Annales Academiæ Scientiarum Fennicæ Series A. I. Mathematica Volumen 19, 1994, 323–338

SOME HOMEOMORPHISMS OF THE SPHERE CONFORMAL OFF A CURVE

Christopher J. Bishop

SUNY at Stony Brook, Department of Mathematics Stony Brook, NY 11794-3651, U.S.A.; bishop@math.sunysb.edu

Abstract. We give an example of a flexible curve, i.e., a closed curve Γ so that for any other closed curve Γ′ and ε > 0 there is a homeomorphism Φ of the sphere, conformal off Γ, so that Φ(Γ) lies in an ε neighborhood of Γ′. Moreover, for any guage function h such that h(t) = o(t) as t → 0, we may take Γ to have zero Hausdorff measure with respect to h. This extends a result of Robert Kaufman on removable sets. We also survey some known results on removable sets and give a totally disconnected version of the example above.

1. Introduction

A curve Γ in the plane is called conformally rigid (or removable for conformal homeomorphisms) if any homeomorphism of the Riemann sphere C_{∞} which is conformal off Γ must be a Möbius transformation. Morera’s theorem implies that a circle is rigid, for example. Other examples include rectifiable curves and quasicircles. In this note we are interested in curves with the opposite behavior. For convenience we will let CH(E) denote the homeomorphisms of C_{∞} to itself which are conformal off E. We shall say Γ is flexible if given any other curve Γ′ and any ε > 0 there is a homeomorphism Φ ∈ CH(Γ) of C_{∞} to itself which is conformal off Γ and so that

ϱ(Φ(Γ), Γ′) < ε,

where ϱ(E, F) is the Hausdorff metric,

ϱ(E, F) = sup_{z∈E} inf_{w∈F} |z − w| + sup_{w∈F} inf_{z∈E} |z − w|.

The main purpose of this note is to show such things exist.

Theorem 1. There exists a closed Jordan curve Γ which is flexible.

1991 Mathematics Subject Classification: Primary 30C35; Secondary 30C60, 30C85.

The author is partially supported by a NSF Grant DMS 92-04092 and an Alfred P. Sloan Research Fellowship.

Equivalently, there is a curve Γ such that the restriction of CH(Γ) to Γ is dense in the collection of all homeomorphisms to Γ into C (up to reparameterizations of the curve). In fact our proof will actually show something stronger: there is a Γ so that given any conformal mappings Φ_{1}, Φ_{2} on the two complementary components Ω_{1}, Ω_{2} of Γ which have disjoint images and any ε > 0, there is a Φ ∈ CH(Γ) which approximates Φ_{i} to within ε on { z ∈ Ω_{i} : dist(z, Γ) > ε) } for i = 1, 2. A modification of the proof also shows that flexible arcs exist. If CH(E) consists of just the Möbius transformations we shall call CH(E) trivial or say E is CH-removable. Otherwise E is called non-removable. Building non-removable curves is not hard. For example, a curve of positive area has this property, as can be seen by taking a quasiconformal mapping whose dilatation lives on the curve. At the other extreme a curve with σ -finite length is removable. Robert Kaufman has constructed examples of non-removable sets which are “close” to σ - finite length and removable sets which are “close” to positive area in the sense of Hausdorff measures (the precise results will be described in the next section). Our proof will also show that flexible curves can be taken as close to σ -finite length as we wish:

Theorem 2. For any Hausdorff measure function h such that h(t) = o(t) as t → 0, there is a flexible curve Γ such that Λ_{h}(Γ) = 0.

The construction will also show how to build a curve Γ and non-Möbius Φ so that both Λ_{h}(Γ) = Λ_{h}(Φ(Γ)) = 0. Moreover, Γ itself may be taken in an ε neighborhood of any preassigned closed curve. I do not know if an example of a non-removable curve where both Γ and Φ(Γ) have small dimension has been recorded before. In Kaufman’s example, Φ(Γ) has positive area, though possibly his argument could be modified to give Φ(Γ) small as well. Zheng-Xu He asked me about an analogue of this for totally disconnected sets. A modification of the proof of Theorems 1 and 2 gives

Theorem 3. For any Hausdorff measure function h such that h(t) = o(t) as t → 0, there is a totally disconnected E and a non-Möbius Φ ∈ CH(E) so that Λ_{h}(E) = Λ_{h}(Φ(E)) = 0.

