Moreover, φ is not Hölder in any neighborhood of 0, and J_{φ} ∉ L_{loc}^{p} for p > 1.

The main goal of this note is to show how David maps differ from quasiconformal maps in the way they distort Hausdorff dimension of sets. Recall that the Hausdorff s-measure of E ⊂ C is defined by

H^{s}(E) := lim_{ε→0} inf_{𝒰} ∑_{i}(diam U_{i})^{s},

where the infimum is taken over all countable covers 𝒰 = {U_{i}} of E by sets of Euclidean diameter at most ε. The Hausdorff dimension of E is defined by

dim_{H} E := inf{s : H^{s}(E) = 0}.

Quasiconformal maps can change Hausdorff dimension of sets only by a bounded factor depending on their maximal dilatation. This was first proved by Gehring and Väisälä [GV] who showed that if φ: U → V is K -quasiconformal, E ⊂ U, dim_{H} E = α and dim_{H} φ(E) = β, then

\frac{2(p(K) − 1)α}{2p(K) − α} ≤ β ≤ \frac{2p(K)α}{2(p(K) − 1) + α}.

Here p(K) > 1 is the constant defined in (1.2). By Astala’s result (1.3), one obtains

\frac{2α}{2K − (K − 1)α} ≤ β ≤ \frac{2Kα}{2 + (K − 1)α}

which can be put in the symmetric form

(1.6)  \frac{1}{K} (\frac{1}{α} − \frac{1}{2}) ≤ \frac{1}{β} − \frac{1}{2} ≤ K(\frac{1}{α} − \frac{1}{2}).

It follows in particular that quasiconformal maps preserve sets of Hausdorff dimension 0 and 2.

By contrast, we prove

Theorem A. Given any two numbers α and β in [0, 2], there exists a David map φ: C → C and a compact set Λ ⊂ C such that dim_{H} Λ = α and dim_{H} φ(Λ) = β.

The proof shows that the parameters of φ can be taken independent of α and β.

In the special case of a K -quasicircle, i.e., the image Γ of the round circle under a K -quasiconformal map, the estimate (1.6) gives

1 ≤ dim_{H} Γ ≤ \frac{2K}{K + 1}

(the lower bound comes from topological considerations). It is well known that dim_{H} Γ can in fact take all values in [1, 2). We show that the upper bound 2 is attained by a David image of the round circle. Let us call a Jordan curve Γ ⊂ C a David circle if there exists a David map φ: C → C such that Γ = φ(S^{1}), where S^{1} is the unit circle {z ∈ C : |z| = 1}.

Theorem B. There exist David circles of Hausdorff dimension 2.

One corollary of this result is that there are Jordan curves of Hausdorff dimension 2 that are (quasi)conformally removable (see Section 4).

Both results are bad (or exciting?) news for applications in holomorphic dynamics, where one often wants to estimate the Hausdorff dimension of invariant sets by computing the dimension in a conjugate dynamical system. The dichotomy of having dimension < 2 or = 2 for such invariant sets, which is respected by quasiconformal conjugacies, is no longer preserved by David conjugacies. For example, by performing a quasiconformal surgery on a Blaschke product, Petersen proved that the Julia set of the quadratic polynomial Q_{θ}: z ↦ e^{2πiθ}z + z^{2} is locally-connected and has measure zero whenever θ is an irrational of bounded type [P]. In this case, the boundary of the Siegel disk of Q_{θ} is a quasicircle whose Hausdorff dimension is strictly between 1 and 2 (compare [GJ]). On the other hand, by performing a trans-quasiconformal surgery and using David’s theorem, Petersen and the author extended the above result to almost every θ [PZ]. It follows that there exists a full-measure set of rotation numbers θ for which the boundary of the Siegel disk of Q_{θ} is a David circle but not a quasicircle. Thus, Theorem B opens the possibility that this boundary alone might have dimension 2, which would be a rather curious phenomenon.

2. Preliminary constructions

For two positive numbers a and b, we write

a ≼ b

if there is a universal constant C > 0 such that a ≤ Cb. We write

a ≍ b

if a ≼ b and b ≼ a, i.e., if there is a universal constant C > 0 such that C^{−1}b ≤ a ≤ Cb. In this case, we say that a and b are comparable.

