Weak chord-arc curves and double-dome quasisymmetric spheres

Let $\Omega$ be a planar Jordan domain and $\alpha>0$. We consider double-dome-like surfaces $\Sigma(\Omega,t^{\alpha})$ over $\overline{\Omega}$ where the height of the surface over any point $x\in\overline{\Omega}$ equals $\text{dist}(x,\partial\Omega)^{\alpha}$. We identify the necessary and sufficient conditions in terms of $\Omega$ and $\alpha$ so that these surfaces are quasisymmetric to $\mathbb{S}^2$ and we show that $\Sigma(\Omega,t^{\alpha})$ is quasisymmetric to the unit sphere $\mathbb{S}^2$ if and only if it is linearly locally connected and Ahlfors $2$-regular.


Introduction
A metric space which is quasisymmetric to the standard n-sphere S n is called a quasisymmetric n-sphere. Quasisymmetric circles were completely characterized by Tukia and Väisälä in [9]. Bonk and Kleiner [5] identified a necessary and sufficient condition for metric 2-spheres to be quasisymmetric spheres. A consequence of their main theorem is that if a metric 2-sphere is linearly locally connected (or LLC) and Ahlfors 2-regular then it is a quasisymmetric 2-sphere. Although the LLC property is necessary, there are examples of quasisymmetric 2-spheres constructed by snowflaking procedures that fail the 2-regularity [4], [6], [8].
As it turns out, for these surfaces the 2-regularity is necessary for quasisymmetric parametrization by S 2 .
Theorem 1.1. The surface Σ(Ω, t α ) is quasisymmetric to S 2 if and only if it is linearly locally connected and 2-regular.
What conditions on Ω and α ensure that Σ(Ω, t α ) is a quasisymmetric 2-sphere? When α > 1, Theorem 1.1 is trivial since, for any Jordan domain Ω, the surface Σ(Ω, t α ) is not linearly locally connected and hence not quasisymmetric to S 2 . If α = 1, Σ(Ω, t) is quasisymmetric to S 2 if and only if ∂Ω is a quasicircle [12,Theorem 1.1]. This result, combined with the fact that the projection of Σ(Ω, t) on Ω is a bi-Lipschitz mapping, and the fact that Ω is 2-regular if ∂Ω is a quasicircle, gives Theorem 1.1 when α = 1.
In the case α ∈ (0, 1), the part of Σ(Ω, t α ) near ∂Ω × {0} resembles the product of ∂Ω with an interval. Väisälä [10] has shown that the product γ × I of a Jordan arc γ and an interval I is quasisymmetric embeddable in R 2 if and only if γ satisfies the chord-arc condition (2.2). In view of this result, it is expected that the chord-arc property of ∂Ω is necessary for Σ(Ω, t α ) to be a quasisymmetric sphere. Moreover, as the double-dome-like surface envelops the interior Ω above and below, the following condition on Ω is needed to ensure the linear local connectedness of the surfaces Σ(Ω, t α ). We say that Ω has the level quasicircle property (or LQC property) if there exist 0 > 0 and K > 1 such that for all ∈ [0, 0 ], the -level set of Ω γ = {x ∈ Ω : dist(x, ∂Ω) = } is a K-quasicircle. A consequence of Theorem 1.2 in [12] is that if a planar Jordan domain Ω has the LQC property and ∂Ω is a chord-arc curve then Σ(Ω, t α ) is quasisymmetric to S 2 for all α ∈ (0, 1).
For these surfaces, the LQC property of Ω is essential: if a Jordan domain Ω does not have the LQC property then Σ(Ω, t α ) is not quasisymmetric to S 2 for any α ∈ (0, 1); see Lemma 4.6. However, the chord-arc condition of ∂Ω is not necessary. Contrary to the intuition based on Väisälä's result [10], we construct in Section 5.1 a Jordan domain Ω whose boundary is a nonrectifiable curve and Σ(Ω, t α ) is a quasisymmetric sphere for all α ∈ (0, 1).
Instead, only a weak form of the chord-arc condition is needed: a Jordan curve Γ is said to have the weak chord-arc condition if there exists N 0 > 1 such that every subarc Γ ⊂ Γ with diam Γ ≤ 1 can be covered by at most N 0 (diam Γ ) −1 subarcs Γ i ⊂ Γ of diameter at most (diam Γ ) 2 . Under this terminology we have the following.
(3) Ω has the LQC property and ∂Ω is a weak chord-arc curve.
Although not neccessarily rectifiable, any weak chord-arc quasicircle has Hausdorff dimension equal to 1. In fact, a slightly stronger result holds true. Theorem 1.3. If a quasicircle satisfies the weak chord-arc property then it has Assouad dimension equal to 1.
The example in Section 5.2, however, shows that the converse is false. Moreover, since the Assouad dimension is larger than the Hausdorff, upper box counting and lower box-counting dimensions, the conclusion of Theorem 1.3 holds for any of these dimensions. A consequence of Theorem 1.3 is that if Ω is a Jordan domain and ∂Ω has Assouad dimension greater than 1, then the surface Σ(Ω, t α ) is not quasisymmetric to S 2 for any α ∈ (0, 1).
In Section 3 we define an index that measures how much a curve Γ deviates from being a chord-arc curve on a certain scale, and we discuss the weak chord-arc condition. The proofs of Theorem 1.2 and Theorem 1.3 are given in Section 4. Finally, two examples, based on homogeneous snowflakes, illustrating the weak chord-arc condition are presented in Section 5.
Acknowledgments: The author is grateful to Pekka Pankka, Kai Rajala and his adviser Jang-Mei Wu for stimulating discussions and valuable comments on the manuscript.

