aa r X i v : . [ m a t h . M G ] J a n CONFORMAL GEOMETRY AND DYNAMICSAn Electronic Journal of the American Mathematical SocietyVolume 00, Pages 000–000 (Xxxx XX, XXXX)S 1088-4173(XX)0000-0
TOPOLOGICAL CONFORMAL DIMENSION
CLAUDIO A. DIMARCO
Abstract.
We investigate a quasisymmetrically invariant counterpart of thetopological Hausdorff dimension of a metric space. This invariant, called the topological conformal dimension , gives a lower bound on the topological Haus-dorff dimension of quasisymmetric images of the space. We obtain results con-cerning the behavior of this quantity under products and unions, and computeit for some classical fractals. The range of possible values of the topologicalconformal dimension is also considered, and we show that this quantity can befractional. Introduction
For a metric space (
X, d ) , the topological dimension can be defined inductivelyas dim t X = inf { c : X has a basis U such that dim t ∂U ≤ c − U ∈ U} , where dim t ∅ = − . Since topological dimension is bounded above by Hausdorffdimension, we write dim t X ≤ dim H X and say that X is fractal if dim t X < dim H X [12].Topological dimension is invariant under homeomorphisms, while Hausdorff di-mension is bi-Lipschitz invariant. An intermediate interesting class of maps be-tween the first two is quasisymmetric maps [8, 14]. In order to classify spaces upto quasisymmetric equivalence, it is useful to have a concept of dimension that isquasisymmetrically invariant. One such example is conformal dimension [11], whichwe denote dim C X. Conformal dimension is modeled after Hausdorff dimension, which measures thesize of a metric space. Recently, Balka, Buczolich and Elekes [1] introduced the topological Hausdorff dimension , which is a bi-Lipschitz invariant sensitive to con-nectivity. Unlike conformal dimension, this dimension is not quasisymmetricallyinvariant (Theorem 6.2). Indeed, it can be arbitrarily increased by quasisymmetricmaps (Fact 6.1). At present, there are no nontrivial lower bounds on topologicalHausdorff dimension of quasisymmetric images, unlike for Hausdorff dimension.In this paper, we introduce a quasisymmetric invariant that delivers such abound: the topological conformal dimension , denoted dim tC X. Writing dim tH X for the topological Hausdorff dimension,dim tH f ( X ) ≥ dim tC X Received by the editors September 5, 2018.2010
Mathematics Subject Classification.
Primary 28A80, 30L10; Secondary 28A78, 54F45.
Key words and phrases. metric space, conformal dimension, topological dimension, quasisym-metric map, Cantor sets. c (cid:13) XXXX American Mathematical Society for every quasisymmetric mapping f of X. Our investigation uncovers a parallelbetween conformal dimension and topological conformal dimension. The formerdetects rich families of curves, while the latter detects rich families of surfaces. Bydefinition,dim tC X = inf { c : X has a basis U such that dim C ∂U ≤ c − U ∈ U} . We obtain results concerning the behavior of this new quantity under productsand unions, along with the range of its possible values. The topological confor-mal dimension is computed for some classical fractals, often via comparison to theHausdorff dimension. The following theorem, proven in section 4, gives a two-sidedestimate for the topological conformal dimension of a space that contains a “dif-fuse” family of surfaces. This result is similar to that of Pansu; lower bounds forconformal dimension are based on the presence of a diffuse family of curves. TheAssouad dimension of X is denoted dim A X [8]. Theorem 4.5.