A family of Cantor sets. Given a strictly decreasing sequence d = {d_{n}}_{n≥0} of positive numbers with d_{0} = 1, we construct a Cantor set Λ(d) as the intersection of a nested sequence {Λ_{n}}_{n≥0} of compact sets in the unit square Λ_{0} := [−\frac{1}{2}, \frac{1}{2}] × [−\frac{1}{2}, \frac{1}{2}] defined inductively as follows. Set a_{1} := 2^{−2}(d_{0} − d_{1}) and define Λ_{1} as the disjoint union of the four closed squares of side-length 2^{−1}d_{1} in Λ_{0} which have distance a_{1} to the boundary of Λ_{0} (see Figure 1). Suppose Λ_{n−1} is constructed for some n ≥ 2 so that it is the disjoint union of 4^{n−1} closed squares of side-length 2^{−(n−1)}d_{n−1} . Define

(2.1)  a_{n} := 2^{−(n+1)}(d_{n−1} − d_{n}).

For any square S in Λ_{n−1} , consider the disjoint union of the four closed squares in S of side-length 2^{−n}d_{n} which have distance a_{n} to the boundary of S. The union of all these squares for all such S will then be called Λ_{n}. Clearly Λ_{n} is

Figure 1. First two steps in the construction of Λ(d).

the disjoint union of 4^{n} closed squares of side-length 2^{−n}d_{n}, and the inductive definition is complete.

The Cantor set Λ(d) is defined as ⋂_{n≥0} Λ_{n}. We have

area Λ(d) = lim_{n→∞} area Λ_{n} = lim_{n→∞} d_{n}^{2}.

Lemma 2.1. The Hausdorff dimension of the Cantor set Λ = Λ(d) satisfies

(2.2) 2 − lim sup_{n→∞} \frac{−2 log d_{n+1}}{− log d_{n} + n log 2} ≤ dim_{H} Λ ≤ 2 − lim inf_{n→∞} \frac{−2 log d_{n}}{− log d_{n} + n log 2}.

Proof. For each n ≥ 0, there are 4^{n} squares of diameter 2^{(1/2)−n}d_{n} covering Λ. Hence the Hausdorff s-measure of Λ is bounded above by

lim inf_{n→∞} 4^{n}(2^{(1/2)−n}d_{n})^{s} = 2^{s/2} lim inf_{n→∞} 2^{n(2−s)}d_{n}^{s},

which is zero if s > 2 − lim inf_{n→∞}(−2 log d_{n})/(− log d_{n} + n log 2). This proves the upper bound in (2.2).

The lower bound follows from a standard mass distribution argument: Construct a probability measure μ on Λ which gives equal mass 4^{−n} to each square in Λ_{n}, so that

μ(S) = \frac{area (S)}{d_{n}^{2}} if S is a square in Λ_{n}.

Let x ∈ Λ and ε > 0, and choose n so that 2^{−n}d_{n} < ε ≤ 2^{−(n−1)}d_{n−1} . The disk D(x, ε) intersects at most πε^{2}/(4^{−n}d_{n}^{2}) squares in Λ_{n} each having μ-mass of 4^{−n}. It follows that

μ(D(x, ε)) ≼ \frac{ε^{2}}{d_{n}^{2}} = ε^{s} \frac{ε^{2−s}}{d_{n}^{2}} ≼ ε^{s} \frac{2^{−n(2−s)}d_{n−1}^{2−s}}{d_{n}^{2}} .

If s < 2−lim sup_{n→∞}(−2 log d_{n+1})/(− log d_{n}+n log 2), the term 2^{−n(2−s)}d_{n−1}^{2−s}/d_{n}^{2} will tend to zero as n → ∞, so that

μ(D(x, ε)) ≼ ε^{s}.

It follows from Frostman’s lemma (see for example [M]) that dim_{H} Λ ≥ s. This gives the lower bound in (2.2).

Standard homeomorphisms between Cantor sets. We construct standard homeomorphisms with controlled dilatation between Cantor sets of the form Λ(d) defined above. The construction will depend on the following lemma:

Lemma 2.2. Fix 0 < a ≤ b < \frac{1}{2} . Let A_{a} be the closed annulus bounded by the squares

\{(x, y) ∈ R^{2} : max\{|x|, |y|\} = \frac{1}{2}\} and \{(x, y) ∈ R^{2} : max\{|x|, |y|\} = \frac{1}{2} − a\},

and similarly define A_{b}. Let φ: ∂A_{a} → ∂A_{b} be a homeomorphism which is the identity on the outer boundary component and acts affinely on the inner boundary component, respecting the horizontal and vertical sides. Then φ can be extended to a K -quasiconformal homeomorphism A_{a} → A_{b}, with

(2.3)  K ≍ \frac{b(1 − 2a)}{a(1 − 2b)}.

