PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 128, Number 11, Pages 3361{3367 S 0002-9939(00)05433-2 Article electronically published on May 18, 2000 SETS OF MINIMAL HAUSDORFF DIMENSION FOR QUASICONFORMAL MAPS JEREMY T. TYSON (Communicated by Albert Baernstein II) Abstract. For any 1 ≤ α ≤ n, there is a compact set E ⊂ Rn of (Hausdorff) dimension α whose dimension cannot be lowered by any quasiconformal map f : Rn ! Rn. We conjecture that no such set exists in the case α<1. More generally, we identify a broad class of metric spaces whose Hausdorff dimension is minimal among quasisymmetric images. 1. Introduction Quasiconformal homeomorphisms of a Euclidean space Rn, n ≥ 2, can distort the Hausdorff dimensions of subsets. While the dimensions of sets of Hausdorff dimension zero or n must be preserved, Gehring and V¨ais¨al¨a [5, pp. 505-506] n construct, for any β 2 (0;n), a compact set Eβ ⊂ R with dim Eβ = β and, for 0 n n any β,β 2 (0;n), a quasiconformal map f : R ! R with Eβ0 = fEβ. Recently, Bishop [2] showed that the dimension of any compact set E ⊂ Rn of positive dimension can be raised arbitrarily close to n by a quasiconformal (quasisymmetric if n = 1) homeomorphism of Rn. On the other hand, for certain values of α 2 (0;n), one can construct compact sets E ⊂ Rn of dimension α whose dimension cannot be lowered by any quasicon- formal mapping of the ambient space. For example, consider the closed unit k-cube [0; 1]k, k =1; 2;::: ;n−1. These examples have Hausdorff dimension equal to their topological dimension and thus the Hausdorff dimension cannot be lowered by any homeomorphism of Rn. A different example (arising from the theory of removable sets for quasiconformal maps) was given by Bishop [1] in the case n =3,α =2. Bishop [2] asks whether there exist examples of this phenomenon for all 0 <α<n. The following result is a simple consequence of the main theorem in this paper: Theorem 1.1. Let 1 ≤ α ≤ n. There is a compact set E ⊂ Rn with Hausdorff dimension α so that dim fE ≥ α for all quasiconformal maps f : Rn ! Rn. When α>1 we may take the set E to be the Cartesian product of any Ahlfors (α − 1)-regular set in Rn−1 with the unit interval [0; 1]; for the definition of Ahlfors Received by the editors October 15, 1998 and, in revised form, January 15, 1999. 2000 Mathematics Subject Classification. Primary 30C65; Secondary 28A78. Key words and phrases. Hausdorff dimension, quasiconformal maps, generalized modulus. The results of this paper form part of the author's doctoral dissertation at the University of Michigan. The author was supported by a National Science Foundation Graduate Research Fellowship and a Sloan Doctoral Dissertation Fellowship. c 2000 American Mathematical Society 3361 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 3362 JEREMY T. TYSON regularity, see section 2 (for the existence of such sets, note that the set Eβ in the above example of Gehring and V¨ais¨al¨aisβ-regular). We will prove a more general result (Theorem 3.4) using the machinery of the generalized (or discrete) modulus introduced in [9] and developed in [13]. The restriction to subsets of a Euclidean space is not necessary; we give a condition on metric spaces which is sufficient to guarantee that the Hausdorff dimension is minimal among quasisymmetric images (recall that in Rn, n ≥ 2, the classes of quasiconformal and quasisymmetric maps agree). It should be emphasized that Theorem 1.1 is not new; the result appears in works of Pansu [9] and Bourdon [3] (see, e.g., [9, Example 3.4] and [3, Example 1.7(b)]). These authors give a family of conditions (Lemma 3.9) on an arbitrary metric space which imply nontrivial lower bounds on the Hausdorff dimensions of quasisymmetric images. With an additional (weak) assumption, we show in Proposition 3.11 that the hypotheses of Theorem 3.4 are implied by the \best" condition of Bourdon and Pansu. The above-mentioned examples which illustrate Theorem 1.1 satisfy this \best" condition of Bourdon-Pansu. The restriction to dimensions larger than one is necessary in our proof, which relies on the geometric characterization of quasiconformality in terms of curve fam- ilies. We conjecture that no such example exists for dimensions strictly smaller than one, more precisely: Conjecture 1.2. Let n ≥ 1 and >0.IfacompactsetE ⊂ Rn has Hausdorff dimension strictly smaller than one, then there exists a quasiconformal (quasisym- metric if n =1)mapf : Rn ! Rn so that fE has Hausdorff dimension smaller than . Note that such a set must be totally disconnected. For related results, see [12]. 