CONFORMAL GEOMETRY AND DYNAMICSAn Electronic Journal of the American Mathematical SocietyVolume 00, Pages 000–000 (Xxxx XX, XXXX)S 1088-4173(XX)0000-0

GLOBAL CONFORMAL ASSOUAD DIMENSION IN THEHEISENBERG GROUP

JEREMY T. TYSON

Abstract. We study global conformal Assouad dimension in the Heisen-berg group Hn. For each α ∈ 0 ∪ [1, 2n + 2], there is a bounded set inHn with Assouad dimension α whose Assouad dimension cannot be low-ered by any quasiconformal map of Hn. On the other hand, for any set Sin Hn with Assouad dimension strictly less than one, the infimum of theAssouad dimensions of sets F (S), taken over all quasiconformal mapsF of Hn, equals zero. We also consider dilatation-dependent bounds forquasiconformal distortion of Assouad dimension. The proofs use recentadvances in self-similar fractal geometry and tilings in Hn and regularityof the Carnot-Caratheodory distance from smooth hypersurfaces.

1. Introduction

The conformal (Hausdorff) dimension of a metric space (X, d) is

(1.1) C dimH X = inf dimH Y,

the infimum taken over all metric spaces (Y, d′) which are quasisymmetricallyequivalent to X. Here dimH denotes Hausdorff dimension. Conformal di-mension was introduced by Pansu [40] in connection with the quasi-isometricclassification problem in geometric group theory, and has since found sig-nificant application in uniformization and rigidity questions in analysis onmetric spaces and the theory of Gromov hyperbolic groups, see e.g., Bourdon[14], [13], Keith–Laakso [29], Bonk–Kleiner [12].

Now assume that (X, d) is a fixed metric space, which we regard as anambient environment. For a given subset S ⊂ X, we define the globalconformal (Hausdorff) dimension of S to be

(1.2) GC dimH S = GC dimXH S = inf dimH F (S),

the infimum taken over all quasisymmetric homeomorphisms F of X ontoitself. Global conformal dimension has been considered by the author andvarious collaborators [45], [10], [11], [47] in the case when the ambient is a

Received by the editors January 25, 2008.1991 Mathematics Subject Classification. Primary 30C65; Secondary 28A78, 43A80.Key words and phrases. Quasiconformal map, conformal dimension, Assouad dimen-

sion, Heisenberg group, self-affine tiling.Research supported by NSF grant DMS 0555869.

c©1997 American Mathematical Society

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Euclidean space, and by Balogh [4] in the case when the ambient is the firstHeisenberg group equipped with a sub-Riemannian metric.

Recently, Kovalev [33] answered in the affirmative a question posed bythe author on the range of conformal and global conformal dimension. Moreprecisely (see Theorems 1.1 and 1.2 in [33]), he showed that

(1.3) C dimH X ∈ 0 ∪ [1,∞]

for all metric spaces (X, d), and

(1.4) GC dimVH S ∈ 0 ∪ [1,dim V ]

for any S ⊂ V , where V is any separable Banach space. In short, conformalHausdorff dimension takes on no value strictly between zero and one.

Kovalev’s result (1.4) relies on the linear structure of the ambient space,and admits no obvious extension to more general ambients. For example, itleaves open the study of global conformal dimensions of subsets of the sub-Riemannian Heisenberg group or more general Carnot groups. In [46], weestablished (1.4) in the case V = Rn for a different notion of dimension, theso-called Assouad dimension dimA. Thus we showed that global conformalAssouad dimension GC dimRn

A (defined as in (1.2) but with dimA replacingdimH) takes on no value strictly between zero and one. In this paper,we extend the results of [46] to the Heisenberg group Hn. The range ofGC dimHn

H remains unknown.Let us briefly recall the definition of the Assouad dimension. Let (X, d)

be a metric space. The Assouad dimension of X is

dimA X = infs ≥ 0 : X is s-homogeneous,where we say that X is s-homogeneous if the s-homogeneity constant of X,

Cs(X) := supΛ⊂X

(#Λ)(

mind(x, y) : x, y ∈ Λ, x 6= ydiam Λ

)s

,

is finite, the supremum taken over all finite sets Λ ⊂ X. Here #Λ denotesthe cardinality of Λ. We say that dimA X = ∞ if X is not s-homogeneousfor any finite s. We always have dimH X ≤ dimA X [35, section 3]. For anyopen set U ⊂ Rn, dimA U = n.

Assouad dimension is a global construct which records the growth rateof the cardinality of uniformly separated subsets of X of a given diameter.It was introduced by Assouad [3] (under a different name) in connectionwith bi-Lipschitz embedding problems for doubling metric spaces. Indeed,(X, d) is a doubling space (e.g., a space of homogeneous type in the senseof Coifman and Weiss) if and only if dimA X is finite. See Heinonen [25,Chapter 10] and Luukkainen [35] for further information.

By an observation of Heinonen [25, Theorem 14.16], the conformal As-souad dimension of a uniformly perfect metric space agrees with the Ahlforsregular conformal dimension, defined as in (1.1) but with the infimum takenover all Ahlfors regular metric spaces (Y, d′) quasisymmetrically equivalentwith (X, d). Bonk and Kleiner [12] have indicated the relevance of Ahlfors

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regular conformal dimension for Cannon’s conjecture and the quasisymmet-ric uniformization problem for metric 2-spheres.

By analogy with (1.1) and (1.2), we define the conformal Assouad dimen-sion of a metric space X, resp. the global conformal Assouad dimension ofa set S ⊂ X, to be

(1.5) C dimA X = inf dimA Y,

the infimum taken over all metric spaces (Y, d′) which are quasisymmetricallyequivalent to X, resp.

(1.6) GC dimA S = GC dimXA S = inf dimA F (S),

the infimum taken over all quasisymmetric homeomorphisms F : X → X.The inequalities C dimH X ≤ C dimA X and GC dimH S ≤ GC dimA S areobvious. In [46] (see also Proposition 2.4 in this paper) we showed thatC dimA X ∈ 0 ∪ [1,∞] for every metric space X and GC dimRn

A S ∈ 0 ∪[1, n] for every S ⊂ Rn.

We now state our main result. Let Hn, n ∈ N, be the Heisenberg groupequipped with a sub-Riemannian metric of Carnot-Caratheodory type. Werecall that Hn is Ahlfors regular with homogeneous dimension Q = 2n + 2,in particular, dimA Hn = dimH Hn = 2n + 2.

Theorem 1.1. For any set S ⊂ Hn,

GC dimHn

A S ∈ 0 ∪ [1, 2n + 2],

and each value in the set 0 ∪ [1, 2n + 2] is realized as the global conformalAssouad dimension of some bounded subset of Hn.

The existence of subsets of Hn with prescribed global conformal Assouaddimension in [1, 2n + 2] is a modification, and extension to arbitrary n, ofTheorem 1.3 in [4], which concerns global conformal Hausdorff dimension inH1. The hard part of Theorem 1.1 is the proof that GC dimHn

A takes on novalue strictly between zero and one. The proof of this result is similar tothat of its Euclidean predecessor in [46], but also incorporates elements ofthe theory of self-similar fractal geometry and tilings in Heisenberg groups.

We also consider dilatation-dependent estimates for the distortion of As-souad dimension.

Theorem 1.2. Let S ⊂ Hn be s-homogeneous for some s < 2n + 2 andlet F : Hn → Hn be a K-quasiconformal homeomorphism. Then F (S) iss′-homogeneous for some s′ = s′(s, n, K) < 2n + 2.

Corollary 1.3. If S ⊂ Hn satisfies 0 < dimA S < 2n + 2 and F : Hn → Hn

is quasiconformal, then 0 < dimA F (S) < 2n + 2.

Corollary 1.3 can be made quantitative but the precise statement is some-what technical. See Theorem 4.6.

In section 2 we review basic notions and definitions and establish thesetting for the paper. Section 3 presents relevant elements of the theory

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of self-similar fractal geometry and tilings in Hn. In section 4 we relateAssouad dimension with porosity and prove Theorem 1.2 and Corollary 1.3.

Section 5 contains the proof of our main result, Theorem 1.1. In subsec-tion 5.2 we show that sets S of Assouad dimension strictly less than onehave global conformal Assouad dimension zero. To this end, we use the con-nection between Assouad dimension and porosity to identify annuli whichavoid S at all scales and locations, and implement a Koranyi–Reimann flowof quasiconformal maps which increases the conformal moduli of the sepa-rating annuli. This leads to a reduction in the Assouad dimension.

The second part of the proof of Theorem 1.1 consists in the constructionof sets which are minimal for global conformal Assouad dimension in thedimension range [1, 2n + 2]. The low codimension case [2n + 1, 2n + 2] isparticularly challenging. We consider Cantor sets of smooth closed non-characteristic hypersurfaces in Hn. Lower bounds on the global conformaldimension of such sets are established in subsection 5.1 using a method from[4]. In a concluding appendix, we establish the Ahlfors regularity of theseexamples via the regularity theory for the CC distance recently developedby Arcozzi and Ferrari [1], [2].

Acknowledgements. I am grateful to Nicola Arcozzi for providing me withan early copy of [1] and discussing the forthcoming work [2].

2. Review of definitions and basic results

2.1. Assouad dimension. We recall some basic properties of the Assouaddimension. For a more comprehensive discussion, see Luukkainen [35] andChapter 10 of [25].

The following theorem combines (6.6) and Theorem A.5 of [35] with Ex-ercise 10.16 of [25].

Proposition 2.1. Let (X, d) be a metric space, and let Y, Y ′ ⊂ X. Then

(i) (bi-Lipschitz invariance) if d′ is a metric on X which is bi-Lipschitzequivalent with d, then dimA(Y, d) = dimA(Y, d′),

(ii) (monotonicity) dimA Z ≤ dimA Y if Z ⊂ Y ,(iii) dimA Y = dimA Y , where Y denotes the closure of Y ,(iv) (finite stability) dimA Y ∪ Y ′ = maxdimA Y, dimA Y ′,(v) dimA Y = 0 if Y is finite,(vi) if 0 < ε < 1 and d′ denotes the snowflake metric d′(x, y) = d(x, y)ε

on X, then dimA(Y, d′) = dimA(Y, d)/ε,(vii) dimH Y ≤ dimA Y ,(viii) if Y carries an α-regular measure, then dimA Y = dimH Y = α.

