BILIPSCHITZ HOMOGENEOUS CURVES IN R^{2} ARE QUASICIRCLES

CHRISTOPHER J. BISHOP

Abstract. We show that a biLipschitz homogeneous curve in the plane must satisfy the bounded turning condition, and that this is false in higher dimensions. Combined with results of Herron and Mayer this gives several characterizations of such curves in the plane.

Date: July 2, 1998. 1991 Mathematics Subject Classification. Primary: 30C65 . Key words and phrases. BiLipschitz mappings, homogeneous continua, quasicircles, bounded

turning, quasihomogeneous embeddings, chord-arc, quasiconformal mappings, Hausdorff dimension.

The author is partially supported by NSF Grant DMS 95-00557.

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1. Introduction

A set X ⊂ ℝ^{d} is called homogeneous with respect to a set of mappings M if for any two points x, y ∈ X there is an f ∈ M such that f(x) = y. In this paper we are primarily concerned with the case when M is the set K-BL(X) of K-biLipschitz mappings of X to itself. Recall that K-biLipschitz means that K^{−1}|x − y| ≤ |f(x) − f(y)| ≤ K|x − y|. In this case we say X is K-biLipschitz homogeneous and for convenience write “X is K-BLH”. We say X is biLipschitz homogeneous if we mean there is some K < ∞ for which this holds and write “X is BLH”. If X ⊂ ℝ^{d} is compact then by Baire’s theorem one can show that if X is BLH , then it must be K-BLH for some K < ∞ (e.g., the argument of Theorem 3.1 of [10], or Theorem 6.1 of [7]). However, D. Herron has constructed an example of a non-compact curve in the plane which is BLH, but not for a fixed K (personal communication). Our main result extends work of Ghamsari and Herron [4] and Herron and Mayer [6] by showing

Theorem 1.1. If Γ is a BLH Jordan curve in ℝ^{2}, then it is a quasicircle.

There are many equivalent definitions of quasicircles in the plane, but for our purposes it is most convenient to say Γ is a quasicircle iff it satisfies the bounded turning condition: there is a C < ∞ so that if x, y ∈ Γ then the diameter of the smaller arc connecting x and y is less than C|x−y|. The class of curves with bounded turning will be denoted by BT. In [4] Ghamsari and Herron showed Theorem 1.1 was true if Γ was also rectifiable. In [6], Herron and Mayer proved it when Γ was homogeneous with repect to a group of uniformly biLipschitz mappings. They also independently give an example of a curve in ℝ^{3} which is BLH but not BT.

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By combining Theorem 1.1 with the results of [6] we can deduce the following characterizations of BLH curves in the plane. The relevant definitions will be given following the statment of the theorem.

Corollary 1.2. Suppose Γ is a closed Jordan curVE IN ℝ^{2}. Then the following are equivalent

1. Γ is BLH.

2. There is a h : S^{1} → Γ which conjugates rotations to biLipschitz selfmaps of Γ.

3. Γ is homogeneous with respect to BL(ℝ^{2}).

4. Γ is the orbit of a 1 parameter group in BL(ℝ^{2}).

5. Γ has a very weak quasihomogeneous parameterization.

6. Γ has a weak quasihomogeneous parameterization.

7. Γ has a quasihomogeneous parameterization.

8. There is a φ so that Γ is φ-chord-arc.

9. Γ is pointwise bi-BLH.

10. Γ is BLH on arcs.

In ℝ^{d}, d ≥ 3, conditions (6)-(10) are equivalent and characterize BLH curves which are also BT. In ℝ^{3}, we will give examples to show that for the analogous conditions, (1) ⇏ (2), (3) ⇏ (4) and (5) ⇏ (6). The remaining (non-trivial) implications are open so far as I know.

Now for the definitions. In what follows, the notation a ≃_{C} b means that C^{−1} ≤ \frac{a}{b} ≤ C, and we write a ≃ b if this is true for some C < ∞. Similarly for a ≲_{C} b, a ≲ b, a ≳_{C} b and a ≳ b.