Preliminaries
2.1. Definitions and notation. An embedding f of a metric space ( A metric n-sphere S that is quasisymmetrically homeomorphic to S n is called a quasisymmetric sphere when n ≥ 2, and a quasisymmetric circle when n = 1. Beurling and Ahlfors [3] showed that a planar Jordan curve is a quasisymmetric circle if and only if it is a K-quasicircle (K ≥ 1), i.e., the image of the unit circle S 1 under a K-quasiconformal homeomorphism of R 2 . A geometric characterization due to Ahlfors [1] states that a Jordan curve γ is a K-quasicircle if and only if it satisfies the 2-point condition: (2.1) there exists C > 1 such that for all x, y ∈ γ, diam γ(x, y) ≤ C|x−y|, where the distortion K and the 2-point constant C are quantitatively related.
Here and in what follows, given two points x, y on a metric circle γ we denote by γ(x, y) the subarc of γ connecting x and y of smaller diameter, or either subarc when both have the same diameter. Tukia and Väisälä [9] proved that a metric circle is a quasisymmetric circle if and only if it is doubling and satisfies condition (2.1).
For the rest, a Jordan arc is any proper subarc of a Jordan curve and a quasiarc is any proper subarc of a quasicircle.
A rectifiable Jordan curve γ in R 2 is called a chord-arc curve if where γ (x, y) is the shortest component of Γ\{x, y} and (·) denotes length. It is easy to see that a rectifiable curve γ is a chord-arc curve if and only if (γ(x, y)) ≤ C|x − y| for some C > 1; here constants c and C are quantitatively related .
The notion of linear local connectivity generalizes the 2-point condition (2.1) on curves to general spaces. A metric space X is λ-linearly locally connected (or λ−LLC) for λ ≥ 1 if the following two conditions are satisfied.
A metric space X is said to be Ahlfors Q-regular if there is a constant C > 1 such that the Q-dimensional Hausdorff measure H Q of every open ball B(a, r) in X satisfies Bonk and Kleiner found in [5] an intrinsic characterization of quasisymmetric 2-spheres and then derived a readily applicable sufficient condition.
. Let X be an Ahlfors 2-regular metric space homeomorphic to S 2 . Then X is quasisymmetric to S 2 if and only if X is LLC.
For x ∈ R n and r > 0, define B n (x, r) = {y ∈ R n : |x − y| < r}. In addition, let R 3 be the lower half-space. For any a = (a 1 , a 2 , a 3 ) ∈ R 3 , denote by π(a) = (a 1 , a 2 , 0) the projection of a on the plane R 2 × {0}. For x, y ∈ R 2 , denote by [x, y] the line segment having x, y as its end points.
In the following, we write u v (resp. u v) when the ratio u/v is bounded above (resp. bounded above and below) by positive constants. These constants may vary, but are described in each occurrence. In general the sets γ need not be connected and if connected need not be curves see [11, Figure 1]. We say that Ω has the level quasicircle property (or LQC property), if there exist 0 > 0 and K ≥ 1 such that the level set γ is a K-quasicircle for every 0 ≤ ≤ 0 . Sufficient conditions for a domain Ω to satisfy the LQC property have been given in [11] in terms of the chordal flatness of ∂Ω, a scale-invariant parameter measuring the local deviation of subarcs from their chords.
We list some properties of the level sets from [11] which are used in the proof of Theorem 1.2.   Let Ω be a Jordan domain that has the LQC property. If ∂Ω is a c-chord-arc curve then there exist 0 > 0 and c > 1 such that γ is a c -chord-arc curve for all ∈ [0, 0 ].