We recall that a Borel measure µ on a metric space (X, d) is called(Ahlfors) Q-regular (Q ≥ 0) if there exists a constant C < ∞ so that

C−1rQ ≤ µ(B(x, r)) ≤ CrQ

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for every x ∈ X and r ∈ (0,diam X). If X carries a Q-regular measure µ,then µ is comparable with the Hausdorff measure HQ [25, Exercise 8.11].Similarly, when Y is a subset of X, we say that a Borel measure µ supportedon Y is (Ahlfors) α-regular (α ≥ 0) if there exists a constant C < ∞ so that

C−1rα ≤ µ(Y ∩B(y, r)) ≤ Crα

for every y ∈ Y and r ∈ (0,diam Y ).Combining Proposition 2.1(viii) with the above observations gives

Corollary 2.2. If Y is an α-regular subset of (X, d) and C dimH Y ≥ α,then

GC dimXA Y = dimA Y = GC dimX

H Y = dimH Y = α.

We say that X is Ahlfors regular if it is Q-regular for some Q ≥ 0. EveryAhlfors regular space is uniformly perfect which means that there exists c > 0so that cr ≤ diam B(x, r) ≤ 2r for all x ∈ X and r ∈ (0,diam X). Thisfollows easily from volume considerations. The Ahlfors regular conformaldimension of a metric space (X, d) is ARC dim X = inf dimH Y , the infimumtaken over all Ahlfors regular spaces (Y, d′) which are quasisymmetricallyequivalent to X. By Theorem 14.16 of [25] ARC dim X = C dimA X foreach complete, uniformly perfect metric space (X, d).

In the proof of Theorem 1.1 we will employ the following lemma whichgoes back to Bourdon [13] and Pansu [41].

Lemma 2.3 (Bourdon, Pansu). Let X be a compact α-regular metric space,and suppose that X contains a family of curves Γ which is equipped with aprobability measure µ so that

(i) infγ∈Γ diam γ > 0, and(ii) supx∈X µγ ∈ Γ : γ ∩B(x, r) 6= ∅ ≤ Crα−1 for all r > 0.

Then C dimH X ≥ α and hence C dimA X = C dimH X = α.

The range of global conformal Assouad dimension in Rn was studied in[46]. In Theorem 3.1 of [46] we showed the following result:1

Proposition 2.4. For any S ⊂ Rn, GC dimRn

A S ∈ 0 ∪ [1, n].

2.2. The Heisenberg group Hn. Our setting is the Heisenberg groupHn = Cn × R = R2n+1 equipped with the non-abelian group law

(z, t) · (z′, t′) = (z + z′, t + t′ + 2〈z, z′〉),where 〈z, z′〉 = Im(

∑nj=1 zjz′j) denotes the standard symplectic inner prod-

uct in Cn. We denote by o = (0, 0) ∈ Cn × R the identity element in Hn.Writing z = (z1, . . . , zn) = (x1 + iy1, . . . , xn + iyn) for an arbitrary point ofCn, we denote by Xj = ∂

∂xj+ 2yj

∂∂t and Yj = ∂

∂yj− 2xj

∂∂t , j = 1, . . . , n,

1In [46, Theorem 3.1] we assumed S bounded. The extension to arbitrary S is easilyachieved using the invariance of Assouad dimension under conformal inversion; see [35,Theorem A.10(1)] for this result, and Proposition 2.6 for the Heisenberg analog. Comparealso the last paragraph in the proof of Proposition 5.2.

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a basis for the horizontal distribution HHn, which is annihilated by thecontact form ω = dt + 2

∑nj=1(xj dyj − yj dxj). We equip the vector sub-

spaces HpHn with a smoothly varying inner product which makes the vectorfields X1, Y1, . . . , Xn, Yn orthonormal, and denote by dcc the associatedCarnot-Caratheodory (CC) metric. Then (Hn,HHn, dcc) has the structureof a Carnot (nilpotent stratified Lie) group, indeed, the Lie algebra of Hn

is step two nilpotent, with first layer h spanned by X1, Y1, . . . , Xn, Yn andsecond layer v = [h, h] spanned by T = ∂

∂t . The homogeneous dimensionof Hn is 2n + 2; this is the Hausdorff dimension of (Hn, dcc). The Haarmeasure in Hn coincides with the Lebesgue measure in the underlying Eu-clidean space R2n+1 and (up to a constant multiple) with the (2n + 2)-dimensional CC Hausdorff measure. We denote by δr the anisotropic dila-tion δr(z, t) = (rz, r2t) with scale factor r > 0, by RA(z, t) = (Az, t) therotation by A ∈ U(n), and by π : Hn → Cn the projection π(z, t) = z.

For z as above, we write |z|∞ = max|x1|, |y1|, . . . , |xn|, |yn| and recordthe elementary inequality

(2.1) |〈z, z′〉| ≤ 2n|z|∞|z′|∞, z, z′ ∈ Cn.

In place of the Carnot-Caratheodory metric dcc we will use more computa-tionally friendly metrics such as the gauge metric d(p, q) = ||p−1 · q||, where

||p|| = max|z|∞,|t|1/2

√2n, p = (z, t),

and the related Koranyi metric dH(p, q) = ||p−1 · q||H, where

||p||H = (|z|4 + |t|2)1/4, p = (z, t).

The fact that dH is a metric is well-known; that d is a metric is an easyconsequence of (2.1).

All three metrics, dcc, d and dH, are left invariant and evidently homo-geneous of order one with respect to the dilations δr, hence bi-Lipschitzequivalent. We denote by B(p, r), resp. BH(p, r), resp. Bcc(p, r) the ballwith center p and radius r in the metric d, resp. dH, resp. dcc. Diameters ofsets and distances between sets will always be computed in the gauge metricand denoted diam, dist. We write dimH S, resp. dimA S, for the Hausdorff,resp. Assouad, dimension of a set S ⊂ Hn calculated in the gauge metric.Also we write Hα, resp. Hα

δ , 0 < δ ≤ ∞, for the α-dimensional Hausdorffmeasure, resp. premeasure at scale δ, in the gauge metric, and we write Hα

Efor the α-dimensional (Euclidean) Hausdorff measure on R. We note thatHα∞ coincides with the α-dimensional Hausdorff content, defined as follows:

(2.2) Hα∞(A) = inf

∞∑i=1

rαi ,

the infimum taken over all covers of A with balls B(pi, ri) : i = 1, 2, . . ..The CC metric dcc will not be used in any quantitative sense, apart from

a brief occurrence in Example 6.5.6

The Heisenberg group Hn is equipped with a natural conformal inversion

(2.3) jH(z, t) =(

−z

|z|2 − it,

−t

||(z, t)||4H

).

Note that ||jH(·)||H = || · ||−1H and jH δr = δ1/r jH for r > 0.

2.3. Quasiconformal maps. Let Ω be a domain in Hn. A homeomorphismF : Ω → F (Ω) ⊂ Hn is K-quasiconformal (K-QC for short) if

HF (p) := lim supr→0

supdH(F (p), F (q)) : dH(p, q) = rinfdH(F (p), F (q)) : dH(p, q) = r

≤ K

for every p ∈ Ω. If F is K-QC for some K < ∞ we say that F is quasicon-formal or QC. We use conformal as a synonym for 1-quasiconformal.

Each C2 smooth quasiconformal map F is a contact map which meansthat the pullback of the contact form ω under F is a nonvanishing multipleof ω: F ∗ω = λω. F is called orientation-preserving if λ > 0 and orientation-reversing if λ < 0. According to Theorem 8 of [31], sufficiently smooth2

orientation-preserving conformal maps on a domain in Hn are precisely therestrictions of compositions of left translations p 7→ p0 · p, p0 ∈ Hn, dilationsp 7→ δr(p), r > 0, rotations p 7→ RA(p), A ∈ U(n), and the conformalinversion3 F (p) = jH(p), defined on Ω = Hn \ o.

The theory of quasiconformal maps in Hn is due to Koranyi and Reimann[31], [32]; see also Heinonen [23], and Heinonen–Koskela [26] for extensionsto metric measure spaces of bounded geometry. Quasiconformal maps inCarnot groups first arose in the context of Mostow’s rigidity theorem [38].It is well-known that the class of quasiconformal homeomorphisms of Hn

coincides with the class of quasisymmetric homeomorphisms. A homeo-morphism F : X → Y between metric spaces (X, d) and (Y, d′) is calledquasisymmetric if there is a homeomorphism η : [0,∞) → [0,∞) so that

d(x1, x2) ≤ td(x1, x3) ⇒ d′(F (x1), F (x2)) ≤ η(t)d′(F (x1), F (x3))

for all x1, x2, x3 ∈ X and t > 0.Koranyi and Reimann [31], [32] identified the infinitesimal generators of

flows of smooth quasiconformal homeomorphisms of Hn. According to [32,§5], for each u ∈ C∞(Hn) the vector field

(2.4) V := u · T +14

n∑j=1

((Xju)Yj − (Yju)Xj)

generates a flow (Fs)s∈R of C∞ quasiconformal homeomorphisms of Hn.The coefficient of quasiconformality Ks for Fs can be bounded above interms of n, s and the expressions ||XkXju−YkYju||∞, ||XkYju+YkXju||∞,j, k = 1, . . . , n. (See Proposition 25 in [32].)

2After work of Capogna [15], [16] and Capogna–Cowling [17], it is now known thatthe smoothness criterion is automatically satisfied for 1-quasiconformal maps, even inarbitrary Carnot groups.