Let (
X, d, µ ) be a compact, doubling metric measure space withdim A X = D < ∞ . Suppose that there exists a family Q of surfaces with thefollowing properties:(i) there is a family { f α : α ∈ J } of L − bi-Lipschitz maps such that Q α = f α ([0 , ) for each Q α ∈ Q . (ii) there exists a measure ν on Q , a constant C , and r > < ν { Q α ∈ Q : Q α ∩ B ( x, r ) = ∅} ≤ C r d for all x ∈ X and for all r ≤ r . (iii) The union of edges S α ∈ J f α (cid:0) ∂ ([0 , ) (cid:1) is not dense in X. Then 1 + DD − d ≤ dim tC X ≤ D. Such spaces are shown to exist for every 0 < d < D = 2 + d. Since2 + d / ≤ dim tC ≤ d , it follows that the topological conformal dimension attainsinfinitely many fractional values even though they are not explicitly determined.In particular, if 0 < d < D = 2 + d , then 2 < dim tC X <
3, which furnishes acollection of compact metric spaces with fractional topological conformal dimension.A key difference between topological conformal dimension and topological Haus-dorff dimension is that conformal dimension may increase under Lipschitz maps.This makes it difficult to give a lower bound on topological conformal dimension,such as that in Theorem 4.5.2.
Preliminaries and Basic Properties
The subscripts of dim indicate the type of dimension. By convention, everydimension of the empty set is − . For any set A, | A | means the cardinality of A. We write B ( x, ε ) for the open ball centered at x of radius ε. We will often referto a basis of X, by which we mean a basis for the topology on X induced by themetric on X. Our definition of topological conformal dimension is similar to that ofthe classical topological dimension and the topological Hausdorff dimension. The topological Hausdorff dimension is defined in [1] asdim tH X = inf { c : X has a basis U such that dim H ∂U ≤ c − ∀ U ∈ U} . OPOLOGICAL CONFORMAL DIMENSION 3
An embedding f : X → Y is quasisymmetric if there is a homeomorphism η :[0 , ∞ ) → [0 , ∞ ) so that d X ( x, a ) ≤ td X ( x, b ) implies d Y ( f ( x ) , f ( a )) ≤ η ( t ) d Y ( f ( x ) , f ( b ))for all triples a, b, x of points in X, and for all t > p-dimensionalHausdorff measure of X is H p ( X ) = lim δ → inf nX (diam E j ) p : X ⊂ [ E j and diam E j ≤ δ ∀ j o . The
Hausdorff dimension of X is dim H X = inf { p : H p ( X ) = 0 } , and the conformaldimension of X isdim C X = inf { dim H f ( X ) : f is quasisymmetric } . Definition 2.1.
The topological conformal dimension of a metric space (
X, d ) is(2.1)dim tC X = inf { c : X has a basis U such that dim C ∂U ≤ c − U ∈ U} We will have occasion to use the following two facts.dim t X ≤ dim C X ≤ dim H X (2.2) dim t X ≤ dim tC X ≤ dim tH X ≤ dim H X (2.3) Proof.
Inequality (2.2) is well known [11]. For the first inequality in (2.3), let U bea basis of X and U ∈ U . By (2.2) we have dim t ∂U ≤ dim C ∂U, so if dim C ∂U ≤ c − t ∂U ≤ c − . It follows that dim t X ≤ dim tC X. Using (2.2), thesame argument gives dim tC X ≤ dim tH X. The inequality dim tH X ≤ dim H X isTheorem 4.4 in [1]. (cid:3) By Fact 4.1 in [1] we have dim tH X = 0 ⇔ dim t X = 0 , so (2.2) yields(2.4) dim t X = 0 ⇔ dim tC X = 0 ⇔ dim tH X = 0 . Proposition 2.2. If X is a metric space, then dim t X ≤ dim tC X ≤ dim C X. Proof.
The first inequality is (2.3). For the second, let ε > . By definition ofdim C X there is a quasisymmetric map f with dim H f ( X ) ≤ dim C X + ε. Since thetopological conformal dimension is a quasisymmetric invariant, we have (cid:3) (2.5) dim tC X = dim tC f ( X ) ≤ dim tH f ( X ) ≤ dim H f ( X ) ≤ dim C X + ε. Proposition 2.3.
The topological conformal dimension of a countable set is zero,and for open subsets of R d and for smooth d − dimensional manifolds, the topologicalconformal dimension is d. Proof.