A weak-chord arc property
3.1. A chord-arc index. Let Γ be a Jordan curve or arc. A partition P of Γ is a finite set of mutually disjoint (except for their endpoints) subarcs of Γ whose union is all of Γ. We denote with |P | the number of elements a partition P has. A δ-partition of Γ, for δ ∈ (0, 1), is a partition P of Γ such that for each Γ ∈ P Standard compactness arguments show that every Jordan curve or arc has δ-partitions for all δ ∈ (0, 1). Moreover, if P is a δ-partition of Γ then |P | ≥ 1/δ. For a partition P of Γ define  (1) If P is a δ-partition then 1 2 |P |δ ≤ M (Γ, P ) ≤ |P |δ. (2) Suppose that P = {Γ 1 , . . . , Γ N } is a partition of Γ and for each i = 1, . . . , N , P i is a partition of Γ i . Then, (3) Assume that Γ is a K-quasiarc and δ ∈ (0, 1). There exists M 1 > 1 depending on K such that M (Γ, P )/M (Γ, P ) ≤ M 1 for any two δpartitions P , P of Γ.
Proof. Property (1) is an immediate consequence of the definition. For In the proof of Lemma 3.1 we used a covering property of quasicircles which we prove in the following lemma.
Lemma 3.2. For each K > 1 there exists a number C > 1 depending only on K such that, if Γ is a K-quasiarc and λ ∈ (0, 1] then, each partition Proof. We may assume that diam Γ = 1. Fix λ ∈ (0, 1) and let {Γ 1 , . . . , Γ N } be a partition of Γ with diam Γ i ≥ λ. Let A = {x 0 , x 1 , . . . , x N } be the set of endpoints of Γ 1 , . . . , Γ N . The 2-point condition of Γ implies that there exists C > 1 depending on K such that |x − y| ≥ λ/C for all x, y ∈ A. Since all points of A lie in R 2 which is a doubling, space, there is a universal The next lemma is another application of Lemma 3.2.
Proof. We prove the lemma for N 1 = 5M 1 N 0 where M 1 is as in the third part of Lemma 3.1. Contrary to the claim, suppose that there exist a number δ ∈ [δ, 1) and a δ - The following lemma is used in the proof of Theorem 1.3.
Let k 0 ∈ N be the smallest integer that is greater than log (δ/2) log δ . Let Define P * = w P w where the union is taken over all words w = i 1 . . . i k 0 . Note that for each Γ w ∈ P * , Moreover, by our assumptions, for all partitions P w defined, If P is a δ-partition of Γ then, for each Γ ∈ P and each Γ w ∈ P * , diam Γ w < diam Γ . Thus, |P | ≤ |P * | < N which is a contradiction.