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Suppose Γ is a closed Jordan curve in ℝ^{d}. We say We say h is weak quasihomogeneous (or WQH) if there is a K < ∞ so that for x, y, u, v ∈ S^{1}, |x − y| ≤ |u − v| implies |h(x) − h(y)| ≤ K|h(u) − h(v)|. We say h : S^{1} → Γ is quasihomogeneous (or QH) if there is a homeomorphism η of [0, ∞) to itself such that

\frac{|h(x) − h(y)|}{|h(u) − h(v)|} ≤ η(\frac{|x − y|}{|u − v|}),

for all distinct x, y, u, v ∈ S^{1}. These are homogeneous versions of the better known quasisymmetric conditions. A map h : S^{1} → Γ is called weakly quasisymmetric (or WQS) if there is a K < ∞ so that

|y − x| ≤ |z − x| implies |f(y) − f(x)| ≤ K|f(z) − f(x)|,

for x, y, z ∈ S^{1} and is called quasisymmetric (or QS) if there is an homeomorphism η : [0, ∞) → [0, ∞) such that

\frac{|f(y) − f(x)|}{|f(z) − f(x)|} ≤ η(\frac{|y − x|}{|z − x|}),

for all triples in S^{1}. In the plane, both conditions are equivalent to Γ being a quasicircle. See [15]. Finally, we say that h is very weakly quasihomogeneous (or VWQH) if there is a K < ∞ so that for x, y, u, v ∈ S^{1}, |x − y| = |u − v| implies |h(x) − h(y)| ≤ K|h(u) − h(v)|.

Given an increasing function φ on [0, ∞), the corresponding Hausdorff content is defined as

H_{∞}^{φ}(E) = inf{∑_{j} φ(r_{j}) : E ⊂ ∪_{j}B(x_{j}, r_{j})}.

Hausdorff φ-measure is defined as

H^{φ}(E) = lim_{δ→0} inf{∑_{j} φ(r_{j}) : E ⊂ ∪_{j}B(x_{j}, r_{j}), r_{j} ≤ δ}.

Both quantities have the same null sets, but only the latter is a measure. If φ(t) = t^{α} then H^{φ} is called α-dimensional measure and is denoted H^{α}. If α = 1 then it is

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called linear measure. If α = d is the dimension of the space, then H^{d} and H_{∞}^{d} are comparable.

Given an increasing function φ, a curve Γ is called φ-chord-arc if given x, y ∈ Γ the smaller diameter subarc γ of Γ connecting these points has H^{φ} measure ≃ φ(|x − y|). The usual notion of chord-arc curve corresponds to the special case φ(t) = t. The α-chord-arc condition with φ(t) = t^{α} has been carefully studied by Ghamsari and Herron in [5].

The curve Γ is pointwise bi-BLH if given any C < ∞ there is a K < ∞ so that for any x, y, u, v ∈ Γ with |x − y| ≃_{C} |u − v| there is a K-biLipschitz self-map f of Γ so that f(x) = u and f(y) = v. Similarly, Γ is BLH on arcs if given any C < ∞ there is a K < ∞ so that for any two arcs γ_{1}, γ_{2} ⊂ Γ with diam(γ_{1}) ≃_{C} diam(γ_{2}), there is a K-biLipschitz self-map g of Γ so that f(γ_{1}) = γ_{2}.

Curves homogeneous with respect to quasiconformal maps are considered in [2],

[3] and [10]. Bi-homogeneous sets for quasiconformal mappings are considered in [9]. Also see [11] for further results on BLH curves in higher dimensions. I thank David Herron for his comments on a earlier version of this manuscript, and in particular for pointing out the results of [4] and [6].

2. Proof of Theorem 1.1.

The proof of Theorem 1.1 is by contradiction. We shall assume Γ is a BLH closed Jordan curve in ℝ^{d} which fails the bounded turning condition and we shall deduce that for every point x ∈ Γ and arbitrarily small ϵ > 0 there is a radius r > 0 so that the ball B(x, r) contains ≃ ϵ^{−2} points of Γ which are pairwise ≳ ϵr apart. Using this we can rescale Γ and pass to a limit, obtaining a BLH closed set which has positive area, but which is not the whole plane. Finally we prove that a closed BLH

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set must either be the whole plane or have zero area. The contradiction proves that Γ must have been a quasicircle after all.