Proof. Let us first make a simple observation: If z and w are points in the upper half-plane and L: R^{2} → R^{2} is the affine map such that L(0) = 0, L(1) = 1 and L(z) = w (see Figure 2), then the real dilatation of L is given by

(2.4)  K_{L} = \frac{|z − \overline{w}| + |z − w|}{|z − \overline{w}| − |z − w|}.

To prove the lemma, take the triangulations of A_{a} and A_{b} shown in Figure 2 and extend φ affinely to each triangle. After appropriate rescaling, it follows from (2.4) that on a triangle of type I in the figure, the dilatation of φ is comparable to b/a, while on a triangle of type II, the dilatation of φ is comparable to b(1 − 2a)/(a(1 − 2b)). Since b(1 − 2a)/(a(1 − 2b)) ≥ b/a, we obtain (2.3).

Now take a decreasing sequence d = {d_{n}} of positive numbers with d_{0} = 1, let {a_{n}} be defined as in (2.1), and consider the Cantor set Λ(d) = ⋂ Λ_{n}. Take another such sequence d′ = {d′_{n}} and let a′_{n}, Λ′_{n}, Λ(d′) denote the corresponding data. We construct a homeomorphism φ: C → C which maps the Cantor set Λ = Λ(d) to Λ′ = Λ(d′). This φ is the uniform limit of a sequence of quasiconformal maps φ_{n}: C → C with φ_{n}(Λ_{n}) = Λ′_{n}, defined inductively as follows. Let φ_{0} be the identity map on C. Suppose φ_{n−1} is constructed for some n ≥ 1 and

Figure 2.

that it maps each square in Λ_{n−1} affinely to the corresponding square in Λ′_{n−1} . Define φ_{n} = φ_{n−1} on C \ Λ_{n−1} and let φ_{n} map each square in Λ_{n} affinely to the corresponding square in Λ′_{n}. The remaining set Λ_{n−1} \ Λ_{n} is the union of 4^{n} annuli on the boundary of which φ_{n} can be defined affinely. By rescaling each annulus in Λ_{n−1} \ Λ_{n} and the corresponding annulus in Λ′_{n−1} \ Λ′_{n}, we are in the situation of Lemma 2.2, so we can extend φ_{n} in a piecewise affine fashion to each such annulus. This defines φ_{n} everywhere, and the inductive definition is complete.

To estimate the maximal dilatation of φ_{n}, note that by the above construction φ_{n} is conformal in Λ_{n} and has the same dilatation as φ_{n−1} on C \ Λ_{n−1} . On each of the 4^{n} annuli in Λ_{n−1} \ Λ_{n}, the dilatation of φ_{n} can be estimated using (2.3) in Lemma 2.2. In fact, rescaling each such annulus by a factor 2^{n}/d_{n−1} and the corresponding annulus in Λ′_{n−1} \ Λ′_{n} by a factor 2^{n}/d′_{n−1} , it follows from (2.3) that the dilatation of φ_{n} on each such annulus is comparable to

max \{ \frac{\frac{a′_{n}}{2^{−n}d′_{n−1}} (1 − 2\frac{a_{n}}{2^{−n}d_{n−1}} )}{\frac{a_{n}}{2^{−n}d_{n−1}} (1 − 2\frac{a′_{n}}{2^{−n}d′_{n−1}} )}, \frac{\frac{a_{n}}{2^{−n}d_{n−1}} (1 − 2\frac{a′_{n}}{2^{−n}d′_{n−1}} )}{\frac{a′_{n}}{2^{−n}d′_{n−1}} (1 − 2\frac{a_{n}}{2^{−n}d_{n−1}} )}\}

= max\{\frac{a′_{n}(d_{n−1} − 2^{n+1}a_{n})}{a_{n}(d′_{n−1} − 2^{n+1}a′_{n})}, \frac{a_{n}(d′_{n−1} − 2^{n+1}a′_{n})}{a′_{n}(d_{n−1} − 2^{n+1}a_{n})}\}

= max\{\frac{a′_{n}d_{n}}{a_{n}d′_{n}} , \frac{a_{n}d′_{n}}{a′_{n}d_{n}} \}.

