QUASICONFORMAL MAPPINGS WHICH INCREASE DIMENSION

CHRISTOPHER J. BISHOP

October 21, 1999

Abstract. For any compact set E ⊂ ℝ^{d}, d ≥ 1, with Hausdorff dimension 0 < dim(E) < d and for any ϵ > 0, there is a quasiconformal mapping (quasisymmetric if d = 1) f of ℝ^{d} to itself such that dim(f(E)) > d − ϵ.

1991 Mathematics Subject Classification. 30C65 . Key words and phrases. Quasiconformal mappings, Hausdorff dimension . The author is partially supported by NSF Grant DMS 95-00557.

QUASICONFORMAL MAPPINGS WHICH INCREASE DIMENSION 1

1. Introduction

It is well know that quasiconformal maps can change the Hausdorff dimension of sets. For example, the von Koch snowflake is a quasiconformal image of the circle, but has dimension log 4/ log 3. In this note we answer a question of Juha Heinonen by showing

Theorem 1.1. For any compact set E in ℝ^{d} with dim(E) > 0 and any 0 < γ < d there is a quasisymmetric map h : ℝ^{d} → ℝ^{d} so that dim(h(E)) > γ.

When d ≥ 2 quasisymmetric maps are the same as quasiconformal, so this says that we can always increase dimension by quasiconformal mappings. We must take dim(E) > 0 since quasisymmetric maps are Hölder continuous, and hence they cannot increase the dimension of a set of dimension 0. The theorem is easy in some cases (e.g., E is a line segment) and previously known in others (e.g., the case of self-similar Cantor sets was done by Gehring and Väisälä [11]).

The idea for proving Theorem 1.1 is as follows. We will define a class of “standard” Cantor sets constructed from disjoint collections of b-adic cubes in the usual way and show that given any compact E there is a standard Cantor set F so that E ∩ F has dimension close to that of E. This reduces the proof to the case of standard Cantor sets. We will also show that standard Cantor sets lie on quasiarcs and hence we further reduce to the case of a standard Cantor set on the real line. Finally, given such a set E and a Frostman type measure μ on E, we modify μ to get a doubling measure ν so that the integral h(x) = ∫_{0}^{x} dν is a quasisymmetric map which sends E to a set with dimension close to 1.

Some background is given in Section 2. The definition and basic properties of

2 CHRISTOPHER J. BISHOP

standard Cantor sets is discussed in Section 3. In Section 4 we prove the d = 1 case of Theorem 1.1 by constructing the measure ν described above. In Section 5 we deduce the case d > 1 and in Section 6 we finish with a few comments and questions.

I thank the referee for a very careful reading of the manuscript and numerous suggestions which improved it.

2. Definitions and notation

If E ⊂ ℝ^{d}, then diam(E) will denote its diameter. Lebesgue measure on ℝ^{d} will be denoted by |E|_{d} or just |E| if the dimension is clear from context. Let b ≥ 2 be an integer and let C_{n}^{b,1} denote the nth generation b-adic intervals in ℝ^{1}, i.e., intervals of the form [jb^{−n}, (j + 1)b^{−n}]. Let C_{n}^{b,d} be the b-adic cubes in ℝ^{d}, i.e., products of intervals in C_{n}^{b,1}. Given a dyadic cube Q ∈ C_{n}^{b,d}, we let Q^{∗} denote its “parent”, e.g., the unique cube in C_{n−1}^{b,d} which contains it. Given λ > 0 and a cube Q, we let λQ denote the concentric cube with diam(λQ) = λdiam(Q).

A homeomorphism f : ℝ^{d} → ℝ^{d} is called quasisymmetric, if there exists a homeomorphism η : [0, ∞) → [0, ∞) such that |x − a| ≤ t|x − b| ⇒ |f(x) − f(a)| ≤ η(t)|f(x) − f(b)|. (2.1) For d ≥ 2 this is equivalent to f being quasiconformal, i.e., there is an M such that for all x ∈ Ω,

(2.2) lim sup_{r→0} \frac{sup_{y:|x−y|=r} |f(x) − f(y)|}{inf_{y:|x−y|=r} |f(x) − f(y)|} ≤ M.