In our definition of BLH, we did not assume that the self mappings of the curve Γ preserve orientation. We start by showing that we may assume that they do. The following is the same as Lemma 2.6 of [4], but we include a proof for completeness.

Lemma 2.1. Suppose Γ is Jordan curve which is K-biLipschitz homogeneous. Then Γ is homogeneous with respect to K^{2}-biLipschitz maps which preserve orientation.

Proof. Fix x, y ∈ Γ. Let E_{x} be the set of z ∈ Γ so that there is an orientation preserving K-biLipschitz map sending x to z. Then x ∈ E_{x} (because of the identity map) and E_{x} is closed (by equicontinuity and Ascoli’s theorem we can take limits). Similarly for E_{y}. If E_{x} ∪ E_{y} ≠ Γ, then choose z ∉ E_{x} ∪ E_{y}, and K-biLipschitz, orientation reversing maps f, g which map x to z and y to z respectively. Then g^{−1} ∘ f is the desired map from x to y. If E_{x} ∪ E_{y} = Γ then since Γ is connected we must have E_{x} ∩ E_{y} ≠ ∅. Choose z in the intersection and again the corresponding composition is the desired map.

The following lemma contains the main idea in proving Theorem 1.1. It says that a BLH curve which is not BT must look ‘thick’ in a certain sense.

Lemma 2.2. Suppose Γ is a closed Jordan curve in ℝ^{d} which is K-biLipschitz homogeneous with respect to orientation preserving self maps, but which does not satisfy the bounded turning condition. Then there is a C < ∞ so that for any x ∈ Γ and for any \frac{1}{10} > ϵ > 0 there is a 0 < δ ≤ ϵ and r > 0 so that the ball B(x, r) contains more than C^{−1}δ^{−2} points of Γ, any two of which are at least distance δr/C apart.

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Proof. Since Γ does not satisfy the bounded turning condition, we may assume, by dilating and translating if necessary, that 0 ∈ Γ and there is a point x ∈ Γ with |x| ≤ ϵ and so that the smaller diameter arc γ ⊂ Γ connecting 0 to x leaves the ball B(0, 1). Let γ_{0} be the smallest closed subarc of γ containing 0 with both endpoints on the circle |x| = K^{−2}/10. Similarly, let γ_{1} be the smallest open subarc of γ containing 0 and with both endpoints on {|x| = 1}. Let δ be the distance from γ_{0} to Γ \ γ_{1}. Then δ is clearly positive since we are taking the distance between disjoint compact sets. Moreover, δ ≤ ϵ since 0 ∈ γ_{0} and x ∈ Γ \ γ_{1}. Let y_{0} ∈ γ_{0} and y_{1} ∈ Γ \ γ_{1} be points where the minimum distance is obtained. Note that the shortest subarc of γ connecting γ_{1} to y_{1} has diameter at least 1/2. Let γ_{2} be the minimum closed subarc of Γ which contains γ_{1} and has its endpoints on {|z| = 1/2}. Then clearly y_{1} ∉ γ_{2}.

Let F be the set of orientation preserving, K-biLipschitz maps of Γ to itself. Choose f_{1} ∈ F so that f_{1}(y_{0}) = y_{1}. Let y_{2} = f_{1}(y_{1}). In general, choose f_{n} ∈ F so that f_{n}(y_{0}) = y_{n} and set y_{n+1} = f_{n}(y_{1}). Since f_{n} is K-biLipschitz,

|y_{n+1} − y_{n}| = |f_{n}(y_{1}) − f_{n}(y_{0})| ≤ K|y_{1} − y_{0}| ≤ Kδ.

Thus if n ≤ K^{−2}δ^{−1}, all the points lie in the ball B(0, 2K^{−1}). Since each f_{k} is orientation preserving, we see that the points {y_{k}} are distinct.