3.2.
A weak chord-arc property. Note that, if a curve Γ is a chord-arc curve then there exists M 0 > 1 such that for all subarcs Γ ⊂ Γ, all δ > 0 and all δ-partitions P of Γ , we have M (Γ , P ) < M 0 .
A curve or arc Γ is said to have the weak chord-arc property if there exists In other words, a curve Γ is a weak chord-arc curve if there exists M 0 ≥ 1 such that any subarc Γ of Γ can be partitioned to at most 2M 0 / diam Γ subarcs Γ 1 , . . . , Γ N of diameters comparable to (diam Γ ) 2 . It is clear that a chord-arc curve is a weak chord-arc curve but the converse fails; see Section 5.1. The third claim of Lemma 3.1 implies that, for quasicircles, the weak chord-arc property is equivalent to a stronger condition.
Suppose now that α ∈ (0, 1 2 ) and let k 0 be the smallest integer with Inductively, suppose that Γ w has been defined where w = i 1 · · · i k , i j ∈ N and k < k 0 . Then, let Define P * = w P w where the union is taken over all words w = i 1 · · · i k 0 constructed as above. Note that P * is a By interchanging the roles of 1 2 and α, and applying the arguments in the proof of Lemma 3.5, the following converse can be obtained.

Proofs of the main results
The proof of Theorem 1.2 is given in Section 4.1 and Section 4.2. By the theorem of Bonk and Kleiner, it is clear that (2) implies (1).
To show that (3) implies (2) we first show in Lemma 4.6 that if Ω satisfies the LQC property then Σ(Ω, t α ) is LLC for all α ∈ (0, 1). Then, in Proposition 4.1 we prove that if Ω has the LQC property and ∂Ω is a weak chord-arc curve then Σ(Ω, t α ) is 2-regular.
To prove that (1) implies (3), we show in Lemma 4.6 that if Σ(Ω, t α ) is LLC for some α ∈ (0, 1) then Ω satisfies the LQC property. Then, in Proposition 4.7 we show that if Ω has the LQC property and Σ(Ω, t α ) is quasisymmetric to S 2 then ∂Ω is a weak chord-arc curve.
The proof of Theorem 1.3 is given in Section 4.3.  For the rest of Section 4.1 we assume that there exist C 0 > 1 and 0 ∈ (0, 1 8C 0 ) such that for any ∈ [0, 0 ], the level set γ is a quasicircle and satisfies (2.1) for some C 0 > 1. To show (2.3) we first apply some reductions on a ∈ Σ(Ω, t α ) and r > 0.
for some C > 1 and for all a ∈ Σ(Ω, t α ) + and r > 0 sufficiently small.
for all a ∈ Σ(Ω, t α ) + and r > 0 as above, there exist square pieces D 1 and D 2 such that Thus, by (iii), choosing 0 small enough, we may assume from now on that diam D ≤ (4C 0 ) −1 for all square pieces D. By Lemma 4.2 and the discussion above, Proposition 4.1 is now equivalent to the following lemma.
for all square pieces D on Σ(Ω, t α ) + defined as above.
The lower bound in (4.2) was shown in [12, Section 6.2.1]. For the upper bound, the following lemma is used; we only give a sketch of its proof since it is similar to the discussion in [12, Section 6.2.1].
Lemma 4.4. Let S be a closed subset of Σ(Ω, t α ) + and 0 ≤ t 2 ≤ t 1 be such that π(S) intersects with γ t if and only if t ∈ [t 2 , t 1 ]. Suppose that, for all t, t ∈ [t 2 , t 1 ], the Hausdorff distance between π(S) ∩ γ t and π(S) ∩ γ t is less than c 1 |t − t | for some c 1 > 1, and π(S) ∩ γ t is a c 2 -chord-arc curve for some c 2 > 1. Then H 2 (S) ≤ C(diam S) 2 for some C depending on c 1 , c 2 .
Proof. Fix > 0. Let t 2 = τ 1 < · · · < τ N = t 1 be such that the sets S i = S ∩ (γ τ i × {τ α i }) satisfy /4 ≤ dist(S i , S i+1 ) ≤ /2. By the first assumption of the lemma, it is straightforward to check that N ≤ N 1 diam S/ for some N 1 depending on c 1 . The second assumption implies that each S i contains points x i,1 , . . . , x i,n i satisfying |x i,j − x i,j+1 | ≤ /2 and n i ≤ N 2 diam S i / ≤ N 2 diam S/ for some N 2 depending on c 2 . Thus, S can be covered by at most N 1 N 2 (diam S) 2 / 2 balls of radius and the lemma follows.
For the upper bound of (4.2) we consider two cases. The first case is an application of Lemma 4.4 while in the second case we use the weak chord-arc condition to subdivide D into smaller pieces on which the first case applies.
-partition of Γ i and set P 2 = {Γ ij }. The weak chord-arc condition and Lemma 3.2 imply that