3The 1-quasiconformality of F = jH is an easy exercise using the identity (2.5).7

Example 2.5. Let p0 = (z0, t0) ∈ Hn and define up0 ∈ C∞(Hn) by up0(z, t) =〈z0, z〉 − 1

2(t− t0). By (2.4) the associated vector field is

Vp0 =14

n∑i=1

((x0i − xi)

∂

∂xi+ (y0i − yi)

∂

∂yi

)+

12

(〈z0, z〉 − t + t0)∂

∂t,

where z0 = (x01 + iy01, . . . , x0n + iy0n). The associated flow of mappings is

Fp0,s(p) = p0 · δexp(−s/4)(p−10 · p)

which is a flow of conformal homotheties with common fixed point at p0.Note that XkXjup0 , YkYjup0 and XkYjup0 + YkXjup0 are identically equalto zero for all j, k = 1, . . . , n.

2.4. Assouad dimension redux. Similarities of Hn are bi-Lipschitz andthus preserve Assouad dimension. Luukkainen [35, Theorem A.10(1)] provedthat Assouad dimension of subsets of Rn is invariant under inversion in theunit sphere Sn−1. We establish a Heisenberg analog.

Proposition 2.6. For any set S ⊂ Hn \ o, dimA S = dimA jH(S).

Proof. The key ingredient in the proof is the following analog of a classicalEuclidean inversion relation:

(2.5) dH(jH(p), jH(q)) =dH(p, q)||p||H ||q||H

.

This is a straightforward computation, see [18, (2.15)]. With (2.5) in handthe proof of Proposition 2.6 follows the lines of [35, Theorem A.10(1)]. ByProposition 2.1(i) we may work with the Koranyi metric dH. Since jH isan involution, it suffices to show that S is s-homogeneous whenever jH(S)is s-homogeneous. When s = 0 this is trivial (S is 0-homogeneous if andonly if it is finite), so assume s > 0. Let Λ be a finite subset of S withα ≤ dH(p, q) ≤ β for all distinct p, q ∈ Λ. Set A = minp∈Λ ||p||H andB = maxp∈Λ ||p||H. By (2.5),

α

B2≤ dH(jH(p), jH(q)) ≤ β

A2

for all distinct p, q ∈ Λ. Hence

#Λ = #jH(Λ) ≤ Cs(B/A)2s(β/α)s,

where Cs denotes the s-homogeneity constant for jH(S). If β ≤ A we haveB ≤ A + β ≤ 2A whence #Λ ≤ 4sCs(β/α)s. If β > A we choose k ∈ N with2k−1A ≤ β < 2kA and decompose Λ in a union of k+1 sets Λi by intersectingwith the annuli BH(o, 2iA) \ BH(o, 2i−1A), i = 1, . . . , k + 1. Repeatingthe above argument for each i and summing leads to the estimate #Λ ≤64s/(2s − 1)Cs(β/α)s; see the proof of Theorem A.10(1) in [35] for furtherdetails. Combining the above cases, we see that Λ is s-homogeneous.

In view of Proposition 2.6 and Koranyi and Reimann’s classification ofthe conformal maps in Hn, we have the following

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Corollary 2.7. Assouad dimension is a conformal invariant in Hn.

It is well-known that quasiconformal maps can distort dimension. In theHeisenberg group and for the Assouad dimension we have the following:

Theorem 2.8. For any n ≥ 1 and any 0 < α < β < 2n + 2, there arecompact sets Sα, Sβ ⊂ Hn and a quasiconformal homeomorphism F of Hn

so that dimA Sα = α, dimA Sβ = β, and F (Sα) = Sβ.

The analogous theorem for Hausdorff dimension is due to Balogh [4],following the original result of Gehring and Vaisala in Euclidean space [21].Theorem 2.8 is an immediate consequence of the Hn version of [4, Theorem1.1] in view of the Ahlfors regularity of the relevant examples Sα, Sβ.

For K-dependent bounds on distortion of Assouad dimension by quasi-conformal maps, see Theorem 4.6.

3. Fractal geometry in Hn

3.1. The Heisenberg cube. Throughout this paper we make extensive useof self-similar tilings of Hn. Strichartz [42], [43], first considered dyadic frac-tal tilings in nilpotent stratified Lie groups, see also Gelbrich [22]. Almostsure dimension formulas for horizontal self-similar and self-affine fractals inthe first Heisenberg group were established in [6]; see [5] for further results.Fractal geometry in general Carnot groups, with application to Gromov’sdimension comparison problem, has been studied in [7], [8].

Our primary tool is a class of tilings of Hn which generalizes the construc-tion in [42], [43]. We begin with a description of the basic tile.

Example 3.1. Fix an odd positive integer b ≥ 2n + 1, and consider theiterated function system in Hn generated by the following collection of b2n+2

contractive similarities:

Fk,l : Hn → Hn, Fk,l(p) = pk,l · δ1/b(p),

where pk,l = (zk, tl) with

zk =

(k1 − b+1

2

b+ i

kn+1 − b+12

b, . . . ,

kn − b+12

b+ i

k2n − b+12

b

),

k = (k1, . . . , k2n) ∈ 1, . . . , b2n, and

tl =l − b2+1

2

b2,

l ∈ 1, . . . , b2. Each map Fk,l is a similarity map of Hn with contractionratio 1

b < 1, hence the collection Fk,l defines a unique nonempty compactinvariant set To ⊂ Hn characterized by the identity

(3.1) To =⋃k,l

Fk,l(To).

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Writing fk : Cn → Cn for the first-layer projection of Fk,l: fk(z) = zk + 1bz,

z ∈ Cn, we note that π(To) = [−12 , 1

2 ]2n is the invariant set for the self-similarIFS fk on Cn.

Remark 3.2. We may identify Hn with R2n+1 and view the IFS Fk,l asa self-affine IFS in R2n+1 equipped with the Euclidean metric dE . ThusTo is a Euclidean self-affine set. According to Lemma 3.3 below, o is aninterior point of To. Thus To has (Hausdorff or Assouad) dimension 2n + 2in (Hn, dcc), and has dimension 2n + 1 in (R2n+1, dE). By Theorem 1.1 in[34], To coincides with the closure of its interior, and ∂To ⊂ R2n+1 is a nullset for Lebesgue measure. Observe that

∂To = Q+ ∪Q− ∪W,

where Q± are vertical translates of the Heisenberg square Q which is definedas the invariant set for the IFS Fk,(b2+1)/2k∈1,...,b2n . In fact, Q+ is theinvariant set for the IFS Fk,b2k, while Q− is the invariant set for theIFS Fk,1k. Q is a sub-Riemannian self-similar fractal whose Hausdorffdimension (in either the CC or Euclidean metric) equals 2n. See [5] and [42]for further information. The set W (the walls of To) is a union of portions ofvertical hyperplanes; W has CC Hausdorff dimension 2n + 1 and EuclideanHausdorff dimension 2n.

Lemma 3.3. B(o, rin) ⊂ To ⊂ B(o, rout) where

rin :=

√14n

− 12b + 2

and

rout :=

√14 + 1

2b+2 , n = 1,12 , n ≥ 2.

Note that rin > 0 and rout ≤ 34 for all n, since b ≥ 2n + 1.

In the proof, we use the following representation formula for points in To.Compare (2.8) in [42]. We denote by p : 1, . . . , b2n × 1, . . . , b2 → To thecanonical symbolic representation map:

p((k(1), l(1)),(k(2), l(2)), . . .) = limm→∞

Fk(1),l(1) · · · Fk(m),l(m)(o)

= pk(1),l(1) · δ1/b(pk(2),l(2)) · δ1/b2(pk(3),l(3)) · · · .(3.2)

If

z = limm→∞

fk(1) · · · fk(m)(0) = zk(1) +1bzk(2) +

1b2

zk(3) + · · · =∞∑

j=1

1bj−1

zk(j)

is a point in π(To) = [−12 , 1

2 ]2n, then the t-coordinate of the point in (3.2) is

(3.3)∞∑

j=1

tl(j)

b2j−2+ 2

∑1≤i<j<∞

〈zk(i) , zk(j)〉bi+j−2

.

10

Proof of Lemma 3.3. By (3.3) and Remark 3.2, points in Q+ have symbolicrepresentations as in (3.2) with l(j) = b2 for all j ≥ 1. Then tl(j) = b2−1

2b2for

all j ≥ 1 and the t-coordinate of such a point is equal to

b2 − 12b2

∞∑j=1

1b2j−2

+ 2∑

1≤i<j<∞

〈zk(i) , zk(j)〉bi+j−2

=12

+ 2∑

1≤i<j<∞

〈zk(i) , zk(j)〉bi+j−2

.

By (2.1) this value is at least

12− 4n

∑1≤i<j<∞

|zk(i) |∞|zk(j) |∞bi+j−2

≥ 12− n

(b− 1)2

b2

∑1≤i<j<∞

1bi+j−2

=12− n

b + 1= 2nr2

in > 0

and at most12

+ 4n∑

1≤i<j<∞

|zk(i) |∞|zk(j) |∞bi+j−2

≤ 12

+n

b + 1≤ 2nr2

out

for all n. Here we used the estimate |zk|∞ ≤ (b − 1)/(2b), valid for allk ∈ 1, . . . , b2n. In other words, if (z, t) ∈ Q+, then |z|∞ ≤ 1

2 andt ∈ [2nr2

in, 2nr2out]. Similarly, if (z, t) ∈ Q−, then |z|∞ ≤ 1

2 and t ∈[−2nr2

out,−2nr2in]. Since rin ≤ 1

2 ≤ rout for all n and ∂To = Q+ ∪Q− ∪W ,we conclude that B(o, rin) ⊂ To ⊂ B(o, rout) as desired.