Choose a collection of K^{−2}δ^{−1}/20 points {y_{0}^{j}} on the arc γ_{0} so that y_{0}^{1} = y_{0} and so that any two of them are at least distance δ apart. For example, we can choose y_{0}^{j} in Γ ∩ {|z| = jδ} (at most two of these will have to be omitted because they are too close to y_{0}). For n = 1, …, δ^{−1} let y_{n}^{j} = f_{n}(y_{0}^{j}). Note that y_{0}^{k+1} comes after y_{0}^{k}. This gives us a collection of K^{−2}δ^{−2} points and we claim that for any two of them the distance apart is at least δ/K

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To prove this, consider y_{n}^{j} and y_{m}^{k}. If n = m = 0 then |y_{0}^{j} − y_{0}^{k}| ≥ δ by construction. If n = m ≠ 0 then

|y_{n}^{j} − y_{n}^{k}| = |f_{n}(y_{0}^{j}) − f_{n}(y_{0}^{k})| ≥ K^{−1}|y_{0}^{j} − y_{0}^{k}| ≥ K^{−1}δ

since f_{n} is K-biLipschitz. Finally suppose n ≠ m. Without loss of generality we may assume n < m. Let w_{0} = f_{n}^{−1}(y_{m}^{0}) and w_{1} = f_{n}^{−1}(y_{m}^{k}). Since our maps preserve orientation, w_{0} ∉ γ_{2}. Then w_{1} ∈ γ̃ = f_{n}^{−1}(f_{m}(γ_{0})), which is an arc of diameter at most K^{2}diam(γ_{0}) ≤ 1/10. Since γ̃ contains w_{0} (which is not in γ_{2}), and has small diameter, it cannot intersect γ_{1}. Thus w_{1} ∉ γ_{1}. Thus by our choice of δ,

|y_{n}^{j} − y_{n}^{k}| = |f_{n}(y_{0}^{j}) − f_{n}(w)| ≥ K^{−1}|y_{0}^{j} − w_{1}| ≥ Kδ.

To finish the proof we have to check the collection of points described exists for any z ∈ Γ, instead of just the special point we have chosen. However, if we simply map 0 to z by an element of F the constructed points map to a collection with similar properties (but possibly larger constant C). This completes the proof of the lemma.

We now continue with the proof of Theorem 1.1. First recall the Hausdorff metric on closed sets on ℝ^{d},

d_{H}(E, F) = inf{ϵ : F ⊂ E_{ϵ} and E ⊂ F_{ϵ}},

where X_{ϵ} = {x : dist(x, X) < ϵ}. This makes the closed subsets of ℝ^{d} into a complete metric metric space. We will say a sequence {E_{n}} converges locally in the Hausdorff metric if {E_{n} ∩ B(0, R)} converges for every R. Every sequence of closed sets has a subsequence which converges locally.

Now suppose Γ is a BLH curve in ℝ^{d}. Choose a point z ∈ Γ which is on the boundary of an open disk disjoint from Γ. Choose {ϵ_{n}} → 0 and use the

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previous lemma to find scales {r_{n}} and collections B_{n} of disjoint balls as above. After rescaling to set r_{n} = 1, we have a sequence of curves {Γ_{n}} and a sequence {δ_{n}} → 0 so that for each n there are at least Cδ_{n}^{−2} points of Γ_{n} ∩ B(z, 1) which are at least distance Cδ_{n} apart. Let Γ̃_{n} be the union of Γ_{n} and balls of radius Cδ_{n} around each point. Clearly Γ̃_{n} has area which is bounded uniformly away from zero. By passing to a subsequence if necessary we may assume that {Γ_{n}} converges locally in the Hausdorff metric to a closed set X and it is easy to check that X is also K-biLipschitz homogeneous (use equicontinuity and Ascoli’s theorem to take limits of mappings). The set X is not the entire plane because the point z was on the boundary of a disk disjoint from Γ. This easily implies X omits a half-plane. Moreover, X is also the local Hausdorff limit of a subsequence of {Γ̃_{n}} and hence has positive area by the following easy fact.

Lemma 2.3. If {X_{n}} and X are compact sets in ℝ^{d} and X_{n} → X in the Hausdorff metric. Then for any function φ,

H_{∞}^{φ}(X) ≥ lim inf_{n→∞} H_{∞}^{φ}(X_{n}).