Finally, let us mention an alternative form of Theorem 1. Given a closed curve Γ in C_{∞} with complementary components Ω_{1}, Ω_{2} and Riemann mappings φ_{1}, φ_{2} from the unit disk to Ω_{1}, Ω_{2} , the homeomorphism of the circle to itself defined by ψ = φ_{1}^{−1} ∘ φ_{2} is called a conformal welding associated to Γ (and is determined up to composition with Möbius transformations preserving the disk). A well studied problem is to try to relate properties of Γ to those of ψ and try to determine which homeomorphisms ψ arise as conformal weldings. For “nice” ψ, there is such a Γ and it is unique up to Möbius transformations. Note that two curves Γ, Γ′ have the same welding map if and only if Φ(Γ) = Γ′ for some Φ ∈ CH(Γ). Thus non-removable curves correspond to conformal welding maps for which the corresponding curve is not unique (up to Möbius transformations). In fact, Theorem 1 implies

Corollary 1. There is a homeomorphism ψ of the circle so that the set of corresponding Γ’s is dense in the set of all closed Jordan curves (with the Hausdorff metric).

The idea behind Theorems 1, 2 and 3 is very simple. If the curve Γ was actually a thin strip of some positive width, there would be no problem constructing Φ by letting it be any conformal map on either side of the strip and interpolating these maps continuously across the strip. We will show that if the strip is “filled up” by a highly oscillating curve (e.g., a curve which is ε-dense in the strip) then conformal maps on either side of curve can still be continuously interpolated in some sense. The construction makes this precise.

This paper is a revised version a preprint first written in 1987 and has benefitted from several sources. I would particularly like to thank Peter Jones and Robert Kaufman for a variety of discussions on topics related to removable sets.

2. Summary of related results

In this section I will attempt to summarize what is known (at least to me) about removable sets for conformal mappings. Given a closed set E and its complement Ω it is natural to consider several related spaces of functions

H^{∞}(Ω) = {bounded holomorphic functions on Ω}
S(Ω) = {1 − 1, (Schlicht) analytic maps on Ω}
𝒟(Ω) = W^{1,2}(Ω) = {analytic functions such that ∫∫_{Ω} |f′|^{2} dx dy < ∞}
A_{E} = {elements of C(E) with analytic extensions to C_{∞} \ E}
CH(E) = {homeomorphisms of C_{∞} to itself which are conformal off E}
QCH(E) = {homeomorphisms of C_{∞} to itself which are quasiconformal off E}.

These satisfy some obvious inclusions such as, A_{E} ⊂ H^{∞}(Ω) and CH(E) ⊂ S(Ω). Functions in S(Ω) and CH(E) are not bounded, but if normalized so that ∞ is in Ω and Φ(∞) = ∞, then Φ(z) − z is bounded and also is in 𝒟(Ω). We say that the set E is removable for the class in question if every element of the class agrees with some element of the corresponding class for E = ∅, Ω = C_{∞}, i.e., if the functions in question cannot distinguish E from the empty set.

For none of these classes is there a geometric characterization of the removable sets. What follows is a list of propositions concerning removable sets for the various classes. The references given are not necessarily the original ones.

We call h a measure function if it is a continuous increasing function from [0, ∞) to itself with h(0) = 0 and we define the associated Hausdorff measure by

Λ_{h}^{δ}(Γ) = inf{∑_{j} h(r_{j}) : Γ ⊂ ⋃_{j}D(x_{j}, r_{j}), r_{j} ≤ δ}
Λ_{h}(Γ) = lim_{δ→0} Λ_{h}^{δ}(Γ).

If h(t) = t^{α} we simply denote the measure Λ_{α}. We set

dim(E) = inf{ α : Λ_{α}(E) = 0 }.

A countable union of sets of finite measure is called σ -finite.