This follows immediately from (2.3). (cid:3)
It is clear from the definition that the topological conformal dimension is invari-ant under quasisymmetric maps. In particular, it is invariant under bi-Lipschitzmaps. The following examples are along the lines of those in [1]. The next exampleshows that the topological conformal dimension can increase under Lipschitz mapsin general, and Example 4.12 in [1] gives an example with an injective Lipschitzmap.
CLAUDIO A. DIMARCO
Example 2.4.
Let K ⊂ [0 ,
1] be a Cantor set of positive Lebesgue measure.Lemma 4.10 in [1] gives a surjective Lipschitz map f : K → [0 , , namely f ( x ) = m (( −∞ ,x ) ∩ K ) m ( K ) , where m is Lebesgue measure. Since K has a basis of clopen sets,we have dim t K = dim tC K = 0 . Also 1 = dim tC [0 ,
1] = dim tC f ( K ) , so dim tC K < dim tC f ( K ) . Behavior Under Inclusions, Unions, and Products
As with other types of dimension, tC-dimension is monotone under inclusion.
Proposition 3.1. (Monotonicity under inclusion) If X is a subset of a metricspace Y , then dim tC X ≤ dim tC Y. Proof. If U is a basis of Y then { U ∩ X : U ∈ U} is a basis in X and ∂ X ( U ∩ X ) ⊂ ∂ Y U holds for all U ∈ U . Note that monotonicity of conformal dimension followsfrom monotonicity of Hausdorff dimension. Thus dim C ∂ X ( U ∩ X ) ≤ dim C ∂ Y U, sothe result follows. (cid:3) Conformal dimension is stable under finite disjoint unions of compact sets ([11],Proposition 5.2.3). The same is true of the topological conformal dimension.
Fact 3.2. If S and S are disjoint compact subsets of a metric space and if S = S ∪ S , then dim tC S = max { dim tC S , dim tC S } . Proof.
Let ε > . We find a basis U of S and d ≥ C ∂U ≤ d − U ∈ U , and such that d ≤ max { dim tC S , dim tC S } + ε. By definition ofthe topological conformal dimension, for i = 1 , U i , d i such that U i is abasis for S i , dim C ∂U i ≤ d i − U i ∈ U i , and d i ≤ dim tC S i + ε. Consider the basis U = U ∪ U of S and let d = max { d , d } . For each U = U ∪ U ∈ U , ∂U and ∂U are disjoint compact sets, so ∂U = ∂U ∪ ∂U andProposition 5.2.3 in [11] impliesdim C ∂U = max { dim C ∂U , dim C ∂U }≤ max { d − , d − }≤ max { dim tC S , dim tC S } + ε − . (3.1)Inequality (3.1) yields dim tC S ≤ max { dim tC S , dim tC S } and monotonicity givesthe opposite inequality. (cid:3) Corollary 3.3. If { X , . . . , X n } is a disjoint family of compact subsets of a metricspace, then dim tC S ni =1 X i = max { dim tC X i | i = 1 , . . . , n } . Unlike the Hausdorff dimension, the next example shows that finite stability doesnot hold for non-closed sets.
Example 3.4.
Each of Q and R \ Q has a basis of clopen sets, so their topologicaldimensions are zero. Then (2.4) gives dim tC Q = dim tC ( R \ Q ) = 0 . By (2.3) wehave dim tC R = 1 so thatdim tC R = 1 > { dim tC Q , dim tC ( R \ Q ) } . OPOLOGICAL CONFORMAL DIMENSION 5
Dimension of Products.
For two metric spaces X and Y we consider theproduct space Z = X × Y with metric d Z (( x , y ) , ( x , y )) = max( d X ( x , x ) , d Y ( y , y )) . Under certain favorable conditions, Hausdorff dimension is additive under prod-ucts [13]. This is not the case for conformal dimension [11]. In section 4 we will seethat topological conformal dimension is also not additive under products (Corollary4.7). In this section we provide an upper bound on the tC-dimension of the productof two Jordan arcs. This requires a lemma pertaining to the conformal dimensionof the union of two proper , uniformly perfect subsets of a metric space. Lemma 3.5.