In particular, if d = 2 and φ(t) = t^{2} , then area(X) ≥ lim sup_{n} area(X_{n}).

Proof. For any ϵ > 0 choose a cover of X by open balls {B(x_{j}, r_{j})} so that ∑ φ(r_{j}) ≤ H_{∞}^{φ}(X) + ϵ. Then X_{n} ⊂ U for large enough n and hence

H_{∞}^{φ}(X) + ϵ ≥ H_{∞}^{φ}(U) ≥ lim sup_{n} H_{∞}^{φ}(X_{n}).

Taking ϵ → 0 gives the desired conclusion.

We finish the proof of Theorem 1.1 by observing that the following result gives a contradiction. A set X is porus at scales ≤ R if there is a M < ∞ so that for every x ∈ X and every r ≤ R , there is a point y ∈ ℝ^{d} so that |x − y| ≤ r and

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dist(y, X) ≥ r/M. This condition easily implies X has zero d-dimensional measure since this condition must fail for small r if x is a Lebesgue point of density of X (e.g. Proposition I.2, [14]). (In fact, if X is porus then it has Hausdorff dimension strictly less than d.)

Lemma 2.4. Suppose X ⊂ ℝ^{d} is closed and BLH. Then X is porus at scales less than R, where R is the radius of any ball in the complement of X. In particular, either X = ℝ^{d} or X has zero d-dimensional measure.

Proof. If X ≠ ℝ^{d} then we can find a point z ∈ X which is on the boundary of a open ball B(x, R) which is disjoint from X. Now suppose X is not porus at scales ≤ R. Then for any n there is a point w_{n} ∈ X and a 0 < r_{n} < R so that every point of B(w, r) is within distance r_{n} of a point of X. Define sets Z_{n} = (X − z)/r_{n} and W_{n} = (X − w_{n})/r_{n} and a K-biLipschitz map f_{n} : Z_{n} → W_{n} which maps 0 to 0. Passing to a subsequence, if necessary, we can pass to the limit to obtain closed sets Z and W and a biLipschitz mapping f : Z → W taking 0 to 0. Since 0 is a boundary point of Z, but an interior point of W, this is clearly a contradiction, so we are done.

The argument of this section shows that if Γ is a BLH curve in ℝ^{d}, which is not BT then it has rescalings which converge to a set of positive 2-dimensional measure which is not the whole space. The point is that this is only a contradiction when d = 2. We shall see an example in Section 4 of a BLH curve in ℝ^{3} which has rescalings converging to a cylinder, S^{1} × ℝ.

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3. Proof of Corollary 1.2

Most of this is immediate from Theorem A of [6], once we know that a BLH curve must be a quasicircle. To be more precise, their result says that Γ ⊂ ℝ^{d} is BLH and BT iff it has a quasihomogeneous parameterization, i.e., condition (7) of Corollary 1.2. This trivially implies (6) and (5), and using Theorem C of [6], it implies (4) (and hence (2) and (3)). The equivalence of (1) and (8) is given by their Theorem A. The only conditions which they do not explicity mention are (9) and (10), so we include a short proof that these are equivalent BLH+BT in any dimension.

(6) ⇒ (9), (10): Condition (6) implies Γ is BT, so it suffices to prove either (9) or (10) and other is immediate. We will prove (9). Suppose γ_{1} and γ_{2} are subarcs of Γ so that diam(γ_{1}) ≃ diam(γ_{2}). Let I_{j} = h^{−1}(γ_{j}) for j = 1, 2. We claim I_{1} and I_{2} must have comparable lengths. Suppose not, e.g., suppose |I_{1}| > N|I_{2}|. Then by (6) , γ_{1} contains N disjoint subarcs, all with diameter comparable to diam(γ_{1}). This contradicts

Lemma 3.1. If γ is an arc with the bounded turning property, then it can only contain a bounded number of disjoint subarcs with diameter ≥ diam(γ)/2.

Proof. Suppose {γ_{j}}, j = 1, …, M are disjoint subarcs with diameter ≥ diam(γ) and let x_{j} denote the initial point of each arc. Since all the {x_{j}} in a ball of radius ≃ diam(γ), there must be two of them which are at most Cdiam(γ)/M^{1/n} apart. The arc between these two is contained in γ and contains one of the arcs γ_{j}, hence has diameter ≥ \frac{1}{2}diam(γ). The bounded turning condition now implies M is uniformly bounded.