Proposition 1 ([4]). If Λ_{1}(E) = 0, E is removable for H^{∞}(Ω). If E has σ -finite Λ_{1} measure it is removable for A_{E}. If dim(E) > 0 it is not removable for A_{E} (hence not for H^{∞}(Ω)).

This is essentially a “folk-theorem”. Details are also recorded in [16] among other places. For sets E with dim(E) = 1 or even stronger, 0 < Λ_{1}(E) < ∞, there are a wide variety of results, counterexamples and conjectures. See [12], [16], [19], [22], [23], [24].

One other interesting related fact is that if E has no interior and f ∈ A_{E} then an application of the argument principle (see [9, Lemma 3.6.4]) shows that f(E) = f(C_{∞}), i.e., f takes on E every value it takes anywhere on the sphere. (For a closed curve the proof is particularly easy: suppose f has no zeros on Γ. Then the winding number of f(Γ) around zero is well defined and counts the number of zeros of f on one side of Γ. But its negative counts the number of zeros on the other side, hence both must be zero, i.e, f has no zeros on C_{∞}.) In particular, if f is non-constant then f(E) covers an open set. If Φ ∈ CH(E) is not Möbius and satisfies Φ(0) = 0, Φ(∞) = ∞ then f(z) = Φ(z)/z ∈ A_{E} and so maps E to a set with interior. This says such examples must be fairly complicated. If dim(E) = 1 it says that Φ is at best Hölder of order 1/2.

A better result is possible, as was pointed out to me by Peter Jones. Suppose Γ is a curve and Γ(ε) = { z : dist(z, Γ) < ε }. If Φ is Hölder of order α, then Φ can be aprroximated by a smooth function F which agrees with Φ outside Γ(ε) and satisfies |∇F | ≤ ε^{α−1} . The deviation of F from a Möbius transformation can be measured by the integral,

∫∫ |\overline{∂}F | dx dy ≤ ε^{α−1}|Γ(ε)|.

In our examples the area of Γ(ε) tends to zero faster than ε^{1+δ} for any δ > 0, so non-trivial elements of CH(Γ) cannot be Hölder of any order. For a general curve, Jones’ argument would imply that any non-trivial Φ is Hölder of order at most dim(Γ) − 1. If Γ has positive area, this is correct; by a result of Nguyen Xuan Uy [29], there is a Lipschitz function f, analytic on the complement of Γ. Thus Φ(z) = z + εf(z) is a bi-Lipschitz homeomorphism, analytic off Γ, if ε is small enough. In general though, the construction in this note can be modified to give a curve Γ with any Hausdorff dimension between 1 and 2 so that any non-trivial Φ ∈ CH(Γ) is not Hölder of any positive order. (See the remark at the end of Section 5.)

Proposition 2. CH(E) and QCH(E) have the same removable sets. Quasicircles are removable for both.

A quasicircle is the image of the unit circle under a quasiconformal map. These curves are geometrically characterized by the following condition: γ is a quasicircle if there is a C > 0 so that the shorter arc between two points z, w ∈ γ has diameter at most C|z − w|. Proposition 2 is another well known result and follows easily from the measurable Riemann mapping theorem, e.g., [1, Theorem V.3]. Since quasicircles can have dimension > 1 they are not necessarily removable for A_{E}.

Proposition 3 ([28], [2],[3, Theorem IV.2.d]). The removable sets for the classes S(Ω) and 𝒟(Ω) are the same.

Proposition 4 ([2], [3, Theorem IV.2.b]). E is removable for 𝒟(Ω) if and only if it has absolute measure zero, i.e., for every Φ ∈ S(Ω) the complement of Φ(Ω) has zero measure.

Proposition 5 ([18]). If U is a John domain (see below) and E = ∂U then E is removable for 𝒟(Ω) ∩ C(C_{∞}) (and hence removable for CH(E)).

A John domain U is a connected open set so that there exists ε > 0 and a base point z_{0} ∈ U so that any point z ∈ U can be connected to z_{0} by a curve γ in U which satisfies

dist(w, ∂U) ≥ ε dist(w, z), w ∈ γ.