Let ( X, d ) be a metric space with proper uniformly perfect subsets A and B such that A ∪ B = X. Then dim C X = max { dim C A, dim C B } . The proof of Lemma 3.5 relies heavily on Theorem 1 in [7] several times, so it isincluded below. A metric space X is called proper if closed and bounded subsetsare compact. It is called uniformly perfect if there is a constant C ≥ x ∈ X and for each r > B ( x, r ) \ B ( x, rC ) is nonempty whenever theset X \ B ( x, r ) is nonempty [8]. Theorem 3.6. (Haissinsky)
Let ( X, d X ) be a proper metric space containing atleast two points and ( Y, d Y ) a proper uniformly perfect space. Suppose there is aquasisymmetric embedding f : Y → X. Then there is a metric ˆ d on X such that (i) id : ( X, d X ) → ( X, ˆ d ) is quasisymmetric; (ii) id : ( X \ f ( Y ) , d X ) → ( X \ f ( Y ) , ˆ d ) is locally bi-Lipschitz; (iii) f : ( Y, d Y ) → ( X, ˆ d ) is bi-Lipschitz onto its image.Proof of Lemma 3.5. Put c = dim C ( A, d ) and c ′ = dim C ( B, d ) . Let ε > . By defi-nition of conformal dimension there is a quasisymmetric map g with dim H g ( A, d ) ≤ c + ε. Then f = g − : g ( A, d ) → X is a quasisymmetric embedding. Since A and B are proper, it follows that X is also proper. Indeed, let E ⊂ X be closed andbounded. Then E = ( E ∩ A ) ∪ ( E ∩ B ) and since E is closed in X, E ∩ A and E ∩ B are closed in A and B, respectively. Since E is bounded, E ∩ A and E ∩ B arealso bounded. Therefore E ∩ A and E ∩ B are compact since A and B are proper.Then E is a union of two compact sets and hence is compact, so X is proper. ByTheorem 3.6 there is a metric d on X such thatid : ( X, d ) → ( X, d ) is quasisymmetric(3.2) id : ( X \ A, d ) → ( X \ A, d ) is locally bi-Lipschitz(3.3) f : g ( A, d ) → ( A, d ) is bi-Lipschitz(3.4)Since the restriction of a quasisymmetric map is again quasisymmetric, (3.2)implies dim C ( B, d ) = dim C ( B, d ) = c ′ . Therefore there is a quasisymmetric map r such that dim H r ( B, d ) ≤ c ′ + ε. Then h = r − : r ( B, d ) → ( X, d ) is a qua-sisymmetric embedding with image ( B, d ) . Since r is a homeomorphism, r ( B, d )is proper and uniformly perfect. Another application of Theorem 3.6 gives a metric d on X such thatid : ( X, d ) → ( X, d ) is quasisymmetric(3.5) id : ( X \ B, d ) → ( X \ B, d ) is locally bi-Lipschitz(3.6) h : r ( B, d ) → ( B, d ) is bi-Lipschitz(3.7) CLAUDIO A. DIMARCO
Locally bi-Lipschitz maps preserve Hausdorff dimension, so (3.6) and (3.4) yielddim H ( X \ B, d ) = dim H ( X \ B, d ) ≤ dim H ( A, d )= dim H g ( A, d ) . (3.8)Also (3.7) shows that dim H ( B, d ) = dim H r ( B, d ) , so by (3.8) we havedim H ( X, d ) = max { dim H ( X \ B, d ) , dim H ( B, d ) }≤ max { dim H g ( A, d ) , dim H r ( B, d ) }≤ max { c + ε, c ′ + ε } = max { c, c ′ } + ε. (3.9)Finally (3.5), (3.9) and monotonicity give dim C ( X, d ) = max { c, c ′ } so that (3.2)implies dim C ( X, d ) = max { c, c ′ } . (cid:3) Theorem 3.7. If Γ and Λ are Jordan arcs then ≤ dim tC (Γ × Λ) ≤ max { dim C Γ , dim C Λ } + 1 . Proof.