(See [12] or Theorem 34.1 of [19].) When d = 1, quasisymmetry is equivalent to saying there is an M such that

\frac{1}{M} ≤ \frac{|f(I)|}{|f(J)|} ≤ M, (2.3)

QUASICONFORMAL MAPPINGS WHICH INCREASE DIMENSION 3

whenever I and J are adjacent intervals of the same length. If f satisfies this condition when I and J are also b-adic intervals (for some fixed b) then it automatically satisfies it for all I and J (possibly with a larger constant). Clearly, an increasing homeomorphism f is quasisymmetric iff it is the integral of a doubling measure, i.e., a positive measure μ on ℝ^{1} which satisfies

\frac{1}{M} ≤ \frac{μ(I)}{μ(J)} ≤ M,

where I and J are adjacent intervals of the same length. Again, it suffices to check this only for b-adic intervals. (Note that we check any pair of adjacent b-adic intervals of the same size, instead of just those with the same parent. The latter condition corresponds to “b-adic doubling measures” and gives rise to a different class of measures, e.g. [20]).)

There are two further properties of these maps which we will use. First, quasiconformal mappings are Hölder continuous [9], and have Hölder inverses, and hence map sets of positive dimension to positive dimension. Secondly, quasisymmetric maps of the line to itself extend to be quasiconformal maps of ℝ^{d} [18], [17]. (see also [2], [1] and [8] for the special cases d = 2, 3.)

Define the Hausdorff content

H_{α}^{∞}(K) = inf{∑ diam(U_{j})^{α} : K ⊂ ∪_{j}U_{j})},

and

dim(K) = inf{α : H_{α}^{∞}(K) = 0}.

This is the Hausdorff dimension of K. The mass distribution principle says that if K supports a positive measure μ such that μ(E) ≤ Cdiam(E)^{α}, (2.4)

4 CHRISTOPHER J. BISHOP

for each E ⊂ K, then dim(K) ≥ α. By standard covering arguments it actually suffices for (2.4) to hold only for b-adic cubes. This is because any cube I can be covered by a fixed number M = M(b) of b-adic cubes {J_{k}} of smaller size, so

μ(I) ≤ ∑ μ(J_{k}) ≤ C ∑ diam(J_{k})^{α} ≤ CMdiam(I)^{α}.

If h : ℝ^{1} → ℝ^{1} is quasisymmetric, and (2.4) holds for all intervals of the form h(I), with I b-adic, then dim(E) ≥ α still holds (since the same covering property still holds, but with a constant M depending on h). In the converse direction, Frostman’s theorem implies there is a positive measure μ satisfying (2.4) for every α < dim(K) (e.g., Theorem 8.8 of [15] or Theorem 1, page 7, [7]).

3. Standard Cantor sets

We will say F is a standard Cantor set if there is an integer b ≥ 2 and a λ > 1 so that F can be written as F = ∩F_{n} where each F_{n} is a union of cubes Q in C_{n}^{b,d} so that λQ are pairwise disjoint. We will need the following simple results.

Lemma 3.1. Any standard Cantor set lies on a quasiarc, i.e., for any standard Cantor set F ⊂ ℝ^{d} there is a quasiconformal f : ℝ^{d} → ℝ^{d} so that F ⊂ f(ℝ^{1}).

Lemma 3.2. For any compact E ⊂ ℝ^{d} and any ϵ > 0 there is a standard Cantor set F with dim(E ∩ F) > dim(E) − ϵ. Moreover, we may take the λ in the definition of standard Cantor set as large as we wish.