A curve Γ is a quasicircle if and only if both its complementary domains are John domains so that Proposition 5 contains Proposition 2 as a special case. It also covers sets E which are not curves but which have some sort of self similarity, such as certain Julia sets arising in iteration theory.

Proposition 6 ([11]). If F ⊂ R is compact then E = F × [0, 1] is removable for QCH if and only if F is uncountable.

The proof involves considering a quasiconformal mapping of the form

z → z + g(y)μ({−∞, x}),

where μ is a continuous measure on F and g is a smooth function supported on [0, 1]. The image of E under such a map clearly has positive area, so E cannot be removable. Using a more sophisticated version of this argument Robert Kaufman proved

Proposition 7 ([20]). If F is uncountable then F × [0, 1] contains a graph E which is not removable for QCH.

A graph means that for each x ∈ F, E contains at most one point with first coordinate x. By choosing F correctly we can find E so that Λ_{h}(E) = 0 for any prechosen h with h(t) = o(t). Kaufman has also shown (personal communication): suppose E has the property that for any δ > 0 there is a covering of E by squares {Q_{j}} of size ≤ δ so that the doubles {2Q_{j}} are disjoint. Then E is removable for S(Ω). (This is a special case of a more general result of his on removable sets for meromorphic functions. This version for conformal maps also follows from Proposition 4.) Using this, and given any increasing function h so that h(t)/t^{2} ↗ ∞ as t → 0, he builds a totally disconnected set E with Λ_{h}(E) > 0, but E removable for S(E) (and so removable for CH). Thus in terms of Hausdorff measures alone, positive area is the best possible sufficient condition for non-removability and σ -finite length is the best possible sufficient condition for removability.

The idea of a “flexible” curve in the introduction says that CH(Γ) contains, in some sense, as many homeomorphisms as possible. The relation of this property to the class CH is thus analogous to the relation between Dirichlet sets and the class A_{E}. Recall that A_{E} is called a Dirichlet algebra if the real parts of functions in A_{E} are uniformly dense in C_{R}(E), the collection of all real valued functions on E. E is a Dirichlet set if A_{E} is a Dirichlet algebra. Since a holomorphic function is determined (up to constants) by it real parts, the Dirichlet algebras are thus those which are as “large as possible” in the sense that almost any real function can occur (up to epsilon) as the real part of something in the algebra. Oddly, although we cannot characterize the curves for which A_{Γ} is non-trivial, there is a characterization of the curves which are Dirichlet sets.

Proposition 8. The following are equivalent for a closed curve Γ:

(1) A_{Γ} is a Dirichlet algebra.

(2) If ω_{1} and ω_{2} are the harmonic measures with respect to? points on opposite sides of Γ then ω_{1} ⊥ ω_{2} (i.e., there exists E ⊂ Γ with ω_{1}(E) = ω_{2}(E^{c}) = 0).

(3) If Φ_{1}, Φ_{2} are conformal maps from the unit disk to the two complementary components of Γ then ψ = Φ_{1}^{−1} ∘ Φ_{2} is a singular homeomorphism of the circle to itself (i.e., there exists E ⊂ {|z| = 1} so that |E| = |ψ(E^{c})| = 1).

(4) The set of tangent points of Γ has zero Λ_{1} measure.

A point x ∈ Γ is a tangent of Γ if there is a line L passing though x such that

dist(Γ ∩ D(x, r), L) = o(r).

The equivalence of (2) and (3) is just a matter of unwinding the definitions. (1) if and only if (2) is due to Browder and Wermer [10] and (2) if and only if (4) is in [8]. Also see [6], [7], [14] for generalizations from curves to arbitrary compact sets. As in the remark following Proposition 7, Dirichlet curves can be constructed with Λ_{h}(Γ) = 0 for any preassigned h. There are additional characterizations of Dirichlet sets in terms of continuous analytic capacity and pointwise bounded approximations (see [6] and its references). It is not too hard to see that a flexible curve must also be Dirichlet.

Numerous questions about removable sets remain open, the most basic to be to characterize as geometrically as possible the removable sets for each class. Below we list some other questions (old and new) which may be more accessible (some may even be easy or known).