BILIPSCHITZ HOMOGENEOUS CURVES 11

Now back to the proof. We now have that |I_{1}| ≃ |I_{2}|. Thus there is a biLipschitz mapping of the circle which takes I_{1} to I_{2}. Clearly h conjugates this to a biLipschitz mapping of Γ taking γ_{1} to γ_{2} as desired.

(9) ⇒ (1): The BLH condition is trivial, so we only have to show that (9) implies BT. Suppose γ is the smaller arc between x and y on Γ. Orient γ so x is the initial point and let z be the first point on γ where |x − z| = |x − y|. Let γ_{0} ⊂ γ be the arc from x to z. By (9) there is a biLipschitz map of γ taking γ_{0} to γ, hence diam(γ) ≲ diam(γ_{0}) ≲ |x − y|, as desired.

(10) ⇒ (1): As above, we need only show that (10) implies BT. Suppose γ is the smaller arc between x and y on Γ. Choose a point z ∈ γ so that |x − z| ≃ |z − y| ≃ diam(γ) and let γ_{0} be the arc from x to z. Then by (10) there is a biLipschitz map of Γ fixing x and taking z to y, so diam(γ) ≃ |x − z| ≃ |x − y|, as desired.

4. BLH curves which are not BT.

The following is essentially the same as Example 5.7 of [6]. We include it so that we can make a few further observations about the example.

Example 4.1. There is a BLH curve in ℝ^{3} which is not BT.

Proof. We will define Γ as a limit of curves {Γ_{n}}. Let Γ_{0} be a circle. Choose ϵ_{0} small and let T_{0} be {z ∈ ℝ^{3} : dist(z, Γ_{0}) = ϵ_{0}}, i.e., T_{0} is a long, thin torus with Γ_{0} as an axis. Choose δ_{0} small and let Γ_{1} be a smooth spiral curve on T_{0} with period δ_{0}ϵ_{0}. In general, given the smooth curve Γ_{n}, choose ϵ_{n} so small that T_{n} = {z ∈ ℝ^{3} : dist(z, Γ_{n}) = ϵ_{n}} locally looks like a flat cylinder (up to a very small distortion). Choose δ_{n} small and let Γ_{n+1} be a spiral on T_{n} with period δ_{n}ϵ_{n}. See Figure 4. it is clear that the normalized arclength parameterization satisfies the

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very weak homogeneity condition. and hence so does the limiting curve Γ. Thus Γ has a transitive 1 parameter biLipschitz group, but is clearly not bounded turning if δ_{n} → 0.

Example 4.2. Given ϵ > 0, the curve Γ in Example 4.1 can be constructed so that every element of the circle group acting on Γ has a (1 + ϵ) biLipschitz extension to all of ℝ^{3}.

Proof. At each stage of the construction, we can choose ϵ_{n} so small that the group acting on Γ_{n} also preserves T_{n} and preserves distances up to a factor of ϵ2^{−n} for points inside T_{n} that are at most ϵ_{n} apart. This group does not act on Γ_{n+1}, but for any x, y ∈ Γ_{n+1} can be used to move x to a point that is at most “one twist” of Γ_{n+1} away from y. We can then compose with a rotation and translation of T_{n} along its axis that maps x to y. Use the previously constructed map at points more than Mϵ_{n} from T_{n} and in the region {ϵ_{n} ≤ dist(z, T_{n}) ≤ Mϵ_{n}} interpolate between the two maps by a twist. Passing to the limit shows that Γ is homogeneous with respect to (1 + ϵ)-biLipschitz maps on all of ℝ^{3}.

Example 4.3. The curve in the previous two examples can be constructed so that it is not the orbit of uniformly biLipschitz group acting on ℝ^{3}.