3.2. Strichartz-type tilings in Hn. Let HnZ = (z, t) ∈ Hn : z ∈ Zn +

iZn, t ∈ Z denote the integral Heisenberg group. The defining equation(3.1) for To can be rewritten as the tiling equation

δb(To) =⋃p∈D

p · To,

where p · S := p · q : q ∈ S and D denotes the set of points p = (z, t) inHn

Z for which |z|∞ ≤ b−12 and |t| ≤ b2−1

2 . Iterating and passing to the limit,we obtain

Hn =⋃

p∈HnZ

p · To;

thus To generates a self-similar tiling of (Hn, d). If p, q ∈ HnZ are distinct,

then p · To and q · To have disjoint interiors.4 For m ∈ Z and p ∈ δbmHnZ, let

T (p, bm) := p · δbm(To);

then

(3.4) Hn =⋃

p∈δbmHnZ

T (p, bm)

with disjoint interiors. Note that To = T (o, 1). We write

Tm = T (p, bm) : p ∈ δbmHnZ

4Since π((z, t) · To) ⊂ z + [− 12, 1

2]2n and

Sι∈Z(0, ι) · To = [− 1

2, 1

2]2n × R.

11

for the set of tiles in (3.4), and T∗ =⋃

m∈Z Tm.Tm refines Tm+1 in the sense that each element of Tm is contained entirely

within a unique element of Tm+1. We call T ∈ Tm a child of T ′ ∈ Tm+1 (andT ′ the parent of T ) if T ⊂ T ′. From Lemma 3.3 we deduce the inclusions

(3.5) B(p, rinbm) ⊂ T (p, bm) ⊂ B(p, routbm)

for all m ∈ Z and all p ∈ δbmHnZ.

We will use the following relation defined on Tm. We say that T ((z, t), bm)and T ((z′, t′), bm) are adjacent if |z− z′|∞ ≤ bm and |t− t′| ≤ 18nb2m. Eachtile in Tm is adjacent to

N0 = 32n(36n + 1)

neighbors in Tm. The significance of this notion is clarified in the following

Lemma 3.4. The distance between any two non-adjacent tiles in Tm is atleast 2(1 − rout)bm ≥ 1

2bm, and the distance between any two adjacent tilesin Tm is at most 2rout(36n + 1)bm ≤ 60nbm.

Proof. Let T = T (p, bm) = T ((z, t), bm) and T ′ = T (p′, bm) = T ((z′, t′), bm)be two non-adjacent tiles in Tm. If |z − z′|∞ ≥ 2bm then

dist(T, T ′) ≥ d(p, p′)− 2routbm

≥ |z − z′|∞ − 2routbm ≥ 2(1− rout)bm,

while if |z − z′|∞ ≤ bm and |t− t′| > 18nb2m then also

dist(T, T ′) ≥ d(p, p′)− 2routbm

≥ d(p, (z, t′))− d((z, t′), p′)− 2routbm

≥ |t− t′|1/2

√2n

− |z − z′|∞ − 2routbm ≥ 2(1− rout)bm.

On the other hand, if T and T ′ are adjacent, then there exists a chain oftiles T = T0, T1, . . . , TN = T ′ in Tm with N ≤ 36n + 2 and Ti ∩ Ti−1 6= ∅ forall i = 1, . . . , N . Then

dist(T, T ′) ≤N−1∑i=1

diam Ti ≤ (N − 1)(2routbm)

as desired.

Let T be an element of Tm. We define the star of T to be the set

star(T ) =⋃T ′ ∈ Tm : T ′ is adjacent to T.

By Lemma 3.4,

(3.6) diam(star(T )) ≤ 4rout(36n + 1)bm ≤ 120nbm

for all T ∈ Tm.12

4. Porosity, sparseness, and Assouad dimension

The notion of porosity is a common theme in geometric function theoryand complex dynamics. A set S in a metric space (X, d) is called porousin X if there exists a constant c > 0 so that every ball B(x, r) contains asmaller ball B(z, cr) which is disjoint from S.

Roughly speaking, porous sets avoid subballs of comparable size withinmetric balls at every location and scale. For our purposes, it will be helpfulto refine the notion of porosity to obtain greater control over the location ofthe omitted subball. The following definition is formulated using Strichartztilings.

Definition 4.1. Let b ≥ 2n + 1 be an odd integer and let 1 ≤ k < b2n+2. Aset S ⊂ Hn is (b, k)-sparse if for T ∈ T∗, the set S has nonempty intersectionwith at most k children of T .

See Definition 3.2 in [46] for the analogous notion in Euclidean space. A(b, k)-sparse set is evidently (bj , kj)-sparse for each j ∈ N.

Remark 4.2. It is an easy exercise to show that S is porous in Hn if andonly if it is (b, b2n+2 − 1)-sparse for some sufficiently large b. Moreover, theporosity constant c and the parameter b can be chosen to depend only oneach other and on n.

We now relate bounds for the Assouad dimension with the sparsenesscriterion. Compare Proposition 3.3 in [46].

Proposition 4.3. If S ⊂ Hn is (b, k)-sparse, then S is s-homogeneous withs = log k/ log b. Conversely, if dimA S < 2n + 2 and s ∈ (dimA S, 2n + 2),then there exists b ≥ 2n + 1 sufficiently large so that S is (b, k)-sparse forsome k ≤ bs ≤ b2n+2 − 1.

As a corollary we obtain the equivalence of porosity with a bound on theAssouad dimension. Compare Theorem 5.2 in [35].

Corollary 4.4. S ⊂ Hn is porous if and only if dimA S < 2n + 2. Theporosity constant c depends only a choice of s ∈ (dimA S, 2n + 2) and thes-homogeneity constant Cs. Conversely, dimA S is bounded above by a con-stant depending only on n and the porosity constant c.

Proof of Corollary 4.4. First, suppose that dimA S < 2n + 2. Then S is s-homogeneous for some s < 2n + 2. By Proposition 4.3, S is (b, k)-sparse forsome sufficiently large integer b and k ≤ bs < b2n+2. Then S is (b, b2n+2−1)-sparse and hence porous by Remark 4.2.

Conversely, assume that S is porous and hence (b, b2n+2−1)-sparse. ThenS is s-homogeneous with

s =log(b2n+2 − 1)

log b< 2n + 2

and hence dimA S < 2n + 2. 13

Before proving Proposition 4.3, we state an application to dilatation-dependent bounds for quasisymmetric distortion of Assouad dimension. Werequire, in addition, the quasisymmetric invariance of porosity. The latterresult holds in the general setting of Ahlfors regular Loewner metric measurespaces, as in the following theorem. Recall that a metric measure space(X, d, µ) is called Loewner (with exponent Q) if there exists a decreasingfunction ϕ : (0,∞) → (0,∞) so that the Q-modulus of the family Γ ofcurves joining two nondegenerate continua E,F ⊂ X verifies the inequality

ModQ Γ ≥ ϕ

(dist(E,F )

mindiam E,diam F

)for every pair E,F . See [26] or [25] for more information and the definitionof the modulus of a curve family. The Heisenberg group Hn (or any Carnotgroup equipped with Carnot-Caratheodory metric and Haar measure) is anAhlfors regular Loewner space, see [26, section 6.2], [23] or [24].

Theorem 4.5. Let (X, d, µ) and (Y, d′, µ′) be locally compact Ahlfors Q-regular Loewner metric measure spaces, Q > 1, and let F : X → Y beη-quasisymmetric. If S ⊂ X is c-porous in X for some c > 0, then F (S)is c′-porous in Y for some c′ depending only on c, η, Q, and the regularitydata of X and Y .

Theorem 4.5 is a straightforward application of the Loewner conditionand modulus estimates. It is rather more difficult to prove the invariance ofporosity under quasisymmetric embeddings F : S → Y defined only on S.See [48] for the Euclidean case. We do not need such stronger results in ourcontext.

Proof. We assume that X and Y are unbounded which suffices for our ap-plication to the Heisenberg group. The bounded case is similar. The in-verse map F−1 : Y → X is η′-quasisymmetric for a suitable η′ dependingonly on η. Suppose that S is c-porous in X but F (S) is not c′-porous inY . We will show that c′ admits a uniform lower bound depending only onthe data specified in the statement of the theorem. We may assume thatc′ ≤ (4η(1))−1 ≤ 1

4 . There exists a ball B(y, r′) in Y so that F (S)∩B(y, r′)is c′r′-dense in B(y, r′). Choose r > 0 so that

B(x, r) ⊂ F−1B(y, r′/2) ⊂ F−1B(y, r′) ⊂ B(x,H ′r),

where x = F−1(y) and H ′ = η′(2). By porosity of S, there exists a subballB(z, cr) ⊂ B(x, 1

2r) which is disjoint from S. Choose t > 0 so that

B(w, t) ⊂ FB(z, cr) ⊂ B(w,Ht),

where w = F (z) ∈ B(y, r′/2) and H = η(1). Since B(w, t) is disjoint fromF (S) and B(w, c′r′) meets F (S), we must have t < c′r′, so

B(w, t) ⊂ FB(z, cr) ⊂ B(w,Hc′r′).14

Let Γ denote the family of curves joining B(z, cr) to X \B(x,H ′r). By theLoewner property of (X, d, µ) and the comparability of diam B(x, r) and r,

ModQ Γ ≥ ϕ

(2H ′r/c1

c1cr

)= ϕ

(2H ′

cc21

)> 0

for some c1 > 0 depending only on the regularity data of X. On the otherhand, the curve family F (Γ) is minorized by the curve family Γ′ joiningB(w,Hc′r′) to Y \B(w, r′/2). Then

ModQ F (Γ) ≤ ModQ Γ′ ≤ C

(log

12Hc′

)1−Q

for some constant C depending only on the regularity data for Y , by astandard estimate for the Q-modulus (see [26, Lemma 3.14]).

Every η-quasisymmetric homeomorphism between locally compact Q-regular metric measure spaces, Q > 1, satisfies the geometric quasicon-formality condition

K−1 ModQ Γ ≤ ModQ F (Γ) ≤ K ModQ Γ

for all curve families Γ and some constant K ≥ 1 depending only on η, Q,and the regularity data of X and Y . See [44]. Thus

0 < ϕ(2H ′

cc21

) ≤ KC(log(1

2Hc′)1−Q

so

c′ ≥ 12H

exp

(−(

1KC

ϕ(2H ′

cc21

))1/(1−Q)

).

This completes the proof.