Question 1 (P. Jones). If E is removable for 𝒟(Ω)∩C(C_{∞}) then it is known to be removable for CH(E) (see [18]). Is the converse true?

Question 2. If E is not removable for S then Proposition 4 says there is a conformal map of E^{c} to the complement of a set of positive area. If E is non-removable for CH is there an element Φ ∈ CH(E) so that Φ(E) has positive area?

Question 3. Every planar domain can be conformally mapped to the complement of a set of area zero (but this map need not be continuous on the entire sphere). If E has positive area and no interior is there a Φ ∈ CH(E) so that Φ(E) has zero area?

Question 4. Suppose Γ is a non-removable curve for CH. Does Γ contain a totally disconnected subset which is not removable for CH? Does Γ contain a proper closed subset which is not removable? Is the property of removability (flexibility) local? It would be very surprising if this failed.

Question 5. Is there a characterization of the conformal welding maps which correspond to flexible curves? It should say that the circle homeomorphism is very singular. If the mapping is? even quasisymmetric then the curve Γ is a quasicircle, and thus removable for CH. (Quasisymmetric means |ψ(x + t) − ψ(x)| ∼ |ψ(x) − ψ(x − t)| with constants independent of x and t.)

Question 6. Is a subarc of a flexible arc another flexible arc? (They are for the examples constructed here.)

Question 7 (R. Kaufman). In [20] Kaufman showed that the graph of a continuous, real-valued function could be non-removable for CH. He has also asked if such a function f can be taken arbitrarily close to Lipschitz. More precisely, if ω is an increasing function on [0, ∞) such that ω(t) = o(t) is there a function f with modulus of continuity ω whose graph is a non-removable set?

3. Flexible curves

Before stating our main lemma we introduce some notation. First, for a set S and an ε > 0 we let

S(ε) = { z : dist(z, S) < ε }.

Next, if φ_{1} and φ_{2} are uniformly continuous mappings of Ω_{1} and Ω_{2} respectively onto the complementary components of a curve Γ′, we set

jump(φ_{1}, φ_{2}) = sup_{x∈Γ} dist_{Γ′}(φ_{1}(x), φ_{2}(x)),

where the distance is measured by arclength along Γ′. We will prove the theorem by an iterative construction using:

Lemma 1. Suppose Γ is a smooth, closed Jordan curve with complementary components Ω_{1} and Ω_{2} . Suppose ℱ is a compact family of pairs of smooth conformal maps on \overline{Ω}_{1}, \overline{Ω}_{2} whose images have disjoint interiors. Also suppose α > 0, δ > 0 and η > 0 are given. Then there exists a smooth, closed Jordan curve γ with complementary components ω_{1} , ω_{2} so that the following holds. Suppose (Φ_{1}, Φ_{2}) ∈ ℱ normalized so A = C \ (Φ_{1}(Ω_{1}) ∪ Φ_{2}(Ω_{2})) had diameter ∼ 1. Then there exist conformal mappings φ_{1} and φ_{2} on ω_{1} and ω_{2} such that

(1) γ′ = φ_{1}(γ) = φ_{2}(γ) is a smooth curve.

(2) γ ⊂ Γ(α), γ′ ⊂ A(α).

(3) |Φ_{i}(z) − φ_{i}(z)| < δ for z ∈ Ω_{i} \ Γ(α) and i = 1, 2.

(4) jump_{γ′}(φ_{1}, φ_{2}) < η.

First we will show how to build a non-removable curve for CH with “small” dimension using the lemma. Then we will show how to build a flexible one. Suppose {ε_{n}} is a sequence of positive numbers decreasing to 0 (to be chosen later). Start with any smooth curve Γ_{0} and univalent, conformal maps Φ_{1}^{0} and Φ_{2}^{0} on the complementary components with disjoint, smooth images. (In this case we are taking ℱ to just be one pair of functions.) Using the lemma, approximate them with a curve Γ_{1} and maps Φ_{1}^{1} and Φ_{2}^{1} such that Γ_{1} ⊂ Γ_{0}(ε_{1}) and

jump(Φ_{1}^{1}, Φ_{2}^{1}) < \frac{1}{2}.