To sum up, the construction gives a sequence {φ_{n}} with the following properties:

(i) φ_{n} = φ_{n−1} on C \ Λ_{n−1} . (ii) φ_{n} maps each square in Λ_{n} affinely to the corresponding square in Λ′_{n}. (iii) φ_{n} is K_{n} -quasiconformal, where

(2.5)  K_{n} ≍ max\{K_{n−1}, \frac{a′_{n}d_{n}}{a_{n}d′_{n}} , \frac{a_{n}d′_{n}}{a′_{n}d_{n}} \}

and K_{0} = 1. Evidently, φ := lim_{n→∞} φ_{n} is a homeomorphism which agrees with φ_{n} on C \ Λ_{n} for every n and satisfies φ(Λ) = Λ′. We call this φ the standard homeomorphism from Λ to Λ′. Observe that by the construction, the inverse map φ^{−1} is the standard homeomorphism from Λ′ to Λ.

3. Proof of Theorem A

We are now ready to prove Theorem A cited in Section 1.

Proof of Theorem A. If 0 < α, β < 2, it is well known that there is a K - quasiconformal map φ: C → C mapping a set of dimension α to a set of dimension β (see for example [GV]). Moreover, by (1.6), the minimum K this would require is

max \{ \frac{\frac{1}{β} − \frac{1}{2}}{\frac{1}{α} − \frac{1}{2}} , \frac{\frac{1}{α} − \frac{1}{2}}{\frac{1}{β} − \frac{1}{2}} \} .

In what follows we consider the remaining cases where α and β are distinct and at least one of them is 0 or 2.

Consider the sequences d = {d_{n}}, d′ = {d′_{n}} and d′′ = {d′′_{n}} defined by

d_{n} := 2^{−n/log n},  d′_{n} := 2^{−νn},  d′′_{n} := 2^{−n log n},

where ν > 0, and construct the Cantor sets Λ = Λ(d), Λ′ = Λ(d′) and Λ′′ = Λ(d′′) as in Section 2. By Lemma 2.1,

dim_{H}(Λ) = 2,  dim_{H}(Λ′) = \frac{2}{ν + 1},  dim_{H}(Λ′′) = 0.

We prove that the standard homeomorphisms between these three Cantor sets and their inverses are all David maps; this will prove the theorem. In view of Tukia’s theorem quoted in Section 1, it suffices to check that the sequence of approximating homeomorphisms are David maps with uniform parameters (C, t, K_{0}). In fact, the estimates below show that we can always take C = t = 1.

Case 1. Mapping Λ to Λ′. Suppose {φ_{n}} is the sequence of quasiconformal maps which approximates the standard homeomorphism φ from Λ to Λ′. To estimate the dilatation of φ_{n}, note that

(3.1) a_{n} = 2^{−(n+1)}(d_{n−1} −d_{n}) ≍ 2^{−n}(2^{−(n−1)/log(n−1)} −2^{−n/log n}) ≍ \frac{2^{−n−n/log n}}{log n}

and

(3.2)  a′_{n} = 2^{−(n+1)}(d′_{n−1} − d′_{n}) ≍ 2^{−n}(2^{−ν(n−1)} − 2^{−νn}) ≍ 2^{−(ν+1)n}.

Hence

\frac{a′_{n}d_{n}}{a_{n}d′_{n}} ≍ \frac{2^{−(ν+1)n} · 2^{−n/log n}}{\frac{2^{−n−n/log n}}{log n} · 2^{−νn}} ≍ log n.

It follows from (2.5) that there is a sequence 1 < K_{1} < K_{2} < ⋯ < K_{n} < ⋯ with K_{n} ≍ log n such that φ_{n} is K_{n} -quasiconformal. Fix the index n and a number K > 1. Choose j so that K_{j} ≤ K < K_{j+1} . Then

area {z : K_{φ_{n}}(z) > K} ≤ area {z : K_{φ_{n}}(z) > K_{j}} ≤ area (Λ_{j}) = d_{j}^{2} = 4^{−j/log j}.

Since K ≍ K_{j} ≍ log j, we obtain

area {z : K_{φ_{n}}(z) > K} ≤ e^{−K},

provided that K is bigger than some K_{0} independent of n. It follows that the φ_{n} are all (1, 1, K_{0})-David maps.