2. Notation and definitions In a metric space X,wedenotebyB(x; r) the closed ball with center x 2 X and radius r>0. For Y ⊂ X,wewritediamY for the diameter of Y .IfB = B(x; r) is a ball in X,wewriteCB for the dilated ball B(x; Cr). For x 2 X and r>0, we have diam B(x; r) ≤ 2r, but no lower bound for diam B(x; r) need hold in general. We call X (c1-)uniformly perfect,0<c1 ≤ 2, if diam B(x; r) ≥ c1r for every x 2 X and 0 <r≤ diam X. Every connected space is 1-uniformly perfect. For α>0, we denote by Hα the Hausdorff α-measure on X and by dim X the Hausdorff dimension of X [4, x2.10.2(1)]. For connected sets E,wehavetherelation diam E ≤H1(E)[4,x2.10.12]. For α>0, we call a Borel measure µ on X (Ahlfors) α-regular if there exists a constant C0 ≥ 1 (called a regularity constant for X)sothat 1 α α (2.1) r ≤ µ(B(x; r)) ≤ C0r C0 for all x 2 X and 0 <r<diam X.Ifµ is a Borel measure on X satisfying (2.1), then Hα also satisfies (2.1) (possibly with a different regularity constant) and µ and Hα are equivalent. Thus α-regular spaces have Hausdorff dimension α. Every α-regular space with regularity constant C0 is c1-uniformly perfect for each −2/α c1 <C0 . License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use SETS OF MINIMAL HAUSDORFF DIMENSION 3363 We call a homeomorphism f : X ! Y between metric spaces (η-)quasisymmetric, where η :[0; 1) ! [0; 1) is a homeomorphism, if (2.2) jx − y|≤t jx − zj)jf(x) − f(y)|≤η(t) jf(x) − f(z)j for any x; y; z 2 X and t>0 [11]. By making η larger if necessary, we can (and will) always assume that η(t) ≥ t for t>0 and that both f and f −1 satisfy (2.2). If X ! 0 is c1-uniformly perfect and f : X Y is η-quasisymmetric, then Y is c1-uniformly 0 0 perfect for some c1 = c1(c1,η) > 0. The Hausdorff dimension of a metric space is not a quasisymmetric invariant. Following Pansu [9, D´efinition 3.1], we define the conformal dimension of a metric space X to be C dim X =infdim Y; Y where the infimum is taken over all spaces Y which are quasisymmetrically equiv- alent to X. For any space X, C dim X ≥ dimT X, where dimT X denotes the topological dimension of X [7]. 3. The main result Let (X; µ) be a metric measure space. For a family Γ of curves in X and p ≥ 1, we denote by Modp Γthep-modulus of Γ [14, x6.1], [6, x2.3]. (Throughout this paper, we restrict ourselves to consider only curves whose parametrization is one-to-one.) Definition 3.1. We say that X has nontrivial p-modulus, p ≥ 1, if Modp Γ > 0for some family of curves Γ in X. Example 3.2. Let α>1. Let Z be any compact (α − 1)-regular metric space and let X be the product space Z ×[0; 1] (endowed with any of its standard bi-Lipschitz equivalent metrics). Then X is a compact, α-regular space and the family of curves Γ=fγz :[0; 1] ! Xjγz(t)=(z;t);z 2 Zg has positive α-modulus. Note that for any n ≥ 2and1<α≤ n,wemaychoose Z ⊂ Rn−1 (see the introduction) and so X ⊂ Rn. When 1 <α<2, the preceding example is not connected. We next give con- nected examples for each 1 <α≤ n. Example 3.3. Let 1 <α≤ n and choose a compact (α − 1)-regular Borel set Z ⊂ Sn−1.Letν be an (α − 1)-regular measure on Z and let X = ftz : z 2 Z; 0 ≤ t ≤ 1}⊂Rn n be the cone on Z. Here, for z =(z1;::: ;zn) 2 R and t ≥ 0, we write tz = (tz1;::: ;tzn). We define a measure µ on X as follows: for Borel sets A ⊂ [0; 1] and B ⊂ Z, define the spherical product set A · B = ftz : t 2 A; z 2 Bg and set Z µ(A · B)=ν(B) · rn−1dr: A The collection of spherical product sets generates the Borel σ-algebra on X and µ is an α-regular Borel measure. Moreover, the family of curves Γ=fγz :[1=2; 1] ! Xjγz(t)=tz; z 2 Zg has positive α-modulus, by an argument similar to [14, Example 7.5]. Theorem 1.1 is a consequence of these examples and the following result.