In general, we replace Γ_{n−1} and Φ_{i}^{n−1} , i = 1, 2, with Γ_{n} and Φ_{i}^{n} satisfying Γ_{n} ⊂ Γ_{n−1}(ε_{n}), Φ_{n}(Γ_{n}) ⊂ Φ_{n−1}(Γ_{n−1})(ε_{n}) and

jump(Φ_{1}^{n}, Φ_{2}^{n}) < 2^{−n}.

Then Γ = lim_{n} Γ_{n}, Φ_{i} = lim_{n} Φ_{i}^{n} exist, and Φ_{1} and Φ_{2} agree on Γ so they define a continuous function on the sphere which is analytic off Γ. Since it is the uniform limit of univalent, analytic functions it is also univalent off Γ. If we choose ε_{n} small enough (depending on γ_{n−1} ) one can verify that Γ is a Jordan curve. Since Φ_{1} and Φ_{2} are analytic, they cannot be constant on any subarc of Γ. Using this and continuity, one can prove that they are both injective on Γ, and therefore define a homeomorphism of the sphere which is conformal off Γ. Since this map uniformly approximates Φ_{1}^{0} and Φ_{2}^{0} away from Γ we can easily arrange for Φ not to be Möbius. To show that Λ_{h}(Γ) = Λ_{h}(Φ(Γ)) = 0, take ε_{n} so small that both Γ_{n−1} and Φ_{n−1}(Γ_{n−1}) can be covered by N disks of radius ε_{n} and Nh(ε_{n}) < 2^{−n}. This is possible since the curves are smooth and h(t) = o(t). Then clearly Λ_{h}(Γ) = Λ_{h}(Φ(Γ)) = 0.

To construct a flexible curve, first choose sequences {η_{n}} tending to zero and {δ_{n}} summable. At the nth stage α_{n} is chosen so that Γ_{n−1} can be covered by N disks of radius α_{n} where Nh(α_{n}) < 1/n. This guarantees that the limiting curve will satisfy Λ_{h}(Γ) = 0. There is a compact family of smooth curves Γ′′ so that any closed curve Γ′ of diameter 1 can be approximated to within α_{n}/100 (in the Hausdorff metric) by some member of the family. At the nth stage we will also wish to apply the lemma with a compact family ℱ_{n} of conformal maps determined by mapping Γ_{n−1} onto this compact family of smooth approximating curves.

Define a sequence of curves {Γ_{n}} as above using these sequences. Now we show the resulting curve Γ is flexible. Suppose we are given the target curve Γ′. Rescale so Γ′ has diameter around 1. Choose n so that α_{n} < ε/4. Let Γ′′ be a smooth curve in our compact family which approximates Γ′ to within ε/4 and let let Φ_{1}, Φ_{2} be conformal maps from the complements of Γ_{n−1} to the complementary components of Γ′′. By the lemma there are conformal mappings φ_{1}, φ_{2} on the complementary components of Γ_{n} so that the image of Γ_{n} is a smooth curve approximating Γ′′ to within α_{n} < ε/4 and so that the “jump” is < η_{n}. Now pass to the limit as above, perturbing the image curve by at most ε2^{−m} at the mth stage (which we can do if α_{n} tends to 0 fast enough). The resulting map Φ ∈ CH(Γ) as desired and Φ(Γ) approximates Γ′′ to within ε/2 (and hence approximates Γ′ to within ε). That proves Theorem 2.

4. Proof of the lemma

It only remains to prove the lemma. We start by approximating Γ to within α/100 on either side by two smooth curves, Γ_{1} and Γ_{2} . We choose them so that on small scales Γ, Γ_{1} and Γ_{2} look like three parallel lines (see Figure 1). We normalize so that dist(Γ_{1}, Γ_{2}) ∼ 1 and draw line segments {L_{j}} perpendicular to Γ connecting Γ_{1} and Γ_{2} . The endpoints of adjacent segments should be within ε of each other (ε to be chosen later).