Inductively, we obtain partitions
It is easy to see that the latter series converges. Since diam Γ 0 ≤ diam D, we conclude that H 2 (D) (diam D) 2 and the proof is complete.

4.2.
The LQC property and Väisälä's method. The connection between the LQC property of Ω and the LLC property of Σ(Ω, t α ) is established in the following proposition from [12].  1) and Ω is a Jordan domain whose boundary ∂Ω is a quasicircle. Then Σ(Ω, t α ) is LLC if and only if Ω has the LQC property.
It turns out that the quasicircle assumption of ∂Ω can be dropped in Proposition 4.5.
We now show that the weak chord-arc condition of ∂Ω is necessary for Σ(Ω, t α ) to be quasisymmetric to S 2 . This concludes the proof of Theorem 1.2. The proof follows closely that of [12,Proposition 6.2]. The main idea used is due to Väisälä from [10].
Proof. By our assumptions, there exist 0 > 0 and C > 1 such that, for all ∈ [0, 0 ], the set γ satisfies (2.1) with constant C. Set ∂Ω = Γ. Suppose that the claim is false. Then, by Remark 3.6, for each n ∈ N, there exists a subarc Γ n ⊂ Γ, of diameter less than 1, and a (diam Γ n ) 1 α −1partition P n = {Γ n,1 , . . . , Γ n,Nn } with M (Γ n , P n ) > 4Cn. By Lemma 3.1, the latter implies that Let {x n,0 , x n,1 , . . . , x n,Nn } be the endpoints of the arcs Γ n,1 , . . . , Γ n,Nn , ordered consecutively according to the orientation in Γ n with x n,0 , x n,Nn being the endpoints of Γ n . By adding more points from each subarc Γ n,i , in this collection, we may further assume that

It follows that
Set d n = (10C 2 ) −1 min 1≤i≤Nn |x n,i − x n,i−1 | and note that d α n diam Γ n . The rest of the proof is identical to that of [12, Proposition 5.1] by setting ϕ(t) = t α therein. We sketch the remaining steps for the sake of completeness.
Fix n ∈ N and write N n = N and x n,i = x i . Assume that there exists an η-quasisymmetric map F from Σ(Ω, t α ) onto S 2 . Composing F with a suitable Möbius map we may assume that F (Σ(Ω, t α ) + ) is contained in the unit disc B 2 . Since ∂Ω is a quasicircle, we can find points w 0 , . . . , w N on Γ and points w 0 , . . . , w N on γ d = {x ∈ Ω : dist(x, ∂Ω) = d} which follow the orientation of {x 0 , . . . , Let D be the square-like piece on Σ(Ω, t α ) + whose projection on R 2 × {0} is the Jordan domain bounded by Γ(w 0 , w N ), The partition of Γ(w 0 , w N ) into the subarcs Γ(w i−1 , w i ), i = 1, . . . , N induces a partition of D into N tall and narrow strips D i with height in the magnitude d α and width in the magnitude d. Each D i is further partitioned by planes parallel to R 2 × {0} into k square-like pieces D ij with k d α−1 . The quasisymmetry of F implies that (diam F (D ij )) 2 ≤ c 1 H 2 (D ij ) with c 1 > 1 depending on η, C. Summing first over j and then over i, and applying Hölder's inequality twice, we obtain (diam F (D)) 2 ≤ N kc 1 H 2 (D) ≤ c 2 nH 2 (D) with c 2 > 1 depending on η, C. On the other hand, the quasisymmetry of F on D implies that H 2 (F (D)) ≤ c 3 (diam F (D)) 2 with c 3 > 1 depending on η, C. Since diam F (D) = 0, letting n → ∞, we obtain a contradiction.