Combining Corollary 4.4 and Theorem 4.5 gives Theorem 1.2 and Corol-lary 1.3 from the introduction, which we restate in the following form:

Theorem 4.6. Let F : Hn → Hn be a K-quasiconformal homeomorphism,and let S ⊂ Hn satisfy dimA S ∈ (0, 2n + 2). Then

(4.1) 0 < α ≤ dimA F (S) ≤ β < 2n + 2

where the constants α, β depend only on K, n, a constant s ∈ (dimA S, 2n+2)and the s-homogeneity constant Cs for S.

In fact, α can be chosen to depend only on K and dimA S; this followsfrom Proposition 5.1 in [46] and the fact that F is power quasisymmetric.

The analog of Theorem 4.6 for Hausdorff dimension in Euclidean space isthe well-known result of Gehring and Vaisala [21, Theorem 12]. In the caseof Hausdorff dimension in the Heisenberg group, the relevant result is dueto Balogh [4, Theorem 4.1]. For Assouad dimension in Euclidean space, see[46, Proposition 5.4].

We conclude this section with the proof of Proposition 4.3.15

Proof of Proposition 4.3. Suppose that S is (b, k)-sparse with k < b2n+2; wewill show that S is s-homogeneous for s = log k/ log b < 2n + 2. Choose0 < α ≤ β < ∞ and let Λ be a finite subset of S satisfying α ≤ d(p, q) ≤ βfor all p, q ∈ Λ, p 6= q. We estimate the cardinality of Λ. Choose integers mand M so that 2bm ≤ α < 2bm+1 and 1

2bM−1 ≤ β < 12bM . We claim that

(i) Λ is contained in at most N0 elements of TM , and(ii) any two distinct elements of Λ lie in distinct elements of Tm.

To prove (i), choose p ∈ Λ and a tile T ∈ TM containing p and observe thatΛ ⊂ star(T ). Indeed, if q ∈ Λ ∩ T ′ with T and T ′ nonadjacent in TM , then

12bM ≤ dist(T, T ′) ≤ d(p, q) ≤ β <

12bM .

Similarly, to prove (ii) we note that no two points of Λ can lie in the sameelement of Tm. Indeed, if p, q ∈ Λ ∩ T with T ∈ Tm, then

2bm ≤ α ≤ d(p, q) ≤ diam T ≤ 2routbm < 2bm

by (3.5).By (i), (ii) and the (b, k)-sparsity of S, we have #Λ ≤ N0k

M−m. By thechoice of m and M ,

M −m ≤ 2 +log(4β/α)

log b.

Thus

#Λ ≤ N0k2+log(4β/α)/ log b = N0k

24s(β/α)s ≤ (2b)4n+4N0(β/α)s

which shows that S is s-homogeneous.Next, assume that dimA S < 2n + 2 and s ∈ (dimA S, 2n + 2). Choose

t ∈ (dimA S, s) and let Ct be the t-homogeneity constant of S. Let b ≥ 2n+1be a large odd integer whose exact value will be determined momentarily.We consider the tilings Tm, m ∈ Z. For each T ∈ Tm we may divide thechildren of T into at most N0 families, each containing at least

b2n+2/N0

elements, so that any two tiles in the same family are nonadjacent. Supposethat a collection of tiles T1, . . . , TL from one of these families each intersectS. Choose pi ∈ Ti ∩ S, and let Λ = p1, . . . , pL. Since S is t-homogeneous,

L ≤ Ct

(diam Λ

mind(p, q) : p, q ∈ Λ, p 6= q

)t

≤ Ct

(2routb

m

12bm−1

)t

≤ Ct(3b)t.

Summing over all families, we find that the total number of children of Twhich can intersect S is at most

N0Ct(3b)t ≤ 92n+1(36n + 1)Ctbt.

We now choose b so large that

92n+1(36n + 1)Ctbt ≤ bs ≤ b2n+2 − 1;

then S is (b, k)-sparse with k ≤ bs as desired. 16

5. Global conformal Assouad dimension in Hn

This section is devoted to the proof of Theorem 1.1, which we restate inexpanded form in the following two propositions.

Proposition 5.1. For each α ∈ 0 ∪ [1, 2n + 2] there exists a bounded setS ⊂ Hn with

(5.1) GC dimA S = dimA S = GC dimH S = dimH S = α.

Proposition 5.2. If S ⊂ Hn satisfies dimA S < 1, then GC dimA S = 0.

Proposition 5.1 is a strengthened version of Theorem 1.3 in [4], whichconcerns Hausdorff dimension. When 1 ≤ α ≤ 2n + 1 the relevant examplesfrom [4] (extended to Hn) continue to function for Proposition 5.1. When2n+1 < α < 2n+2 technical complications force us to use different examples.We use recent results from [1] and [2] on the regularity of the CC distancefunction to construct the desired examples. See Example 6.5 for details.

We prove Propositions 5.1 and 5.2 in subsections 5.1 and 5.2, respectively.

5.1. Sets in Hn with prescribed global conformal Assouad dimen-sion. In this subsection, we construct examples of sets S ⊂ Hn satisfying(5.1) for each α ∈ 0 ∪ [1, 2n + 2]. The cases α = 0, α = 1, α = 2n + 1 andα = 2n+2 are easily dealt with, considering finite sets, smooth bounded hor-izontal curves, boundaries of smooth domains (see Example 6.2) and closedballs. We treat the cases 1 < α < 2n + 1 and 2n + 1 < α < 2n + 2 in turn.

Case I: α ∈ (1, 2n + 1). In this case examples from [4] continue to serve;we briefly recall the construction and refer to [4] for the technical details.

We construct an α-regular set S ⊂ Hn which verifies the conditions inLemma 2.3. Let S0 be a self-similar Ahlfors (α − 1)-regular Cantor setcontained in the codimension two subspace P = (z, t) ∈ Hn : z1 = 0.Such a set can be constructed as the invariant set for an iterated functionsystem satisfying the strong open set condition with fixed points in P . See[8] or Example 6.1 for more details. The Hausdorff measure Hα−1 restrictedto S0 (and suitably normalized) is an (α− 1)-regular measure. Set

S = (te1, 0) · p : p ∈ S0, 0 ≤ t ≤ 1,where e1 = (1, 0, . . . , 0) ∈ Cn. We write t ∗ p for the point (te1, 0) · p. Wedefine a measure ν on S by setting

ν(A ∗B) = L1(A)Hα−1(B)

for A ⊂ [0, 1] and B ⊂ S0 (L1 denotes Lebesgue measure on [0, 1]) andextending to a measure on the σ-algebra generated by product sets of thetype A ∗B. Then ν is an α-regular measure on S; see the discussion on pp.305–306 of [4] for the proof in the H1 case. The measure Hα−1 on S0 liftsto a probability measure on the curve family

Γ = γpp∈S0 , γp : [0, 1] → Hn, γp(t) = (te1, 0) · p.17

The hypotheses of Lemma 2.3 are easily verified. This completes the proofin Case I.

Case II: α ∈ (2n+1, 2n+2). As mentioned above, it is unclear whether theexamples from [4] continue to serve in this case. Recall that Lemmas 5.1 and5.2 of [4] show that certain Cantor sets of Heisenberg spheres are minimalfor GC dimHn

H . In attempting to use the same example for GC dimHn

A weencounter difficulties in showing that such sets have the prescribed Assouaddimension. Indeed, although these sets are constructed as a kind of productof Ahlfors regular “radial” and “spherical” sets, it is not clear whether theyare in fact Ahlfors regular. Complications arise in estimating the measurenear characteristic points of the spheres. We avoid this issue by replacing“Cantor sets of Heisenberg spheres” with “Cantor sets of smooth closednoncharacteristic hypersurfaces”. We establish the Ahlfors regularity of suchsets in Example 6.5. In what follows, we show that they are minimal forglobal conformal dimension.

Lemma 5.3. Let S be a C3 noncharacteristic hypersurface bounding a do-main Ω in Hn, let ε > 0 be as in Proposition 6.3, let D ⊂ [0, ε) be an a-regularCantor set (0 < a < 1), and let X =

⋃r∈D P+(S, r) be the (2n + 1 + a)-

regular set constructed in Example 6.5. Then GC dimHn

H X ≥ 2n + 1 + a.

Combining Lemma 5.3 and Example 6.5 we deduce that X is minimal forglobal conformal dimension (Hausdorff or Assouad), of dimension 2n+1+a.

Proof of Lemma 5.3. Again, we adapt an argument from [4]. Let F be aquasiconformal homeomorphism of Hn. Our first goal is to show that

(5.2) infr∈D

H2n+1∞ (F (P+(S, r))) > 0,

where Hα∞ denotes the α-dimensional Hausdorff content, see (2.2).

Adapting an argument of Pansu, see p. 309 of [4], we derive (5.2) fromPansu’s isoperimetric inequality

(5.3) H2n+1(∂Ω) ≥ c|Ω|2n+12n+2

valid for all bounded domains Ω ⊂ Hn with piecewise C2 boundary, forsome absolute constant c > 0. See [39] or [18] for a proof of (5.3) in the firstHeisenberg group. Since F (P+(S, r)) is compact it suffices to consider finitecovers B1, . . . , BN in (5.2). Writing Ω′ = B(S, r)∪B1∪ · · · ∪BN ⊃ Ω andnoting that ∂Ω′ ⊂ ∪N

i=1∂Bi, we estimate

0 < |F (Ω)|(2n+1)/(2n+2) ≤ |F (Ω′)|(2n+1)/(2n+2) ≤ CH2n+1(∂Ω′)

≤ C

N∑i=1

H2n+1(∂Bi) ≤ C

N∑i=1

r2n+1i ,

which yields (5.2). The final inequality here comes from the Ahlfors (2n+1)-regularity of ∂Bi; see Example 6.2.