The first inequality is due to the fact that 2 = dim t (Γ × Λ) ≤ dim tC (Γ × Λ) . For the second inequality, let U = { γ (( c, d )) × λ (( a, b )) : a, b, c, d ∈ [0 , } where γ, λ are the homeomorphisms parametrizing Γ , Λ respectively. Let A × B ∈ U where A = γ ( c, d ) , B = λ ( c, d ) . Since A and B are open, the interior of A × B isagain A × B, so we may write ∂ ( A × B ) as ∂ ( A × B ) = A × B \ ( A × B )=( ∂A × B ) ∪ ( A × ∂B ) ∪ ( ∂A × ∂B )(3.10) = (cid:0) ( A × { λ ( c ) } ) ∪ ( { γ ( b ) } × B ) (cid:1) ∪ (cid:0) A × { λ ( d ) } ) ∪ ( { γ ( a ) } × B ) (cid:1) . Let S = ( A × { λ ( c ) } ) ∪ ( { γ ( b ) } × B ) and T = ( A × { λ ( d ) } ) ∪ ( { γ ( a ) } × B ) . Sinceeach of S and T is a union of two connected sets, Lemma 3.5 givesdim C S = max { dim C ( A × { λ ( c ) } ) , dim C ( { γ ( b ) } × B ) } = max { dim C A, dim C B } dim C T = max { dim C ( A × { λ ( d ) } ) , dim C ( { γ ( a ) } × B ) } = max { dim C A, dim C B } (3.11)Since S and T are connected, Lemma 3.5, (3.10) and (3.11) yielddim C ∂ ( A × B ) = max { dim C S, dim C T } = max { dim C A, dim C B }≤ max { dim C Γ , dim C Λ } . The conclusion now follows from the definition of the topological conformal dimen-sion. (cid:3)
We conclude this section with a simple observation that classifies product spacesconsisting of two factors; one factor is the unit interval, the other has conformaldimension zero.
Fact 3.8. If dim C X = 0 then dim tC ( X × [0 , . OPOLOGICAL CONFORMAL DIMENSION 7
Proof.
Since X × [0 ,
1] contains a line segment, dim tC ( X × [0 , ≥ . On the otherhand, X has a basis U of clopen sets since dim t X ≤ dim C X = 0 . Then for every U ∈ U and 0 ≤ a ≤ b ≤ C ∂ ( U × ( a, b )) = dim C ( U × ∂ ( a, b ))= dim C ( U × { a, b } )= dim C U ≤ dim C X = 0 . (cid:3) Range of Dimension
Taking products of Cantor sets, one sees that the Hausdorff dimension attainsall values in [0 , ∞ ]. Topological Hausdorff dimension attains all values in [1 , ∞ ]by Theorem 4.24 in [1]. It was shown in [15] that conformal dimension attains allvalues in [1 , ∞ ] , and later in [9] that it does not attain any value in (0 , {− , , } ∪ [2 , ∞ ] . Fact 4.1.
The topological conformal dimension cannot take any value in (1 , . Proof. If X = ∅ then dim tC X = − . Let X be a nonempty metric space withdim tC X = c and suppose 1 < c < . By definition of the topological conformaldimension there is a number d and a basis U for X such that 1 < c < d < C ∂U ≤ d − U ∈ U . Then [9] implies dim C ∂U = 0 for all U ∈ U so thatdim tC X ≤ , a contradiction. (cid:3) For tC-dimension, all integer values in [2 , ∞ ] are realized by Euclidean spaces.It is more involved to obtain examples that attain non-integer values. Theorem 4.5gives one method for accomplishing this task. In particular, it shows that there arecompact subsets of [0 , with fractional topological conformal dimension. Definition 4.2.
A metric space X is Ahlfors d -regular if it admits a Borel regularmeasure µ such that(4.1) 1 K r d ≤ µ (cid:0) B ( x, r ) (cid:1) ≤ Kr d for some constant K ≥ B ( x, r ) of radius 0 < r < X. The primary tool we use to prove Theorem 4.5 is Proposition 4.1.3 in [11], so itis included here for sake of completeness.