The curves Γ_{1} and Γ_{2} bound disjoint domains Ω̂_{1} and Ω̂_{2} . Let K_{1} consist

Figure 1.

Figure 2.

Figure 3.

of the union of the closure of Ω̂_{1} and an alternating collection of the {L_{j}} (see Figure 2). K_{2} is defined similarly using Ω̂_{2} and the remaining {L_{j}}.

We will now define a bi-Lipschitz function f on K = K_{1} ∪ K_{2} such that f = Φ_{1} on Ω̂_{1} and f = Φ_{2} on Ω̂_{2} and for

z, w ∈ K, |z − w| < 2ε ⇒ |f(z) − f(w)| < η/10.

If z, w ∈ Ω̂_{1}∪Ω̂_{2} this holds for ε small enough, so we only have to define f on each on the segments L_{j}. One (of many) ways to do this is to note that the region A

Figure 4.

between Γ_{1} and Γ_{2} is a topological annulus, so there is a smooth diffeomorphism H from A \ L_{0} to a rectangle R with an identification of the segment L_{0} with two opposite sides of R. If we let a_{j} = Φ_{1}(L_{j} ∩ Γ_{1}) and b_{j} = Φ_{2}(L_{j} ∩ Γ_{2}) and parameterize L_{j} by t, 0 ≤ t ≤ 1 (going from Γ_{1} to Γ_{2} ) then we can set

f(t) = H^{−1}((1 − t)H(a_{j}) + tH(b_{j})).

Then f is bi-Lipschitz on K and will satisfy the condition above if ε is small enough. See Figure 3.

We now observe that f can be uniformly approximated on K_{1} and K_{2} by functions g_{1} and g_{2} which are holomorphic and univalent on open neighborhoods of K_{1} and K_{2} respectively, and which have disjoint images. One way to see this is to use Mergelyan’s theorem (e.g., [25], [27, Chapter 20]). It says that if K is a compact set not dividing the plane then any function continuous on K and holomorphic on its interior can be uniformly approximated on K by polynomials. Thus we can approximate f′ uniformly on K_{i} by a function holomorphic on a neighborhood of K_{i}. Taking then an appropriate primitive of that function gives the desired approximation since f is bi-Lipschitz.

Also note that if we are given not just one function f, but a compact family of functions, then we can choose a fixed neighborhood of K_{1} and K_{2} on which each member of the family can be approximated. Given the α in the lemma, any curve of diameter 1 can be approximated to within α by a member of some compact family of smooth curves (which depends on α of course). Thus we take our neighborhoods with this property. Next we define two smooth curves Γ̃_{1} and Γ̃_{2} by adjoining to Γ_{1} and Γ_{2} “thickened” versions of the line segments {L_{j}} on which g_{1} and g_{2} are defined (see Figure 4). We let Ω̃_{i}, for i = 1, 2, denote the domains bounded by these curves. We are essentially done now, except that Ω̃_{1} and Ω̃_{2} are not bounded by a common curve. However, this is easy to fix. Consider the curve γ in Figure 5 and let ω_{1} and ω_{2} denote the complementary components. The point is that “conformally” ω_{1} and ω_{2} look like Ω̃_{1} and Ω̃_{2} . More precisely, we can find domains K_{i} ⊂ ω̃_{i} ⊂ Ω̃_{i} (see Figure 6) and conformal maps Ψ_{i}: ω_{i} → ω̃_{i}, for i = 1, 2, such that

z, w ∈ ω_{i}, |z − w| < ν ⇒ |Ψ_{i}(z) − Ψ_{i}(w)| ≤ 2ε, i = 1, 2

for ν small enough. Thus φ_{i} = g_{i} ∘ Ψ_{i} satisfy the lemma.

Figure 5.

Figure 6.

Finally, the images φ_{1}(ω_{1}) and φ_{2}(ω_{2}) bound disjoint smooth domains, but are not bounded by a common curve. This is easily fixed by “filling in” the region between the domains by a curve, each side of which can be mapped conformally to the corresponding domain by a map which approximates the identity away from the curve. See Figure 7.

Figure 7.