Lemma 3.3. If E ⊂ ℝ^{1}, an integer d ≥ 2 and ϵ > 0 are given, then there there is a quasiconformal mapping f : ℝ^{d} → ℝ^{d} so that dim(f(E)) ≥ (d − ϵ) dim(E).

Proof of Lemma 3.1: The proof is a picture which we draw in the plane. Write F = ∩F_{n} as in the definition and consider a component cube Q of F_{n}. Also recall

QUASICONFORMAL MAPPINGS WHICH INCREASE DIMENSION 5

that the dilation λQ is disjoint from similar dilations of the other components of F_{n}. Consider the components {Q_{j}}_{1}^{N} of F_{n+1} which lie inside Q. There is a mapping of λQ to the unit cube in ℝ^{d} which

(1) maps the λQ′_{j}s to N disjoint cubes centered on the real line,

(2) is a Euclidean similarity on ∂λQ and inside each λQ_{j},

(3) is quasiconformal in A_{Q} = λQ \ ∪_{j}λQ_{j}. See Figure 3.1. Since the mapping is non-conformal only in A_{Q} and these regions are

Figure 3.1. Defining a quasiarc which hits E.

all pairwise disjoint for all cubes in F_{n} and for all n, we can obviously compose them and pass to a limit to obtain a quasiconformal mapping of ℝ^{d} which maps F into the real line.

The same proof shows that given a b_{1}-adic standard Cantor set E ⊂ ℝ^{1} and a b_{2}-adic standard Cantor set F ⊂ ℝ^{d} so that for both sets each generational cube has the same number of children, there is a quasiconformal mapping of ℝ^{d} which maps E to F and satisfies

C^{−1} ≤ \frac{|f(x) − f(y)|}{|x − y|^{γ}} ≤ C, x, y ∈ E, γ = \frac{log b_{2}}{log b_{1}} . (3.1)

We will use this in the proof of Lemma 3.3.

6 CHRISTOPHER J. BISHOP

In ℝ^{2}, Lemma 3.1 is a special case of a recent result of P. MacManus: if E is a nowhere dense, compact set whose complement is a uniform domain then E lies on a quasicircle. See [14].

Proof of Lemma 3.2: Without loss of generality assume E ⊂ F_{0} = [0, 1]^{d} and dim(E) > 0. Choose 0 < ϵ < dim(E) and set α = dim(E) − \frac{ϵ}{2}. Then by Frostman’s theorem, there is positive measure μ supported on E which satisfies

μ(K)) ≤ Cdiam(K)^{α},

for every compact set K.

Assume that the Frostman measure μ on E has mass 1. Suppose b is a large even integer and λ > 0. There is a constant A (depending on d and λ, but not on b) so that we can write F_{0} as the union of A sets, each of which is the union of at least two b-adic cubes Q with λQ pairwise disjoint. One of these sets has μ measure ≥ A^{−1}, so choose one and call it F_{1}.

Define a measure ν_{1} on F_{1} by taking the restriction of μ to F_{1} and normalizing it to have mass 1. Since the normalizing constant is less than A, we get for every b-adic cube Q

ν_{1}(Q) ≤ Aμ(Q) ≤ ACdiam(Q)^{α} ≤ Cdiam(Q)^{β},

where

β = α − \frac{log A}{log b} ≤ α − \frac{log A}{log \sqrt{d} + n log b} = α − \frac{log A}{log diam(Q)^{−1}} .

Assume b is so large that β > dim(E) − ϵ.

In general, suppose F_{n} is a union of pairwise non-adjacent cubes in C_{n}^{b,d} and that we have a probability measure ν_{n} on F_{n} which satisfies

ν_{n}(Q) ≤ A^{min(k,n)}μ(Q) (3.2)

QUASICONFORMAL MAPPINGS WHICH INCREASE DIMENSION 7

for all Q ∈ C_{k}^{b,d} for all k. Note that (3.2) implies (3.3) ν_{n}(Q) ≤ A^{k}Cdiam(Q)^{α} = C(\frac{A}{b^{α}})^{k} ≤ Cdiam(Q)^{β} for all Q ∈ C_{k}^{b,d} and for all k.