Assouad dimension. The Assouad dimension of a metric space (X, d), introduced in [2]
, is the infimum of all s > 0 that satisfy the following property: there exists C > 1 such that for any Y ⊂ X and δ ∈ (0, 1), the set Y can be covered by at most Cδ −s subsets of diameter at most δ diam Y . In a sense, the main difference between Hausdorff and Assouad dimension of a space X is that that the former is related to the average small scale structure of X, while the latter measures the size of X in all scales. See [7] for a detailed survey of the concept. The claim of the remark becomes evident after noticing that for all subsets Y of a K-quasicircle Γ there exists a subarc Γ ⊂ Γ containing Y , such that diam Γ ≤ C −1 diam Y for some C > 1 depending on K.
We now turn to the proof of Theorem 1.3.
Proof of Theorem 1.3. Suppose that Γ is a K-quasicircle with Assouad dimension greater than 1; in particular, greater than 1 + for some fixed ∈ (0, 1). We claim that Γ does not have the weak chord-arc property. Contrary to the claim, assume that Γ satisfies the weak chord-arc condition for some M 0 > 1. By Lemma 3.3 there exists N 0 > 1 depending on M 0 , K such that for all Γ ⊂ Γ with diam Γ < 1 and all diam Γ -partitions P of Γ we have |P | < N 0 . By our assumption on the Assouad dimension of Γ, there exists a subarc Γ ⊂ Γ and a number δ ∈ (0, 1) such that all β . Assume first that δ > diam Γ . Let P be a δ-partition of Γ . The weak chord-arc property of Γ and Lemma 3.3 yield |P | ≤ N 0 δ −1 < M δ −1− which is false.
Assume now that δ ≤ diam Γ . Let α ∈ (0, 1) be such that (diam Γ ) The assumption on diam Γ , δ and the fact that diam Γ < 1/2 yield β . By our assumptions on δ and diam Γ we have Apply Lemma 3.4 for Γ with δ = diam Γ and N = δ −1− . There exists a subarc Γ ⊂ Γ and a diam Γ -partition P such that We create a diam Γ -partition of Γ as follows. For each σ ∈ P let P (σ) be a (diam Γ ) 2 diam σ -partition of σ; for those σ ∈ P that satisfy diam σ < (diam Γ ) 2 set P (σ) = {σ}. Define P to be the union of all partitions P (σ). It is easy to see that P is a diam Γ -partition of Γ and Lemma 3.1 gives The latter, however, is false by Lemma 3.1 and the proof is complete.

Examples from homogeneous snowflakes
Let N ≥ 4 be a natural number and p ∈ (1/4, 1/2). A homogeneous (N, p)-snowflake is constructed as follows. Let S 0 be a regular N -gon, of diameter equal to 1/2. At the n-th step, the polygon S n+1 is constructed by replacing all of the N 4 n edges of S n by the same rescaled and rotated copy of one of the two polygonal arcs of Figure 1, in such a way that the polygonal regions are expanding. The curve S is obtained by taking the limit of S n , just as in the construction of the usual von Koch snowflake. It is easy to verify that every homogeneous snowflake satisfies (2.1) for some C depending on N, p, and as a result is a quasicircle.
Let E be an edge of some S n towards the construction of S. Denote with S E the subarc of S, of smaller diameter, having the same endpoints as E. The next lemma will ease some of the computations in the rest.  Proof. The necessity is clear. For the sufficiency, fix a subarc Γ ⊂ S and let n 0 be the greatest integer n for which Γ is contained in S E for some edge E of S n . Assume for now that n 0 > 0. Denote by E i , i = 1, . . . , 4, the oriented four segments constructed after E in the n 0 + 1 step, that is . . , 4}, let E wi , be the oriented four segments constructed after E w in the n 0 + k + 1 step.
If n 0 = 0 then apply the arguments above for Γ ∩ S E 1 , . . . , Γ ∩ S E N where E 1 , . . . , E N are the edges of S 0 .