18

Let 0 < η < a/2 and let B′i = B(p′i, r

′i) : i = 1, 2, . . . be an arbitrary

cover of F (X). We may assume that the dilates B(p′i,15r′i) are pairwise

disjoint. Setting pi = F−1(p′i), we deduce from the quasisymmetry of F andthe equivalence of d and dcc the existence of a constant H = H(F ) ≥ 1 andradii ri > 0 so that

Bcc(pi, ri) ⊂ F−1(B(p′i,15r′i)) ⊂ F−1(B(p′i, r

′i)) ⊂ Bcc(pi,Hri)

for all i. The balls Bcc(pi,Hri) : i = 1, 2, . . . cover X, while the dilatedballs Bcc(pi,

12ri) : i = 1, 2, . . . have disjoint closures. By (5.2),

0 < c ≤ H2n+1∞ (F (P+(S, r))) ≤

∑i:P+(S,r)∩F−1(B′i) 6=∅

(r′i)2n+1

for each r ∈ D. Integrating with respect to the a-regular measure HaE on D

yields

0 < c ≤∑

i

(r′i)2n+1Ha

E(r ∈ D : P+(S, r) ∩ F−1(B′i) 6= ∅)

≤∑

i

(r′i)2n+1Ha

E(r ∈ D : P+(S, r) ∩Bcc(pi,Hri) 6= ∅)

=∑

i

(r′i)2n+1Ha

E(r ∈ D : |r − distcc(pi, S)| ≤ Hri)

≤ C∑

i

(r′i)2n+1ra

i .

Set b = 1+(2n+1)/(a−η) > 1+(2n+1)/a. Since ba > 2n+1+a = dimA Xwe may choose δ > 0 sufficiently small so that ri ≤ δ implies

∑i r

bai ≤ 1.

See [36, section 5.9] for details. By local uniform continuity of F−1, wemay choose ε > 0 sufficiently small so that ri ≤ δ whenever r′i ≤ ε. Anapplication of Holder’s inequality gives

0 < c ≤

(∑i

(r′i)(2n+1)b

b−1

)1−1/b(∑i

rbai

)1/b

≤

(∑i

(r′i)2n+1+a−η

)1−1/b

from which we conclude

H2n+1+a−η(F (X)) ≥ H2n+1+a−ηε (F (X)) ≥ cb/(b−1) > 0.

Thus dimH F (X) ≥ 2n+1+a−η. Taking the infimum over all quasiconfor-mal homeomorphisms F of Hn and letting η 0 completes the proof.

5.2. Global conformal Assouad dimension in Hn takes on no valuebetween zero and one. Suppose that S ⊂ Hn satisfies dimA S < 1. Weaim to show that GC dimA S = 0. According to Proposition 4.3, S is (b, k)-porous for some large integer b and k ≤ bs, where dimA S < s < 1. Bychoosing b ≥ b0 for some sufficiently large b0 = b0(s), we may arrange thatk/b is as small as we please.

19

The crux of the proof of Proposition 5.2 resides in the following technicallemma.

Lemma 5.4. For each s < 1 there exists b0 = b0(s) so that the followingholds for all b ≥ b0: Let S be a subset of the standard b-adic Strichartz tileTo, and assume that S is (b, k)-sparse for some k ≤ bs. For each m ≤ 0,there exists an integer Nm and for each j = 1, . . . , Nm there exist threecollections A(m, j),B(m, j), C(m, j), each consisting of finitely many tiles inTm, satisfying the following conditions:

(i) S is contained in⋃

j ∪A(m, j),(ii) for each m and j,

(5.4) ∪star(T ) : T ∈ A(m, j) = ∪B(m, j)

and

(5.5) ∪star(T ) : T ∈ B(m, j) = ∪C(m, j),

(iii) for each m ≤ −1 and each j, there exists j′ so that ∪C(m, j) ⊂B(m + 1, j′),

(iv) for each m and i 6= j, ∪C(m, i) and ∪C(m, j) are disjoint,(v) for each m and j, there exists T ∈ Tm+1 so that ∪B(m, j) ⊂ star(T ),(vi) for each m and each T ∈ Tm with T ⊂ To, S has nonvoid intersection

with at most k of the sets ∪A(m, j), j = 1, . . . , Nm.

Here for a collection C of sets, we write ∪C = p : p ∈ C for some C ∈ C.The separating annuli ∪C(m, j) \∪B(m, j) which appear in the preceding

proof will enable us to shrink the Assouad dimension of S by implementing aKoranyi–Reimann flow which conformally shrinks each of the sets ∪B(m, j)while leaving the complement of

⋃j ∪C(m, j) invariant. For technical reasons

we work also with the associated annulus ∪B(m, j)\∪A(m, j); the reason forthis added complication in the proof stems from the appearance of star(T )in part (v) of the proof.

Before beginning the proof, we make some brief remarks. First, by (5.5)and Lemma 5.4(iv), no element of B(m, i) is adjacent to any element ofB(m, j) when i 6= j. By Lemma 3.4,5

dist(∪B(m, i),∪B(m, j)) ≥ 12bm ∀i 6= j.

Similarly, by Lemma 5.4(v) and (3.6),

diam(∪B(m, i)) ≤ 120nbm+1 ∀i.

Proof. Choose b0 = b0(s) ≥ 1000n and b ≥ b0 so that k ≤ b/(1000n). Weprove the result by induction on m ≤ 0.

The case m = 0 is trivial: choose N0 = 1 and

A(0, 1) = To, B(0, 1) = star(To), C(0, 1) = star(star(To)).

5We remind the reader that we are using the gauge metric d.

20

Suppose that the conclusion holds for some m + 1 ≤ 0; we show it for m.Fix a tile T in A(m + 1, j) for some j ≤ Nm+1, and let T1, . . . , Tq, q ≤ k, bethe children of T which meet S. Fix i, 1 ≤ i ≤ q, and define a sequence ofannular shells Fj = Fj(T, Ti), j ∈ 0, 1, 2, . . ., as follows: F0 = Ti and foreach j ∈ N, Fj consists of all elements of Tm\Ti which are adjacent to butnot contained in an element of F0 ∪ · · · ∪ Fj−1. Since k ≤ b/(1000n) thereexists j(i), 1 ≤ j(i) ≤ b/(250n), so that Fj(i), Fj(i)+1 and Fj(i)+2 consistentirely of tiles disjoint from S. We claim that

(5.6) (∪F0) ∪ · · · ∪ (∪Fj(i)) ⊂ star(T )

for each i. To see this, suppose that there exists a tile T ′ in F0 ∪ · · · ∪ Fj(i)

whose parent is not adjacent to T . Then12bm+1 ≤ dist(T,parent of T ′) ≤ dist(Ti, T

′).

This distance is bounded above by the sum of j(i) terms, each of which isthe diameter of the star of an element of Tm. By (3.6),

12bm+1 ≤ j(i)(120nbm) ≤ 12

25bm+1,

which is a contradiction. We conclude that all tiles in F0 ∪ · · · ∪ Fj(i) haveparent in star(T ), which immediately implies (5.6).

We say that a pair of tiles T ∈ Tm+1, Ti ∈ Tm are an admissible pair if• T is contained in A(m + 1, j) for some j ≤ Nm+1,• Ti ∩ S 6= ∅, and• Ti is a child of T .

Repeat the construction in the preceding paragraph for each admissible pair(T, Ti), obtaining an annular shell of tiles Fj(i) = Fj(i)(T, Ti). For eachadmissible pair (T, Ti), consider the collection of tiles Tm which are containedin F0∪· · ·∪Fj(i)−1 but are not contained in Fj(i′)(U,Ui′) for any admissiblepair (U,Ui′) 6= (T, Ti). Varying over all admissible pairs (T, Ti), enumeratethe resulting collections as A(m, j), 1 ≤ j ≤ Nm. Define B(m, j) and C(m, j)by the conditions (5.4) and (5.5). Conditions (i), (ii), (iv) and (vi) followdirectly from the construction. Conditions (iii) and (v) are consequences of(5.6). This completes the proof of the inductive step.

Proof of Proposition 5.2. Let S ⊂ Hn satisfy dimA S < 1. In the first stageof the proof, we assume that S is bounded. By a preliminary homothety,we may assume that S ⊂ To.

Choose s ∈ (dimA S, 1). By Proposition 4.3, S is (b, k)-sparse for someinteger b ≥ b0(s) and some k ≤ bs ≤ b/(500n). Apply Lemma 5.4.

For each m and j, choose a tile T(m,j) = T (p(m,j), bm−1) ∈ Tm−1 which is

a child of one of the constituent tiles in A(m, j), and a tile T (m,j) ∈ Tm+1

so that ∪B(m, j) ⊂ star(T (m,j)). Define up(m,j)and (Fp(m,j),s) as in Example

2.5. Our task is to calculate how large s must be to guarantee that

(5.7) Fp(m,j),s(star(T (m,j))) ⊂ T(m,j).21

Since

star(T (m,j)) ⊂ B(p(m,j),diam(star(T (m,j)))

)⊂ B(p(m,j), 120nbm+1)

andT(m,j) ⊃ B(p(m,j), rinbm−1),

we see that (5.7) is satisfied provided

120nbm+1e−s/4 ≤ rinbm−1.

The latter equation is satisfied when s is equal to

s0 := 4 log(

120nb2

rin

).

Now, for fixed m ≤ 0, set

um =∑

j

ϕ(m,j)up(m,j),

where ϕ(m,j) ∈ C∞0 (Hn), 0 ≤ ϕ(m,j) ≤ 1, ϕ(m,j) = 1 on ∪B(m, j), and

suppϕ(m,j) ⊂ ∪C(m, j). Denote by (Gm,s) the corresponding flow of QCmaps of Hn. Choose s = s0 as above. One easily verifies that the mapsGm := Gm,s0 are all quasiconformal with uniformly bounded coefficient ofquasiconformality. Note that Gm acts as a (conformal) dilation in ∪B(m, j)for each j, and is equal to the identity on the complement of

⋃j ∪C(m, j).

Moreover, Gm takes ∪B(m, j) into T(m,j).Now set Fm = GmGm+1· · ·G−1G0. By Lemma 5.4(iii), the maps Fm

are also all quasiconformal with uniformly bounded coefficient of quasicon-formality, and equal to the identity in the complement of T0. Furthermore,Fm takes ∪B(m′, j′) into T(m′,j′) for each m′ ∈ m,m + 1, . . . , 0 and j′.