For each component cube Q of F_{n} we can write Q as the union of A sets, each of which is the union of non-adjacent b-adic cubes in C_{n+1}^{b,d}, and define F_{n+1} by choosing the set with largest ν_{n} measure. Define a probability measure ν_{n+1} on F_{n+1} by restricting ν_{n} and normalizing in each component Q of F_{n} so that for E ⊂ Q, ν_{n+1}(E) = ν_{n}(E)\frac{ν_{n}(Q)}{ν_{n}(F_{n+1})}. Then clearly ν_{n+1} satisfies (3.2) and hence (3.3) for k ≤ n. To see that it also satisfies these estimates for intervals for k = n + 1, note that for Q ∈ C_{n}^{b,d},

ν_{n+1}(Q) ≤ Aν_{n}(Q) ≤ A^{n+1}μ(Q),

which is (3.2) with k ≥ n + 1. As before this implies (3.3) also holds.

Finally, let F = ∩_{n}F_{n}. It is clear that F ⊂ E is a standard Cantor set. Moreover, the measures {ν_{n}} are consistently defined and so define a probability measure ν on F which satisfies ν(Q) ≤ Cdiam(Q)^{β}, for all b-adic cubes. By the mass distribution principle, this implies dim(F) ≥ β ≥ dim(E) − ϵ, as desired.

Proof of Lemma 3.3: Fix ϵ > 0 and choose a large even integer b and set λ = 2. Let F_{1} ⊂ ℝ^{1} be a standard Cantor set constructed using (\frac{b}{2})^{d}-adic cubes as in the proof of Lemma 3.2 so that dim(F_{1} ∩ E) ≥ dim(E) − \frac{ϵ}{2d}. Let F_{d} be a standard Cantor set in ℝ^{d} formed using (\frac{b}{2})^{dn} b-adic cubes at the nth step of the construction. The comment following the proof of Lemma 3.1 shows that there is a quasiconformal mapping f : ℝ^{d} → ℝ^{d} which maps F_{1} onto F_{d}, and satisfies the estimates

C^{−1} ≤ \frac{|f(x) − f(y)|}{|x − y|^{γ}} ≤ C, γ = \frac{log b}{d(log b − log 2)},

8 CHRISTOPHER J. BISHOP

for some C < ∞ and all x, y ∈ F_{1}. It is a standard result that such a map multiplies the dimension of any subset of F_{1} by exactly a factor of γ^{−1}, so

dim(f(E)) ≥ dim(f(E ∩ F_{1})) = \frac{1}{γ} dim(E ∩ F_{1})
= d(1 − \frac{log 2}{log b})(dim(E) − \frac{ϵ}{2d})
≥ (d − ϵ) dim(E),

if b is large enough.

4. Proof of Theorem 1.1 for d = 1

Suppose E ⊂ ℝ^{1} has positive dimension. The easiest way to build a homeomorphism of ℝ^{1} which increases the dimension of E would be to simply integrate a Frostman measure for E, i.e., take the map h(x) = ∫_{−∞}^{x} dμ, where μ is supported on E. Since this maps E to positive Lebesgue measure, it certainly increases the dimension as much as possible, but this map is not generally quasisymmetric because the measure μ need not be doubling.

To construct a quasisymmetric mapping which increases the dimension of E we will modify this idea slightly. Fix 0 < γ < 1 as in the statement of the theorem. Starting with a Frostman measure μ on E we will construct a doubling measure ν so that ν(I)^{γ} ≥ Cμ(I) for all intervals. The integral of this measure will be a quasisymmetric mapping h and the above estimate will imply the push forward of μ under h is a Frostman measure for h(E) with exponent γ.