Proof. Choose two points x and y that are δ_{n}ϵ_{n} apart and joined by an arc of diameter ϵ_{n}/2, i.e., x and y are nearby points on adjacent turnings of Γ at scale

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ϵ_{n}. Let γ be a path of length δ_{n}ϵ_{n} connecting x and y and suppose there is a 1 parameter K-biLipschitz group acting on ℝ^{3} for which Γ is an orbit. Move x forward along Γ by distance ϵ_{n}, i.e., move it around the spiral ϵ_{n}δ_{n+1}^{−1} times and consider what happens to the curve γ. The path cannot cross the middle half of the tube T_{n}. Therefore the image of γ must wrap around the tube ϵ_{n}/(ϵ_{n+1}δ_{n+1}) times and hence have length ≳ ϵ_{n}/δ_{n+1} ≃ diam(γ)/δ_{n+1}. Since δ_{n} → 0 we get a contradiction to the fact that γ and its image must have comparable length. Thus Γ cannot be the orbit of continuous, uniformly biLipschitz group of all of ℝ^{3}.

Example 4.4. There is a BLH curve in ℝ^{3} which is not homogeneous with respect to K-biLipschitz mappings of ℝ^{3} for any K < ∞.

Proof. This is very easy using topology. Construct a curve which has one knot at different scales tending to 0. This can easily be done with a chord-arc curve (which is therefore BT and BLH), but a neighborhood of a flat piece of the curve cannot be biLipschitz mapped to a neighborhood of a knot. See Figure 4.

Example 4.5. There is a BT curve Γ ⊂ ℝ^{3} which is homogeneous with respect to biLipschitz mappings on all of ℝ^{3} but does not have a continuous group of such mappings acting on it.

Proof. Again, this is easy using topology. Build a BT curve by choosing scales ϵ_{n} → quickly and placing knots in Γ about ϵ_{n} apart at scale ϵ_{n} and make the curve look smooth at scales in between. Then it is easy to check that Γ biLipschitz

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homogeneous with mappings that extend to all of ℝ^{3}. To see that there is no 1 parameter group acting on γ take x, y on adjacent knots at scale ϵ_{n} and let γ be a curve in the complement of Γ of length |x − y| and connecting them. Now move x continuously along the curve at least distance Mϵ_{n}. The images of γ can never cross Γ and hence part of γ must stay near the original knot, while the endpoints of γ move farther and farther away. This means the length of the images is getting longer and longer, a contradiction.

5. Further questions

We saw that in ℝ^{2} that every BLH curve has a parameterization which conjugates rotations on the circle to biLipschitz self-maps of Γ. What about higher dimensions?

Question 5.1. Is a curve in ℝ^{d}, d > 2 BLH iff it has a VWQH parameterization?

What happens if biLipschitz maps are replaced by a larger class, e.g. quasisymmetric maps?

Question 5.2. Suppose Γ is a closed Jordan curve in ℝ^{2} which is homogeneous with respect to QS maps. Must Γ be a quasicircle?

Question 5.3. Characterize the BLH compact sets in ℝ^{2}? In ℝ^{1}? In ℝ^{d}, d > 3?

Note that there are non-locally connected examples in ℝ^{2}, e.g., a Cantor set cross a circle. Even in the case of compact, connected plane sets, this may not

BILIPSCHITZ HOMOGENEOUS CURVES 15

be easy. Continua which are homogeneous with respect to self homeomorphisms is a large and active area and even in the plane homogeneous continua have not been completely classified, although examples other than Jordan curves are known (see [13], [12] for a survey of some recent results). It is known, for example, that Jordan curves are the only homogeneous continua in the plane which contain arcs, or are locally connected, or are decomposable (i.e., can be written as a union of two proper sub-continua). Thus in Theorem 1.1 we could replace “Jordan curve” by just “curve” (a continuous image of S^{1}) or “locally connected continuum”. It seems reasonable that none of the more exotic homogeneous planar continua can occur in the biLipschitz case.

Question 5.4. If X is a BLH continua in ℝ^{2}, must it be a closed Jordan curve (and hence a quasicircle)? What if we assume the biLipschitz constant is close to 1?

This in not true in ℝ^{3} because one can build a solenoid which is biLipschitz homogeneous, but not locally connected (locally it look like a Cantor set cross an interval). Does a biLipschitz homogeneous continuum in ℝ^{d} always contain an arc?