By standard compactness theorems for quasiconformal maps, the mapsFm converge pointwise to a limit map F which is again quasiconformal.We check that the Assouad dimension of S is decreased by F . By Lemma5.4(vi), for each tile T ∈ Tm, m ≤ 0, F (S) ∩ T is contained in at most kgrandchildren of T , i.e., F (S) is (b2, k)-porous. By Proposition 4.3,

dimA F (S) ≤ log k

log b2≤ 1

2s.

Thus GC dimA S ≤ 12s. Repeating N times gives GC dimA S ≤ 2−Ns.

Letting N →∞ gives GC dimA S = 0.To remove the assumption that S is bounded, we use the invariance of

Assouad dimension under the inversion jH. Let S be a subset of Hn withdimA S < 1. We may assume that S is closed by Proposition 2.1(iii). Choosep0 6∈ S and ε > 0 so that BH(p0, ε) ∩ S = ∅. Then S′ = (jH δ1/ε)(p

−10 · S)

is bounded and dimA S′ = dimA S by Proposition 2.6. The map Φ(p) =(jHδ1/ε)(p

−10 ·p) is 1-QC from Hn\p0 to Hn\o with limp→p0 ||Φ(p)||H =

+∞. If F is a QC homeomorphism of Hn, then G := jH F Φ is aQC homeomorphism of Hn \ p0 to Hn \ o with limp→p0 ||G(p)||H = 0.

22

Then p0 is a removable singularity for G, and we may extend G to a QChomeomorphism of Hn. Since dimA G(S) = dimA jH(F (S′)) = dimA F (S′)we conclude that GC dimA S′ = GC dimA S. By the first part of the proof,GC dimA S′ = 0. Hence also GC dimA S = 0, which completes the proof.

Remark 5.5. Proposition 5.2 is best possible in the sense that there are setsof Assouad dimension one whose dimension cannot be lowered. However,restricting to subsets of the t-axis allows for a stronger statement, to wit,

S ⊂ T := (z, t) ∈ Hn : z = 0, dimA S < 2 ⇒ GC dimHn

A S = 0.

Indeed, it is well-known that the CC metric in Hn restricts to a multipleof√

dE on T. By Proposition 2.1(vi), the condition dimA(S, dcc) < 2 forS ⊂ T translates to dimA(S, dE) < 1. By [46], there exist quasisymmetricself-maps f of (T, dE) so that dimA(f(S), dE) is arbitrarily small. Conju-gating by the (quasisymmetric) snowflake map id : (T, dE) → (T, dcc) wemay arrange that f is a quasisymmetric self-map of (T, dcc). Moreover, theconstruction of these maps ensures that they may be extended as quasicon-formal self-maps of Hn.6 We conclude that GC dimHn

A S = 0 as asserted.The corresponding result for Hausdorff dimension is likely to be true aswell; one should check that the dimension-decreasing quasisymmetric mapsconstructed by Kovalev [33] in the case V = R, after conjugating with thesnowflake map, admit quasiconformal extensions to Hn. It remains an openproblem, to decide whether every quasisymmetric self-map of the t-axisadmits a quasiconformal extension to Hn. The precise value of the global(Hausdorff or Assouad) conformal dimension of T is also unknown. Heinonenand Semmes [27, Question 25] have asked whether T can be mapped onto arectifiable curve by a quasiconformal self-map of H1; the existence of sucha map would of course imply that GC dimHn T is one. Balogh [4, Corollary3.1] constructed subsets of T with Hausdorff dimension arbitrarily close to2 which can be mapped into rectifiable curves by quasiconformal self-mapsof H1.

Remark 5.6. In the Carnot groups which arise as local models for theGromov boundaries of quaternionic hyperbolic space or the octonionic hy-perbolic plane, all quasiconformal maps are 1-quasiconformal and smooth.(This observation combines Pansu’s rigidity theorem [41] with the main re-sult of Capogna and Cowling [17].) Thus

GC dimH S = dimH S and GC dimA S = dimA S

for every subset S of one of these groups G. For any α ∈ (0, Q) (Q thehomogeneous dimension of G) one can construct a self-similar subset S of Gwhich is Ahlfors regular of dimension α, hence has dimA S = dimH S = α.It follows that the analog of Theorem 1.1 is false in such groups, indeed,GC dimA can take on any value in the interval [0, Q].

6We omit the details, which resemble those in the proof of Proposition 5.2.

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Remark 5.7. Suppose S ⊂ Hn with dimH S > 0. Is sup dimH F (S) =2n + 2, where the supremum is taken over all QC maps F : Hn → Hn? Theanalogous question in Rn has been answered in the affirmative by Bishop [9].His proof relies on the Tukia–Vaisala extension theory for quasiconformalmaps in Rn and admits no straightforward adaptation to Hn. The analogousquestion for the Assouad dimension is open for both Hn and Rn.

6. Appendix

In this appendix, we present examples of Ahlfors α-regular subsets of Hn

for various α. By Proposition 2.1(viii) each of these sets has Assouad andHausdorff dimensions equal to α.

Example 6.1. Let S ⊂ Hn be a set which is self-similar with respect to D,where D is any of the metrics dcc, d or dH. In other words, we assume thatthere exist contractive similarity maps F1, . . . , FM of Hn so that

S = F1(S) ∪ · · · ∪ FM (S).

We denote by ri the contraction ratio for Fi, thus D(Fi(p), Fi(q)) = riD(p, q)for all p, q ∈ Hn. Examples include the Heisenberg cube and Strichartz tiles;further examples can be found in [5] and [8]. If the maps F1, . . . , FM satisfythe open set condition, then S has positive and finite Hausdorff α-measure,where α is the unique nonnegative solution to the equation

M∑i=1

rαi = 1.

Thus dimH S = α. In addition, the natural self-similar measure µ onS, characterized by the identity µ(Fi1 · · · Fim(S)) = rα

1 · · · rαm for all

(i1, . . . , im) ∈ 1, . . . ,Mm, is Ahlfors α-regular. This is part of the contentof the Moran–Hutchinson theorem [37], [28]. See Kigami [30, section 1.5] formore details. By an appropriate choice of the Fi we can arrange Cantor-typeexamples of this sort for each α ∈ [0, 2n + 2].

Example 6.2. Let S be any bounded C1,1 hypersurface in Hn which boundsa domain Ω. By Corollary 1.4 in [19], the perimeter measure PH(Ω, ·) is anAhlfors regular measure of dimension 2n + 1 on S.

In the next example, we construct low codimensional Ahlfors regular sub-sets of Hn which are foliated by smooth hypersurfaces. In subsection 5.1, weshow that these sets are minimal for global conformal Assouad dimension.

Recall that a point p on a C1 hypersurface S ⊂ Hn is called characteristicif TpS = HpHn. The characteristic locus of S is the set of characteristicpoints. For ε > 0 we denote by

P (S, ε) = p ∈ Hn : distcc(p, S) = ε24

the ε-parallel set of S. When S = ∂Ω, we also consider P+(S, ε) = P (S, ε)∩(Hn \ Ω), which is the boundary of the domain

B(S, ε) = p ∈ Hn : distcc(p, S) < ε.Arcozzi and Ferrari [1] studied regularity properties of the distance functionto a smooth hypersurface.7 We recall a special case of some of their results(see Theorems 1.1 and 5.2 and Corollary 5.1 in [1]):

Proposition 6.3 (Arcozzi–Ferrari). Let S be a noncharacteristic Ck hyper-surface in Hn which bounds a domain Ω. Then

(i) distcc(·, S) is of class Ck−1 in an open neighborhood of S.Furthermore, there exists ε > 0 so that

(ii) to every point p ∈ B(S, ε) there corresponds a unique point p′ ∈ Swith dcc(p, p′) = dcc(p, S), and

(iii) p 7→ (p′,distcc(p, S)) is a homeomorphism of B(S, ε) \ Ω onto S ×(0, ε).

It follows that P+(S, r) is a (bounded) Ck−1 hypersurface for r < ε; ifk ≥ 3, then Example 6.2 implies that P+(S, r) is Ahlfors (2n + 1)-regular.

Remark 6.4. Hypersurfaces of the type described in Proposition 6.3 exist.For example, the standard torus S = (z, t) ∈ Hn : (|z| − r′)2 + t2 = r2,0 < r < r′ < ∞, has empty characteristic locus. Indeed, the tangenthyperplane Tp0S at a point p0 = (z0, t0) ∈ S is

(z, t) ∈ R2n+1 : (1− r′

|z0|) Re(zz0) + t0t = 0

,

while the horizontal hyperplane Hp0Hn is(z, t) ∈ R2n+1 : t + 2 Im(zz0) = 0

.

If Tp0S = H0Hn we easily deduce that (1 − r′/|z0|)2 + 4t20 = 0, which isimpossible for points p0 ∈ S.

Example 6.5. Let S be a C3 noncharacteristic hypersurface which boundsa domain in Hn. Let D ⊂ [0, ε) be a standard (compact) Cantor set of(Euclidean) dimension a, 0 < a < 1, equipped with the natural Ahlforsa-regular measure Ha

E . Define

X :=⋃r∈D

P+(S, r).

We claim that X is Ahlfors regular of dimension 2n + 1 + a.Define a measure µ on X by

µ(A) =∫

DH2n+1(A ∩ P+(S, r)) dHa

E(r).

7Note that the paper [1] treats only the case of the first Heisenberg group H1. Thecase of general n will be treated in the forthcoming work [2].

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By [20, Example I], this is the Method II measure constructed from the setfunction

A 7→∫

D(diam A ∩ P+(S, r))2n+1 dHa

E(r),

in particular, it is a metric outer measure and hence a Borel measure. Weverify the Ahlfors regularity condition

(6.1)1C

R2n+1+a ≤ µ(B(p0, R) ∩X) ≤ CR2n+1+a

for all p0 ∈ X and 0 < R < diam X, for some fixed C < ∞.The right hand inequality in (6.1) is easy. If B(p0, R) ∩ P+(S, r) 6= ∅ for

some r ∈ D, choose p1 ∈ B(p0, R) ∩ P+(S, r) and observe that

B(p0, R) ∩ P+(S, r) ⊂ B(p1, 2R) ∩ P+(S, r).