Using Lemma 3.2 we may assume that E = ∩F_{n} is a standard b-adic Cantor set in [0, 1] with a Frostman measure μ which satisfies μ(I) ≤ C|I|^{α}, for some 0 < α ≤ dim(E). We may also assume that λ = 3, i.e., that the components of F_{n} have disjoint

QUASICONFORMAL MAPPINGS WHICH INCREASE DIMENSION 9

triples. This means that any b-adic interval of generation n is either a component of F_{n} or is adjacent to at most one such component.

To define ν it suffices to give the ν measure of each b-adic interval in [0, 1]. We do this by induction on the generation of the interval. For I = [0, 1], let ν(I) = μ(I). For the induction step fix some 1 > ϵ > 0. Suppose we have already defined ν(I) for some nth generation b-adic interval I. Let I_{1}, …, I_{b} be its “children”. To define the measures of these intervals we consider three cases: (1) I is a component of F_{n}, (2) I is adjacent to such a component or (3) I is not adjacent to any component of F_{n}. As part of the induction, we also prove that any two children, I_{j} and I_{k}, of I satisfy

ην(I_{j}) ≤ ν(I_{k}) ≤ η^{−1}ν(I_{j}),

for some η depending only on b and ϵ.

Case 1: Suppose I_{j} is a child of I which is a component of F_{n}. We say I_{i} ⊂ I is “good” if μ(I_{i}) ≥ \frac{ϵ}{b}μ(I). If μ(I_{i}) < \frac{ϵ}{b}μ(I) then we say I_{i} is “bad”. Let B be the number of bad children and note that B > 0 because no two adjacent children can both hit E and thus at least one has zero μ measure. Define

δ = \frac{1}{B} (1 − \frac{1 − ϵ}{μ(I)} ∑_{I_{j} good} μ(I_{j})),

and note that δ ≥ B^{−1}(1 − (1 − ϵ)) ≥ ϵ/b. Next, define

ν(I_{i}) = \begin{cases}δν(I), & if I_{i} is bad \\ (1 − ϵ)ν(I)\frac{μ(I_{i})}{μ(I)}, & if I_{i} is good \end{cases} .

Our choice of δ implies ∑_{i=1}^{b} ν(I_{i}) = ν(I). Each of the bad intervals receives at least mass ν(I)ϵ/b and each of the good intervals receives mass at least ν(I)(1 − ϵ)ϵ/b. Thus all the children receive comparable measure, say ν(I_{j}) ≥ η_{1}ν(I_{k}), j ≠ k, where

10 CHRISTOPHER J. BISHOP

η_{1} = (1 − ϵ)ϵ/b. In particular

\frac{(1 − ϵ)ϵ}{b} ≤ \frac{ν(I_{j})}{ν(I)} ≤ 1 − \frac{ϵ}{b}. (4.1)

Finally, observe that

\frac{ν(I_{j})}{ν(I)} ≤ \frac{μ(I_{j})}{μ(I)},

if I_{j} is good and the reverse inequality holds if I_{j} is bad (i.e., ν gives a larger fraction of the mass of I to bad intervals than μ does and gives less to the good intervals than μ does).

Case 2: Next suppose I = I_{1} ∪ ⋯ ∪ I_{b} ∈ C_{n}^{b} is adjacent to a component interval J = J_{1} ∪ ⋯ ∪ J_{b} of F_{n} (recall that since components have disjoint triples I can be adjacent to at most one component). With loss of generality, assume I_{b} is adjacent to J_{1}. Set ν(I_{b}) = ν(J_{1})\frac{ν(I)}{ν(J)}. Note that by (4.1) we get

\frac{(1 − ϵ)ϵ}{b}ν(I) ≤ ν(I_{b}) ≤ (1 − \frac{ϵ}{b})ν(I).