5. Non-removable Cantor sets

In this section we modify the preceding discussion to construct non-removable Cantor sets such that both E and Φ(E) are small. We remind the reader that similar examples were first constructed in [20] (but in that paper Φ(E) had positive area).

We start back at the beginning to Section 4 with a curve Γ and smooth conformal maps defined on either side of Γ and mapping onto disjoint domains. As before, approximate Γ on either side by smooth curves Γ_{1}, Γ_{2} and define a collection of “crosscuts ” {L_{j}}. The difference is that instead of dividing the crosscuts into alternating subsets and associating one family with one side of the curve and the other family with the other side, we simply take one set K equal to the closure of Ω̂_{1} ∪ Ω̂_{2} ∪ ⋯ ∪_{j} L_{j}. As before we find a smooth, bi-Lipschitz function f of K which we then approximate by a conformal map on some open neighborhood of K. Instead of invoking Mergelyan’s theorem for sets not dividing the plane (K obviously does divide the plane) we use his theorem that if K has only finitely many complementary components then any function continuous on K and holomorphic on its interior can be uniformly approximated by a rational function with poles of K (e.g., [15]). Thus our original maps Φ_{1}, Φ_{2} have been approximated by a single conformal map Φ which is defined on a neighborhood Ω of K. Each complementary component of Ω can be “filled in” by an oscillating curve, in a way which the reader is by now familiar with. See Figure 8. Similarly the complementary components of Φ(Ω) may be filled in by curves.

Now repeat the construction. The only difference is that we are now starting with a collection of Jordan arcs rather than closed Jordan curves. However, this does not cause any problems. One way to see this is to extend each arc slightly at both endpoints so that it goes into the region where Φ is analytic. Now approximate the arc on either side by smooth arcs Γ_{1}, Γ_{2} and form the new set K by removing the thin strip between Γ_{1}, Γ_{2} and adding the crosscuts. Now proceed as

Figure 8.

before. It is fairly clear that a similar argument allows one to construct flexible arcs (rather than just flexible closed curves).

In Section 2 we mentioned that one could modify our construction to produce curves Γ with dim(Γ) = d for any 1 < d < 2, but such that any Hölder homeomorphism in CH(Γ) must be Möbius. This follows from the techniques of Jones’ paper [18]. The argument sketched in Section 2 showed that if Γ looked like a straight line on many small scales then Γ did not have any non-trivial, Hölder elements of CH(Γ). In some sense this is because straight lines are removable for CH. Quasicircles are also removable for CH and using an idea from [18] we can show that if Γ looks like a quasiarc on many small scales, then it has no non-trivial, Hölder homeomorphims. Jones’ arguments in [18] easily imply that if Γ looks like a quasiarc between scales ε_{0} to ε_{1} , (i.e., there is a C so that any subarc of Γ of diameter < ε_{0} can be approximated to within ε_{1} by a quasiarc of constant C), and if Φ is a homeomorphism conformal off Γ(ε) and Hölder of order α then there is a smooth F which equals Φ off Γ(ε) and which satisfies

∫∫_{C} |∇F |^{2} dx dy ≤ C ∫∫_{C\Γ(ε_{0})} |Φ′|^{2} dx dy + Cε_{1}^{α},

where C depends on the quasicircle constant. If ε_{1} is small enough, the second term is dominated by the first. If this happens on infinitely many scales, then we get that Φ can be approximated in the W^{1,2}(C) norm by smooth functions, which implies Φ is Möbius (this is Weyl’s lemma, see e.g. [18]). Thus to produce the desired Γ we only need insure that there are sufficiently many scales on which Γ looks like a quasicircle. In Section 4, we did this by making Γ look like a straight line on infinitely many scales. Here we simply choose some self-similar quasi-arc of the desired dimension (say one based on some polygonal construction like the von Koch snowflake) and make Γ look like this arc on scales between the scales

Some homeomorphisms of the sphere conformal off a curve 337

where the lemma is used. If these “correction” scales are chosen small enough at each stage, the resulting curve will have the correct dimension and no non-trivial Hölder homeomorphisms in CH(Γ).

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338 Christopher J. Bishop

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Received 9 December 1992