The inverse maps ψ_{n} := φ_{n}^{−1} are also K_{n} -quasiconformal with the same dilatation K_{n} ≍ log n and they converge uniformly to ψ := φ^{−1} . Moreover, if K_{j} ≤ K < K_{j+1} , then

area {z : K_{ψ_{n}}(z) > K} ≤ area {z : K_{ψ_{n}}(z) > K_{j}}
≤ area (Λ′_{j}) = (d′_{j})^{2} = 4^{−νj} ≤ e^{−K},

provided that K is bigger than some K_{0} independent of n. It follows that the ψ_{n} are all (1, 1, K_{0})-David maps.

Case 2. Mapping Λ′ to Λ′′. The argument here is quite similar to the previous case. We have

(3.3) a′′_{n} = 2^{−(n+1)}(d′′_{n−1} − d′′_{n}) ≍ 2^{−n}(2^{−(n−1) log(n−1)} − 2^{−n log n})

≍ 2^{−n−n log n+log n}.

Hence, using (3.2) and (3.3), we obtain

\frac{a′′_{n}d′_{n}}{a′_{n}d′′_{n}} ≍ \frac{2^{−n−n log n+log n} · 2^{−νn}}{2^{−(ν+1)n} · 2^{−n log n}} ≍ 2^{log n}.

Let {φ_{n}} be the sequence of quasiconformal maps which approximates the standard homeomorphism φ from Λ′ to Λ′′. It follows from (2.5) that there is a sequence 1 < K_{1} < K_{2} < ⋯ < K_{n} < ⋯ with K_{n} ≍ 2^{log n} such that φ_{n} is K_{n} -quasiconformal. Fix the index n and a number K > 1, and choose j so that K_{j} ≤ K < K_{j+1} . Then K ≍ K_{j} ≍ 2^{log j} and

area {z : K_{φ_{n}}(z) > K} ≤ area {z : K_{φ_{n}}(z) > K_{j}}
≤ area (Λ′_{j}) = (d′_{j})^{2} = 4^{−νj} ≤ e^{−K},

provided that K is bigger than some K_{0} independent of n.

The inverse maps ψ_{n} := φ_{n}^{−1} are K_{n} -quasiconformal with K_{n} ≍ 2^{log n} and they converge uniformly to ψ := φ^{−1} . Moreover, if K_{j} ≤ K < K_{j+1} , then

area {z : K_{ψ_{n}}(z) > K} ≤ area {z : K_{ψ_{n}}(z) > K_{j}}
≤ area (Λ′′_{j}) = (d′′_{j})^{2} = 4^{−j log j} ≤ e^{−K},

provided that K is bigger than some K_{0} independent of n.

Case 3. Mapping Λ to Λ′′. Using (3.1) and (3.3), we obtain

\frac{a′′_{n}d_{n}}{a_{n}d′′_{n}} ≍ \frac{2^{−n−n log n+log n} · 2^{−n/log n}}{\frac{2^{−n−n/log n}}{log n} · 2^{−n log n}} ≍ 2^{log n} log n = n^{log 2} log n.

Let {φ_{n}} be the sequence of quasiconformal maps which approximates the standard homeomorphism φ from Λ to Λ′′. It follows then from (2.5) that there is a sequence 1 < K_{1} < K_{2} < ⋯ < K_{n} < ⋯ with K_{n} ≍ n^{log 2} log n such that φ_{n} is K_{n} -quasiconformal. Fix n, let K be sufficiently large, and choose j so that K_{j} ≤ K < K_{j+1} . Then

area {z : K_{φ_{n}}(z) > K} ≤ area {z : K_{φ_{n}}(z) > K_{j}} ≤ area (Λ_{j}) = (d_{j})^{2} = 4^{−j/log j}.

But K ≍ K_{j} ≍ j^{log 2} log j, so

area {z : K_{φ_{n}}(z) > K} ≤ e^{−K},

provided that K is bigger than some K_{0} independent of n.

The inverse maps ψ_{n} := φ_{n}^{−1} are K_{n} -quasiconformal with K_{n} ≍ n^{log 2} log n and they converge uniformly to ψ := φ^{−1} . Moreover, if K_{j} ≤ K < K_{j+1} , then

area {z : K_{ψ_{n}}(z) > K} ≤ area {z : K_{ψ_{n}}(z) > K_{j}}
≤ area (Λ′′_{j}) = (d′′_{j})^{2} = 4^{−j log j} ≤ e^{−K},

provided that K is bigger than some K_{0} independent of n.