5.1.
A non-rectifiable Jordan curve that satisfies the weak chordarc property. Let S be the homogeneous (N, p)-snowflake where the first polygonal arc in Figure 1 is used only at the 10 n -th steps, n ∈ N. We also require that diam S < (4p) −1 . Note that at the 10 k step of construction, the length of the polygonal curve S 10 k is equal to (4p) k . Thus, S is not rectifiable. We claim that S has the weak chord-arc property.
By Lemma 5.1, it suffices to check that all subarcs S E are weak chord-arc curves. Fix an edge E built at step n. Then diam E ≥ 4 −n . Let k 0 be the smallest k ∈ N such that diam E ≤ (diam E) 2 for all E ∈ S n+k . We claim that k 0 ≤ 9n. Indeed, the construction of S implies that at step 10n, each edge has diameter equal to (4p)4 −9n diam E ≤ (4p)(diam E) 10 since the first polygonal arc in Figure 1 has been used only once. The claim follows from our assumption that diam E ≤ diam S < (4p) −1 .
Let P be the set of all subarcs S E where E are constructed at step n + k 0 and have E as their common parent. Then, diam E = 4 −k 0 C diam E with C = 4p if the first polygonal arc has been used in the k 0 steps or C = 1 otherwise. Since k 0 is minimal, Therefore, |P | = 4 k 0 ≤ 4C(diam E) −1 = 4C(diam S E ) −1 . By Lemma 3.2, there exists a diam S E -partition of S E that has at most C (diam S E ) −1 elements, for some C > 1 depending on N, p. Hence, S E has the weak chord-arc property.
Corollary 5.2. There exists a Jordan domain Ω with nonrectifiable boundary such that Σ(Ω, t α ) is a quasisymmetric sphere for all α ∈ (0, 1]. Proof. It follows from the discussion in Section 7 of [11] that there exists p 0 ∈ ( 1 4 , 1 2 ) and an integer N 0 ≥ 4 such that every homogeneous (N, p)snowflake with p ≤ p 0 , N ≥ N 0 bounds a domain that satisfies the LQC property. Let Ω be the domain bounded by the snowflake constructed above with p ≤ p 0 , N ≥ N 0 . Since S is a quasicircle, Σ(Ω, t) is a quasisymmetric sphere. Moreover, the weak chord-arc property of S and Theorem 1.2 imply that Σ(Ω, t α ) is a quasisymmetric sphere for all α ∈ (0, 1).

5.2.
A quasicircle of Assouad dimension 1 that does not satisfy the weak chord-arc property. Let S be the homogeneous (N, p)-snowflake where the first polygonal arc in Figure 1 is used only at the n 2 -th steps for n ∈ N. For convenience we also assume that each edge of S 0 has length equal to 1. Then, if E is an edge of S n , diam E = 4 −n (4p) √ n where, x denotes the greatest integer which is smaller than x.
We show that S has Assouad dimension equal to 1 but does not have the weak chord-arc property.
Fix > 0; we claim that S has Assouad dimension less than 1+ . Similarly to Lemma 5.1, it is easy to show that it is enough to verify the Assouad condition only for the subarcs S E . Take δ ∈ (0, 1) and an edge E of the n-th step polygon S n , for some n ∈ N. Let m be the largest integer such that diam E ≥ δ diam E for all edges E of S m and P be the set of all subarcs S E ⊂ S E where E is an edge of S m . Then, 4 n−m (4p) Therefore, |P | = 4 m−n ≤ 4 M (4p) 2 δ −1− and the claim follows. Since was chosen arbitrarily, S has Assouad dimension equal to 1.
We show now that S does not have the weak chord-arc property. Let n ∈ N, E be an edge of the n-th step polygon S n and m ≥ n be the greatest integer such that diam E ≥ (diam E) 2 for each edge E of S m . Let P be the set of all subarcs S E ⊂ S E where E are edges of S m . Then, Note that as n goes to infinity, m goes to infinity and n/m goes arbitrarily close to 1/2. Hence, m > 25 16 n for all sufficiently large n. Therefore, which goes to infinity as n goes to infinity. Thus S is not a weak chord-arc curve.