The result now follows from the Ahlfors (2n + 1)-regularity of P+(S, r), theAhlfors a-regularity of D, and the definition of µ.

To prove the left hand inequality in (6.1), let p0 ∈ P+(S, r0) for somer0 ∈ D, let 0 < R < diam X, and let r1 ∈ D with |r0 − r1| ≤ R/2. Set

p1 = N+p0

(P+(S, r0))(r1 − r0),

where N+p (S) denotes the oriented metric normal to S at p; see Definition

4.1 in [1]. The fact which we require is the identity

(6.2) dcc(N+q F (t), q) = |t|,

valid for any C1 hypersurface F , q ∈ F and t sufficiently close to zero. Inour situation, we obtain p1 ∈ P+(S, r1) and an application of the triangleinequality and (6.2) gives dcc(p0, p1) = |r0 − r1| ≤ R/2. It follows thatB(p1,

12R) ∩ P+(S, r1) ⊂ B(p0, R) for such r1, and so

µ(B(p0, R) ∩X) =∫

DH2n+1(B(p0, R) ∩ P+(S, r)) dHa

E(r)

≥∫r1∈D:|r0−r1|≤R/2

H2n+1(B(p0, R) ∩ P+(S, r)) dHaE(r1)

≥∫r1∈D:|r0−r1|≤R/2

H2n+1(B(p1,12R) ∩ P+(S, r)) dHa

E(r1)

≥ C−1R2n+1HaE(r1 ∈ D : |r0 − r1| ≤ R/2) ≥ C−1R2n+1+a,

where we used the (2n + 1)-regularity of P+(S, r), resp. the a-regularity ofD, in the next-to-last, resp. last, line. Hence X is (2n + 1 + a)-regular.

References

[1] Arcozzi, N., and Ferrari, F. Metric normal and distance function in the Heisen-berg group. Math. Z. 256, 3 (2007), 661–684.

[2] Arcozzi, N., Ferrari, F., and Montefalcone, F. in preparation.[3] Assouad, P. Plongements lipschitziens dans Rn. Bull. Soc. Math. France 111 (1983),

429–448.

26

[4] Balogh, Z. M. Hausdorff dimension distribution of quasiconformal mappings on theHeisenberg group. J. Anal. Math. 83 (2001), 289–312.

[5] Balogh, Z. M., Hoefer-Isenegger, R., and Tyson, J. T. Lifts of Lipschitz mapsand horizontal fractals in the Heisenberg group. Ergodic Theory Dynam. Systems 26,3 (2006).

[6] Balogh, Z. M., and Tyson, J. T. Hausdorff dimensions of self-similar and self-affine fractals in the Heisenberg group. Proc. London Math. Soc. (3) 91, 1 (2005),153–183.

[7] Balogh, Z. M., Tyson, J. T., and Warhurst, B. Gromov’s dimension comparisonproblem on Carnot groups. C. R. Math. Acad. Sci. Paris. to appear.

[8] Balogh, Z. M., Tyson, J. T., and Warhurst, B. Sub-Riemannian vs. Euclideandimension comparison and fractal geometry on Carnot groups. Preprint, 2007.

[9] Bishop, C. J. Quasiconformal mappings which increase dimension. Ann. Acad. Sci.Fenn. Ser. A I Math. 24 (1999), 397–407.

[10] Bishop, C. J., and Tyson, J. T. Conformal dimension of the antenna set. Proc.Amer. Math. Soc. 129 (2001), 3631–3636.

[11] Bishop, C. J., and Tyson, J. T. Locally minimal sets for conformal dimension.Ann. Acad. Sci. Fenn. Ser. A I Math. 26 (2001), 361–373.

[12] Bonk, M., and Kleiner, B. Conformal dimension and Gromov hyperbolic groupswith 2-sphere boundary. Geom. Topol. 9 (2005), 219–246 (electronic).

[13] Bourdon, M. Au bord de certains polyedres hyperboliques. Ann. Inst. Fourier(Grenoble) 45 (1995), 119–141.

[14] Bourdon, M. Immeubles hyperboliques, dimension conforme et rigidite de Mostow.Geom. Funct. Anal. 7 (1997), 245–268.

[15] Capogna, L. Regularity of quasilinear equations in the Heisenberg group. Comm.Pure Appl. Math. 50, 9 (1997), 867–889.

[16] Capogna, L. Regularity for quasilinear equations and 1-quasiconformal maps inCarnot groups. Math. Ann. 313, 2 (1999), 263–295.

[17] Capogna, L., and Cowling, M. Conformality and Q-harmonicity in Carnot groups.Duke Math. J. 135, 3 (2006), 455–479.

[18] Capogna, L., Danielli, D., Pauls, S. D., and Tyson, J. T. An introductionto the Heisenberg group and the sub-Riemannian isoperimetric problem, vol. 259 ofProgress in Mathematics. Birkhauser Verlag, Basel, 2007.

[19] Capogna, L., and Garofalo, N. Ahlfors type estimates for perimeter measures inCarnot-Caratheodory spaces. J. Geom. Anal. 16, 3 (2006), 455–497.

[20] Falconer, K. J., and Mauldin, R. D. Fubini-type theorems for general measureconstructions. Mathematika 47 (2000), 251–265.

[21] Gehring, F. W., and Vaisala, J. Hausdorff dimension and quasiconformal map-pings. J. London Math. Soc. (2) 6 (1973), 504–512.

[22] Gelbrich, G. Self-similar periodic tilings on the Heisenberg group. J. Lie Theory 4,1 (1994), 31–37.

[23] Heinonen, J. Calculus on Carnot groups. In Fall School in Analysis (Jyvaskyla,1994), vol. 68. Ber. Univ. Jyvaskyla Math. Inst., Jyvaskyla, 1995, pp. 1–31.

[24] Heinonen, J. A capacity estimate on Carnot groups. Bull. Sci. Math. 119 (1995),475–484.

[25] Heinonen, J. Lectures on analysis on metric spaces. Springer-Verlag, New York,2001.

[26] Heinonen, J., and Koskela, P. Quasiconformal maps in metric spaces with con-trolled geometry. Acta Math. 181 (1998), 1–61.

[27] Heinonen, J., and Semmes, S. Thirty-three yes or no questions about mappings,measures, and metrics. Conform. Geom. Dyn. 1 (1997), 1–12.

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

27

[29] Keith, S., and Laakso, T. Conformal Assouad dimension and modulus. Geom.Funct. Anal. 14, 6 (2004), 1278–1321.

[30] Kigami, J. Analysis on fractals, vol. 143 of Cambridge Tracts in Mathematics. Cam-bridge University Press, Cambridge, 2001.

[31] Koranyi, A., and Reimann, H. M. Quasiconformal mappings on the Heisenberggroup. Invent. Math. 80 (1985), 309–338.

[32] Koranyi, A., and Reimann, H. M. Foundations for the theory of quasiconformalmappings on the Heisenberg group. Adv. Math. 111 (1995), 1–87.

[33] Kovalev, L. V. Conformal dimension does not assume values between zero and one.Duke Math. J. 134, 1 (2006), 1–13.

[34] Lagarias, J. C., and Wang, Y. Self-affine tiles in Rn. Adv. Math. 121, 1 (1996),21–49.

[35] Luukkainen, J. Assouad dimension: antifractal metrization, porous sets, and homo-geneous measures. J. Korean Math. Soc. 35 (1998), 23–76.

[36] Mattila, P. Geometry of sets and measures in Euclidean spaces, vol. 44 of CambridgeStudies in Advanced Mathematics. Cambridge University Press, Cambridge, 1995.

[37] Moran, P. A. P. Additive functions of intervals and Hausdorff measure. Proc. Cam-bridge Philos. Soc. 42 (1946), 15–23.

[38] Mostow, G. D. Strong rigidity of locally symmetric spaces. Princeton UniversityPress, Princeton, N.J., 1973. Annals of Mathematics Studies, No. 78.

[39] Pansu, P. Une inegalite isoperimetrique sur le groupe de Heisenberg. C. R. Acad.Sci. Paris Ser. I Math. 295, 2 (1982), 127–130.

[40] Pansu, P. Dimension conforme et sphere a l’infini des varietes a courbure negative.Ann. Acad. Sci. Fenn. Ser. A I Math. 14 (1989), 177–212.

[41] Pansu, P. Metriques de Carnot-Caratheodory et quasiisometries des espacessymetriques de rang un. Ann. of Math. (2) 129 (1989), 1–60.

[42] Strichartz, R. S. Self-similarity on nilpotent Lie groups. In Geometric analysis(Philadelphia, PA, 1991), vol. 140 of Contemp. Math. Amer. Math. Soc., Providence,RI, 1992, pp. 123–157.

[43] Strichartz, R. S. Self-similarity in harmonic analysis. J. Fourier Anal. Appl. 1, 1(1994), 1–37.

[44] Tyson, J. T. Quasiconformality and quasisymmetry in metric measure spaces. Ann.Acad. Sci. Fenn. Ser. A I Math. 23 (1998), 525–548.

[45] Tyson, J. T. Sets of minimal Hausdorff dimension for quasiconformal maps. Proc.Amer. Math. Soc. 128, 11 (2000), 3361–3367.

[46] Tyson, J. T. Lowering the Assouad dimension by quasisymmetric mappings. IllinoisJ. Math. 45 (2001), 641–656.

[47] Tyson, J. T., and Wu, J.-M. Quasiconformal dimensions of self-similar fractals.Rev. Mat. Iberoamericana 22 (2006), 205–258.

[48] Vaisala, J. Porous sets and quasisymmetric maps. Trans. Amer. Math. Soc. 299(1987), 525–533.

Department of Mathematics, University of Illinois, 1409 West Green St.,Urbana, IL 61801

E-mail address: tyson@math.uiuc.edu

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