Figure 3. Cell decompositions of A and B.

4. Proof of Theorem B

The idea of the proof of Theorem B is to construct a David map φ: C → C which sends a linear Cantor set Σ ⊂ [−\frac{1}{2}, \frac{1}{2}] to a Cantor set of the form Λ(d) with dimension 2. The image φ([−\frac{1}{2}, \frac{1}{2}]) will then be an embedded arc of dimension 2. Since the construction allows φ = id outside the square [−\frac{1}{2}, \frac{1}{2}] × [−\frac{1}{2}, \frac{1}{2}], we can easily complete this arc to a David circle.

A linear Cantor set. Consider the closed unit square Σ_{0} := [−\frac{1}{2}, \frac{1}{2}] × [−\frac{1}{2}, \frac{1}{2}] in the plane. We construct a nested sequence \{Σ_{n}\}_{n≥0} of compact sets whose intersection is a linear Cantor set. For 1 ≤ j ≤ 4, let f_{j}: C → C be the affine contraction defined by

f_{j}(z) = \frac{1}{8}z + \frac{1}{8}(2j − 5),

and set

Σ_{n} := ⋃_{j_{1},...,j_{n}} f_{j_{1}} ∘ ⋯ ∘ f_{j_{n}}(Σ_{0}),

where the union is taken over all unordered n-tuples j_{1}, …, j_{n} chosen from {1, 2, 3, 4}. It is easy to see that Σ_{n} is the disjoint union of 4^{n} closed squares of side-length 8^{−n} with centers on [−\frac{1}{2}, \frac{1}{2}] and sides parallel to the coordinate axes (compare Figure 5 left). We define the Cantor set Σ as the intersection ⋂_{n=0}^{∞} Σ_{n}. Evidently, Σ is a subset of [−\frac{1}{2}, \frac{1}{2}] which has linear measure zero and Hausdorff dimension \frac{2}{3} .

Figure 4. Extending φ between cells of type IV.

A quasiconformal twist. The proof of Theorem B depends on the following lemma which is a triply-connected version of Lemma 2.2. For simplicity we denote by S(p, r) the open square centered at p whose side-length is r.

Lemma 4.1. Fix 0 < a < \frac{1}{5} and let A and B be the closed triply-connected sets defined by

\begin{aligned}A & := ([0, \frac{1}{2}] × [−\frac{1}{2}, \frac{1}{2}]) \ (S (\frac{1}{8}, \frac{1}{8}) ∪ S (\frac{3}{8}, \frac{1}{8})) , \\ B & := ([0, \frac{1}{2}] × [−\frac{1}{2}, \frac{1}{2}]) \ (S (\frac{1}{4}(1 + i), \frac{1}{2} − 2a) ∪ S (\frac{1}{4}(1 − i), \frac{1}{2} − 2a)) \end{aligned}

(see Figure 3). Let φ: ∂A → ∂B be a homeomorphism which is the identity on the outer boundary component and acts affinely on the inner boundary components, mapping ∂S(\frac{1}{8}, \frac{1}{8}) to ∂S(\frac{1}{4}(1 + i), \frac{1}{2} −2a) and ∂S(\frac{3}{8}, \frac{1}{8}) to ∂S(\frac{1}{4}(1 − i), \frac{1}{2} −2a), respecting the horizontal and vertical sides. Then φ can be extended to a K - quasiconformal map φ: A → B, with

K ≍ \frac{1}{a}.

Proof. We consider the affine cell decompositions of A and B shown in Figure 3 and require φ to map each cell in A to its corresponding cell in B in a piecewise affine fashion. By symmetry, it suffices to define φ piecewise affinely between the cells labelled I, II, III, and IV. We let φ be affine between the triangular cells III. On the cells I and II we subdivide the trapezoids into two triangular cells and define φ to be affine on each of them. An easy computation based on (2.4) then shows that the dilatation of φ on I, II, and III is comparable to 1/a.