The only known homogeneous continua in the plane are a point, a closed Jordan curve, the pseudo-arc and a circle of pseudo-arcs [12]. Thus a special case of the question above is

Question 5.5. Is there a pseudo-arc which is BLH?

See [1] for a definition of pseudo-arc and a proof that it is homogeneous. Briefly, a pseudo-arc is the intersection of sets D_{n} where each D_{n} is a union of a “chain” of open sets D_{n} = {D_{n}^{i}}. Each element D_{n}^{i} of the chain has diameter ≤ 1/n and intersects D_{n}^{j} iff |i − j| = 1 and the chain D_{n} is “crooked” in the previous chain

D_{n−1}. This means that each link of D_{n} is contained in a link of D_{n−1} and that if

i < j and D_{n}^{i} intersects D_{n−1}^{h} and D_{n}^{j} intersects D_{n−1}^{k} with |h − k| > 2, then there

are links D_{n}^{r} and D_{n}^{s} with i < r < s < j so that D_{n}^{r} and D_{n}^{s} are subsets of links of

D_{n−1} which are adjacent to D_{n−1}^{k} and D_{n−1}^{h} respectively.

If the answer to the previous question is no, then what is the best modulus

of continuity that a transitive set of homeomorphisms of a pseudo-arc can have?

If there is a BLH pseudo-arc (or other exotic example), what strengthening of

biLipschitz homogeneity implies implies Jordan curve? Note that in [8] it is proven

that any homogeneous, compact metric space has a compatible metric with respect

to which the space is BLH. Thus the fact the continuum is in the plane must play

an important role.

Some of our examples in Section 4 used the fact that Γ was knotted. Is this

necessary?

Question 5.6. Is there an unknoted Γ which is BLH and BT, but which is not

homogeneous with respect to BL(ℝ^{3})?

References

[1] R.H. Bing. A homogeneous indecomposable plane continuum. Duke Math. J., 15:729–742, 1948.

[2] B. Brechner and T. Erkama. On topologically and quasiconformally homogeneous continua. Ann. Acad. Sci. Fenn. Ser A I Math, 4:207–208, 1978/79.

[3] T. Erkama. Quasiconformally homogeneous curves. Michigan Math. J., 24:489–493, 1977.

[4] M. Ghamsari and D.A. Herron. Bilipschitz homogeneous Jordan curves. Trans. Amer. Math. Soc. to appear.

[5] M. Ghamsari and D.A. Herron. Higher dimensional ahlfors regular sets and chordarc curves in R^{n}. Rocky Mountain J. Math. to appear.

[6] D.A. Herron and V. Mayer. Bilipschitz homoegeneous curves in doubling metric spaces. 1998. preprint.

[7] A. Hohti. On Lipschitz homogenity of the Hilbert cube. Trans. Amer. Math. Soc., 291:75–86, 1985.

[8] A. Hohti and Junnila. A note on homoegeneus metrizable spaces. Houston J. Math., 13:231– 234, 1987.

[9] P. MacManus, R. Näkki, and B. Palka. Quasiconformally bi-homogeneous compacta in the complex plane. 1997. preprint.

[10] P. MacManus, R. Näkki, and B. Palka. Quasiconformally homogeneous compacta in the complex plane. 1997. preprint.

[11] V. Mayer. Trajectoires de groupes á 1-paramétre de quasi-isométries. Revista. Mat. Iber., 11:143–164, 1995.

[12] J.T. Rogers. Homogeneous continuua. Top. Proc., 8:213–233, 1983.

[13] J.T. Rogers. Classifying homogeneous continuua. Top. and Appl., 44:341–352, 1992.

[14] E. Stein. Singular integrals and differentiability properties of functions. Princeton University Press, 1971.

[15] P. Tukia and J. Väisälä. Quasisymmetric embeddings of metric spaces. Ann. Acad. Sci. Fenn., 5:97–114, 1980.

C.J. Bishop, Mathematics Department, SUNY at Stony Brook, Stony Brook, NY 11794-3651

   E-mail address: bishop@math.sunysb.edu