Proposition 4.3. (Pansu)
Let ( Z, d, µ ) be a compact, doubling metric measurespace, and let < p < ∞ and p ′ = pp − . Suppose that there exists a family E ofconnected sets in Z and a probability measure ν on E with the following properties: (i) there exists c > so that diam E ≥ c for all E ∈ E , and (ii) there exists C and r > so that (4.2) ν { E ∈ E : E ∩ B ( x, r ) = ∅ } ≤ Cµ ( B ( x, r )) / p ′ for all balls x ∈ Z and for all r ≤ r . Then dim C Z ≥ p. Lemma 4.4. If A ⊂ C is open, connected, and bounded, then there is a connectedset E ⊂ ∂A such that diam E = diam A. CLAUDIO A. DIMARCO
Proof.
Let { S α | α ∈ J } be the set of connected components of A c and let β ∈ J besuch that S β is unbounded. We will show that E = ∂S β satisfies the conclusion ofthe lemma.The Phragm´en-Brouwer theorem (Theorem VI.2.1 in [17]) shows that ∂S β isconnected. Note that ∂S β ⊂ ∂A. Indeed, if w ∈ ∂S β \ ∂A then either w ∈ A or w ∈ A c . If w ∈ A c then w ∈ S γ for some γ = β. Since S β is connected and ∂S β ⊂ S β , we have S β ⊂ S γ , a contradiction. If w ∈ A then S β ⊂ S β ⊂ A , acontradiction.For z ∈ C let Arg( z ) be the principal argument of z. Without loss of generality,let 0 ∈ A and choose an argument − π < φ ≤ π and a point z ∈ C with | z | > diam A and Arg( z ) = φ. Connect 0 to z by a line segment; call it L. The function w
7→ | w | is continuous on the compact set L ∩ A, so it attains a maximum on L ∩ A, say at z φ . Put F = { z φ : − π < φ ≤ π } . We claim that diam F ≥ diam A . To see this,choose a, b ∈ ∂A such that | a − b | = diam A. We will show that a, b ∈ F. Since0 ∈ A we have | a − b | > | a | and | a − b | > | b | . Note that the function f ( t ) = | ta − b | is strictly convex on [0 , ∞ ) and f (0) = | b | < | a − b | = f (1) . Then f is strictlyincreasing on [1 , ∞ ) so that ta / ∈ A for all t > . It follows that a ∈ F and similarly b ∈ F. Clearly F ⊂ ( ∂A ) ∩ ( ∂S β ). Therefore diam ∂S β ≥ diam F ≥ diam A. Since ∂S β ⊂ A we see that diam ∂S β ≤ diam A = diam A, so diam E = diam A with E = ∂S β . (cid:3) The following theorem is the main result of this section. Recall that f : X → Y is L − bi-Lipschitz if both f and f − are L − Lipschitz. We write dim A X for theAssouad dimension of X, see [8],[11]. Theorem 4.5.
Let ( X, d, µ ) be a compact, doubling metric measure space with dim A X = D < ∞ . Suppose that there exists a family Q of surfaces with the fol-lowing properties: (i) There is a family { f α : α ∈ J } of L − bi-Lipschitz maps such that Q α = f α ([0 , ) for each Q α ∈ Q . (ii) There exists a measure ν on Q , a constant C , and r > so that (4.3) 0 < ν { Q α ∈ Q : Q α ∩ B ( x, r ) = ∅} ≤ C r d for all x ∈ X and for all r ≤ r . (iii) The union of edges S α ∈ J f α (cid:0) ∂ ([0 , ) (cid:1) is not dense in X. Then dim tC X ≥ DD − d . Corollary 4.6.
Let (
C, ν C ) be a compact Ahlfors d − regular metric measure space.If X = C × [0 , then 2 + d / ≤ dim tC X ≤ d. Proof of Theorem 4.5.