It remains to define φ between the cells IV and estimate its dilatation. Note that the cell IV in A has bounded geometry, so there is a K_{1} ≍ 1 and a piecewise affine K_{1} -quasiconformal map f_{1} from this cell to the square with vertices 0, 1, \frac{1}{2}(1 + i), \frac{1}{2}(1 − i) which maps the horizontal edge of this cell to the segment from \frac{1}{2}(1 − i) to 1 (see Figure 4). The cell IV in B, after a conformal change of coordinates T, becomes the 4-gon with vertices

0, 1, z := −\frac{1 − 2a}{4a} + \frac{i}{2}, z′ := 1 − \frac{(1 − 4a)i}{4a}.

Let f_{2} be the piecewise affine map on this 4-gon which maps the triangle Δ(0, 1, z) to Δ(0, 1, \frac{1}{2}(1 + i)) and the triangle Δ(0, 1, z′) to Δ(0, 1, \frac{1}{2}(1 − i)) (see Figure 4). Then a brief calculation based on (2.4) shows that f_{2} is K_{2} -quasiconformal, with K_{2} ≍ 1/a. The map φ can then be defined by T^{−1} ∘ f_{2}^{−1} ∘ f_{1} , whose dilatation K_{1}K_{2} is clearly comparable to 1/a.

We are now ready to prove Theorem B cited in Section 1.

Proof of Theorem B. Consider the Cantor set Σ = ⋂_{n=0}^{∞} Σ_{n} constructed above and the Cantor set Λ = Λ(d) = ⋂_{n=0}^{∞} Λ_{n} constructed in Section 2, where d = \{d_{n}\} is defined by d_{n} := 2^{−\sqrt{n}}. It follows from Lemma 2.1 that dim_{H}(Λ) = 2.

We construct a David map φ: C → C, identity outside the square [−\frac{1}{2}, \frac{1}{2}] × [−\frac{1}{2}, \frac{1}{2}], with the property φ(Σ) = Λ. Then the embedded arc φ([−\frac{1}{2}, \frac{1}{2}]) contains Λ and hence has dimension 2. By pre-composing φ with an appropriate quasiconformal map, we obtain a David map sending the round circle to a Jordan curve of dimension 2.

The map φ will be the uniform limit of a sequence of quasiconformal maps φ_{n}: C → C with φ_{n}(Σ_{n}) = Λ_{n}, defined inductively as follows. Let φ_{0} be the identity map on C. To define φ_{1} , set φ_{1} = φ_{0} on C\Σ_{0} and map each of the four squares in Σ_{1} affinely to the “corresponding” square in Λ_{1} . Here “corresponding” means that the squares in Σ_{0} , from left to right, map respectively to the north west, south west, north east and south east squares in Λ_{1} (compare Figure 5). The remaining set Σ_{0} \ Σ_{1} is the union of two triply-connected regions, on the boundary of which φ_{1} can be defined affinely, so we can extend φ_{1} to each such region as in Lemma 4.1.

In general, suppose φ_{n−1} is constructed for some n ≥ 2 and that it maps each square in Σ_{n−1} affinely to a square in Λ_{n−1} . Define φ_{n} = φ_{n−1} on C \ Σ_{n−1} and let φ_{n} map each square in Σ_{n} affinely to the “corresponding” square in Λ_{n} in the above sense. The remaining set Σ_{n−1} \ Σ_{n} is the union of 2^{2n−1} triply-connected regions on the boundary of which φ_{n} can be defined affinely. By rescaling each such region in Σ_{n−1} \ Σ_{n} by a factor 8^{n−1} and the corresponding region in Λ_{n−1} \ Λ_{n}

Figure 5. First two steps in the construction of the map φ. The solid arcs on the right are φ_{n}(R) for n = 1, 2.

by a factor 2^{n−1}/d_{n−1} , we are in the situation of Lemma 4.1, so we can extend φ_{n} in a piecewise affine fashion as in that lemma, and the dilatation of the resulting extension will be comparable to

\frac{d_{n−1}}{2^{n−1}a_{n}} = \frac{d_{n−1}}{2^{n−1} · 2^{−(n+1)}(d_{n−1} − d_{n})}

= \frac{2^{−\sqrt{n−1}}}{2^{n−1} · 2^{−(n+1)}(2^{−\sqrt{n−1}} − 2^{−\sqrt{n}})}

≍ \sqrt{n} .

The sequence {φ_{n}} obtained this way has the following properties:

(i) φ_{n} = φ_{n−1} on C \ Σ_{n−1} . (ii) φ_{n} maps each square in Σ_{n} affinely to the corresponding square in Λ_{n}. (iii) φ_{n} is K_{n} -quasiconformal, with K_{n} ≍ \sqrt{n} .