Now define ν on the remaining b−1 intervals so that they get equal measure, namely

ν(I_{j}) = (ν(I) − ν(I_{b}))/(b − 1) ≥ \frac{ϵ}{b(b − 1)}ν(I) ≥ η_{1}ν(I),

for j = 1, …, b − 1. Thus any two children of I have comparable ν measure with some constant η_{2} depending only on b and ϵ.

Case 3: Finally, if I is not adjacent to a component of F_{n} simply define ν(I_{i}) = \frac{1}{b}ν(I) (i.e., each subinterval gets equal measure). Thus the ν measure of any two children is the same.

This completes the construction of ν. To complete the proof of Theorem 1.1 we simply have to check that ν is a doubling measure and that the corresponding homeomorphism maps E to a set of dimension ≥ γ.

QUASICONFORMAL MAPPINGS WHICH INCREASE DIMENSION 11

Proof that ν is a doubling measure: It is enough to check the doubling condition for b-adic intervals. We have already done this (during the construction of ν) if we take two intervals with the same parent so suppose I_{1} and I_{2} are adjacent b-adic intervals of the same length and have different parents I_{1}^{∗}, I_{2}^{∗}. Since components of F_{n} have disjoint triples there are only three possibilities (up to relabeling): (1) I_{1}^{∗} is a component of F_{n} and I_{2}^{∗} is not, (2) I_{1}^{∗} is adjacent to a component but I_{2}^{∗} is not, or (3) neither is adjacent to a component of F_{n}. In case (1) ν(I_{1})/ν(I_{2}) = ν(I_{1}^{∗})/ν(I_{2}^{∗}) by definition. By taking parents until we reach a common case 1 ancestor, we see that any pair of this form is comparable with the constant η_{1}. In case (2) ν(I_{1})/ν(I_{2}) is comparable to ν(I_{1}^{∗})/ν(I_{2}^{∗}). Moreover, I_{1}^{∗} is a component of F_{n} and I_{2}^{∗} is adjacent to it, so their ν measures are comparable with an absolute constant. Thus ν(I_{1}) and ν(I_{2}) are comparable by a (different) absolute constant. Finally, in case (3), ν(I_{1})/ν(I_{2}) = ν(I_{1}^{∗})/ν(I_{2}^{∗}) by definition. Keep taking parents until either we reach a common ancestor, or are in case (2). In either case we have comparable ν measures and the same constant works for I_{1} and I_{2}. This completes the proof that ν is a doubling measure.

Proof that h(E) has large dimension: It suffices to show that if

h(x) = ∫_{0}^{x} dν,

then the measure σ defined by σ(F) = H^{∗}μ(F) = μ(h^{−1}(F)) is a Frostman measure on h(E) with exponent γ, i.e.,

σ(J) ≤ C|J|^{γ}.

It suffices to do this for intervals which are h images of b-adic intervals, so using the

12 CHRISTOPHER J. BISHOP

definition of h and σ, we need only show μ(I) ≤ Cν(I)^{γ}, or equivalently ν(I) ≥ Aμ(I)^{1+κ}, (4.2) for all b-adic intervals I with some A > 0 and κ = \frac{1}{γ} − 1.

Fix an interval I ∈ C_{n}^{b} and for each k = 0, …, n let I_{k} ∈ C_{k}^{b} be the unique interval containing I. Let

x_{k} = − log_{b} μ(I_{k}), y_{k} = − log_{b} ν(I_{k}).

Then desired inequality (4.2) becomes y_{k} ≤ (1 + κ)x_{k} + log_{b} A^{−1}. Since μ satisfies μ(I) ≤ C|I|^{α}, we have x_{n} ≥ αn − log_{b} C, or equivalently

n ≤ \frac{x_{n} + log_{b} C}{α} .

If I_{k} is a good interval then

\frac{ν(I_{k})}{ν(I_{k−1})} = (1 − ϵ)\frac{μ(I_{k})}{μ(I_{k−1})},

which implies y_{k} − y_{k−1} = − log_{b}(1 − ϵ) + (x_{k} − x_{k−1}). (4.3) If ϵ < \frac{1}{2}, then log_{b}(1 − ϵ) ≤ 2/(ϵ log b), so this becomes,

y_{k} − y_{k−1} ≤ \frac{2}{ϵ log b} + (x_{k} − x_{k−1}). (4.4)

On the other hand, if I_{k} is bad, then ν gives a larger proportion of its mass to I_{k} than μ did, i.e.,

\frac{ν(I_{k})}{ν(I_{k−1})} ≥ \frac{μ(I_{k})}{μ(I_{k−1})},

which implies y_{k} − y_{k−1} ≤ x_{k} − x_{k−1}. (4.5)

QUASICONFORMAL MAPPINGS WHICH INCREASE DIMENSION 13

Since y_{0} = 0, summing (4.3) and (4.5) for k = 1, …, n gives

y_{n} ≤ x_{n} + \frac{2n}{ϵ log b} ≤ x_{n} + \frac{2 log_{b} C + x_{n}}{αϵ log b} ≤ x_{n}(1 + κ) + A,

if we choose b so large that 2(αϵ log b)^{−1} ≤ κ. This is the desired inequality, and completes the proof.

5. Proof of Theorem 1.1 when d ≥ 2

Suppose E ⊂ ℝ^{d} has dimension α > 0. By Lemma 3.2 there is a standard Cantor set F ⊂ E with dim(F) > α/2, and by Lemma 3.1 F has a quasiconformal image F_{1} = f_{1}(F) which lies on a line. Since quasiconformal mappings are Hölder continuous [9], F_{1} has positive Hausdorff dimension, and hence by Theorem 1.1 there is a quasisymmetric map f_{2} of the line to itself which maps F_{1} to a set F_{2} with dimension as close to 1 as we wish, say ≥ 1 − \frac{γ}{10d}. Since quasisymmetric maps of the line to itself extend to be quasiconformal maps of ℝ^{d} [18], [17], we may assume F_{2} = f_{2}(F_{1}) where f_{2} is quasiconformal on ℝ^{d}. Next use Lemma 3.3 to can find a quasiconformal map f_{3} of ℝ^{d} to itself which maps F_{2} to a set of dimension ≥ γ. Thus f_{3} ∘ f_{2} ∘ f_{1} is the desired quasiconformal map. This completes the proof of Theorem 1.1.

6. Decreasing dimension by QC maps

We have shown how to increase the dimension of a set by a quasiconformal mapping. What about decreasing the dimension? It is easy to see there are sets whose dimension cannot be lowered by any quasiconformal mapping. A trivial example is a line segment. A well known result of Gehring and Väisälä [11], [10] implies that if d ≥ 2, quasiconformal images of sets in ℝ^{d} of dimension d also have dimension d. The latter result fails for d = 1 since a quasisymmetric map may send a set of positive Lebesgue measure to dimension strictly less than 1 (e.g., [5] or [16]). Other examples

of sets whose dimension cannot be decreased can be obtained by considering non-

removable sets for quasiconformal mappings. Any set of dimension less than d − 1

is removable (Theorem 35.1, [19]), and there are examples of of totally disconnected,

non-removable sets of dimension d − 1 (at least when d = 2, 3, [3], [4], [6], [13]). Thus

the quasiconformal image of such a set can never have smaller dimension. There are

also Cantor sets in ℝ^{3}, such as Antoine’s necklace, whose complements are not simply

connected, and hence no homeomorphism of ℝ^{3} can map the set to dimension less

than 1. For 0 < α < d, is there always a set E ⊂ ℝ^{d} of dimension α whose dimension

cannot be lowered by any quasisymmetric map?

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19. J. Väisälä. Lectures on n-dimensional quasiconformal mappings. Lecture Notes in Math. 229. Springer-Verlag, 1971.

20. J.-M. Wu. Doubling measures for different bases. 1996. preprint.

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

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