Evidently, φ := lim_{n→∞} φ_{n} is a homeomorphism which agrees with φ_{n} on C \ Σ_{n} for every n and satisfies φ(Σ) = Λ.

To check that φ is a David map, choose a sequence 1 < K_{1} < K_{2} < ⋯ < K_{n} < ⋯ with K_{n} ≍ \sqrt{n} such that φ_{n} is K_{n} -quasiconformal. Fix some n, let K > 1, and choose j such that K_{j} ≤ K < K_{j+1} . Then

area {z : K_{φ_{n}}(z) > K} ≤ area {z : K_{φ_{n}}(z) > K_{j}} ≤ area (Σ_{j}) = 2^{−4j}.

Since K ≍ K_{j} ≍ \sqrt{j} , we have

area {z : K_{φ_{n}}(z) > K} ≤ e^{−K},

provided that K is bigger than some K_{0} independent of n. It follows that the φ_{n} are all (1, 1, K_{0})-David maps. By Tukia’s theorem in Section 1, we conclude that φ = lim_{n→∞} φ_{n} is a David map.

Removability of David circles. A compact set Γ ⊂ C is called (quasi)conformally removable if every homeomorphism φ: C → C which is (quasi)conformal off Γ is (quasi)conformal in C. It is well known that conformal and quasiconformal removability are identical notions.

Every set of σ -finite 1-dimensional Hausdorff measure, such as a rectifiable curve, is removable. Quasiarcs and quasicircles provide examples of removable sets which can have any dimension in the interval [1, 2). One can even construct removable sets of dimension 2: the Cartesian product of two linear Cantor sets with zero length and dimension 1 is such a set.

At the other extreme, sets of positive area are never removable, as can be seen by an easy application of the measurable Riemann mapping theorem. Also, there exist non-removable sets of Hausdorff dimension 1 (see for example [K]).

To add an item to the above list of examples, we show that David circles are removable, which, combined with Theorem B, proves that there exist removable Jordan curves of Hausdorff dimension 2. First we need the following simple lemma on David maps (compare [PZ]) whose analogue in the quasiconformal case is standard.

Lemma 4.2. Suppose φ: C → C is a homeomorphism whose restrictions to D and C \ \overline{D} are David. Then φ itself is a David map.

Proof. The complex dilatation μ = μ_{φ} is defined almost everywhere in C and satisfies an exponential condition of the form (1.4) in D and in C \ \overline{D} (by making C bigger and t and ε_{0} smaller if necessary, we can assume that the same constants (C, t, ε_{0}) work for both D and C\ \overline{D}). So to prove the lemma, we need only show that φ ∈ W_{loc}^{1,1}(C).

On every compact subset of C \ S^{1} , the ordinary partial derivatives ∂φ and ¯∂φ exist almost everywhere, are integrable, and coincide with the distributional

David maps and Hausdorff dimension 137

partial derivatives of φ. We check that ∂φ, and hence ¯∂φ = μ · ∂φ, is locally integrable near the unit circle S^{1} .

Let D be any small disk centered on S^{1} . We have

|∂φ|^{2} = \frac{J_{φ}}{1 − |μ|^{2}} ≤ \frac{J_{φ}}{1 − |μ|},

so that

(4.1)  |∂φ| ≤ (J_{φ})^{1/2} · (1 − |μ|)^{−1/2}.

Now J_{φ} ∈ L^{1}(D) since ∫_{D} J_{φ} ≤ area (φ(D)) < +∞, and (1 − |μ|)^{−1} ∈ L^{1}(D) because of the exponential condition (1.4). It follows from Hölder inequality applied to (4.1) that ∂φ ∈ L^{1}(D).

Theorem 4.3. David circles are (quasi)conformally removable.

Proof. Let φ: C → C be a David map and Γ = φ(S^{1}). Let f: C → C be a homeomorphism which is conformal in C \ Γ. Then the homeomorphism f ∘ φ is David in D and in C \ \overline{D}. By Lemma 4.2, f ∘ φ: C → C is a David map. Since μ_{f∘φ} = μ_{φ} almost everywhere, it follows from the uniqueness part of David’s theorem [D] that f must be conformal in C.

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Received 5 March 2003