Let U be a basis for X . For each α , call f α ( ∂ C ([0 , )) the edge of Q α . In this proof, ∂G means the boundary of G in Q α . Boundaries in otherspaces are indicated by subscripts. Assumption (iii) shows that there is an open set U ′ ⊂ X that does not meet the edge of Q α for any α. Choose a nonempty U ∈ U such that U ⊂ U ′ . We will show that dim C ∂ X U ≥ DD − d using Proposition 4.3.Choose a compact set K ⊂ U with nonempty interior, and for each α with K ∩ Q α = ∅ choose z α ∈ K ∩ Q α . Put J K = { α ∈ J : K ∩ Q α = ∅ } . Let B α be theconnected component of U ∩ Q α containing z α . Consider A α = f − α ( B α ). Note that A α is open, connected, and does not intersect ∂ C ([0 , ) , so Corollary 4.4 gives a OPOLOGICAL CONFORMAL DIMENSION 9 connected set E α ⊂ ∂ C A α with diam E α = diam A α . Since f α is a homeomorphism, f α ( E α ) is connected.To satisfy the first part of condition (i) of Proposition 4.3, define E = { f α ( E α ) : α ∈ J K } . We need to show that f α ( E α ) ⊂ ∂ X U. Since B α is a component of U ∩ Q α we have ∂B α ⊂ ∂ ( U ∩ Q α ) . Then f α ( E α ) ⊂ f α ( ∂ C A α ) = ∂f α ( A α ) = ∂B α ⊂ ∂ ( U ∩ Q α ) ⊂ ∂ X U. (4.4)To satisfy (4.2) we must show that there is c > f α ( E α ) ≥ c forall α ∈ J K . We have diam f α ( E α ) ≥ L diam E α = 1 L diam A α ≥ L diam B α (4.5)for all α ∈ J K , so it suffices to show that there is c ′ > B α ) ≥ c ′ for all α ∈ J K . Since B α is connected and meets both K and ∂ X U, we see thatdiam B α ≥ dist( K, ∂ X U ) = c ′ >
0. Then (4.5) yieldsdiam f α ( E α ) ≥ L diam( B α ) ≥ c ′ L (4.6)for all α ∈ J K , so condition (i) of Proposition 4.3 is satisfied with c = c ′ / L .To establish (4.2) we let s > D and show that (4.2) holds with p = ss − d and p ′ = sd . Put r = diam ∂ X U. We need to use an existence theorem for doubling measuresproved by Vol’berg and Konyagin, and extended by Luukainen and Saksman [10].Since s > D = dim A X ≥ dim A ∂ X U, the Corollary to Theorem 1 in [10] togetherwith Theorem 13.5 in [8] show that ∂ X U carries a ( t , s ) − homogeneous measurefor some t . So there is a doubling measure µ on ∂ X U satisfying(4.7) µ ( B ( x, λr )) ≤ t λ s µ ( B ( x, r ))for all x ∈ ∂ X U, r > , and λ ≥ . Fix such a ball B ( x, r ) . With λ = diam ∂ X Ur , inequality (4.7) gives(4.8) µ ( B ( x, r )) ≥ µ ( ∂ X U ) t (diam ∂ X U ) s r s = M r s so that(4.9) µ ( B ( x, r )) / p ′ ≥ M / p ′ r d . To finish establishing (4.2) we need to define a measure ν on E . Let F α = f α ( E α ) , and for F ⊂ E define(4.10) ν ( F ) = K ν { Q α : F α ∈ F } , where K is such that ν ( E ) = 1 . Note that F α ∩ B ( x, r ) = ∅ implies Q α ∩ B ( x, r ) = ∅ . Coupled with (4.10) and (4.3), this yields ν { F α : F α ∩ B ( x, r ) = ∅ } ≤ ν { F α : Q α ∩ B ( x, r ) = ∅ } = K ν { Q α : Q α ∩ B ( x, r ) = ∅ }≤ K C r d . (4.11) Now (4.11) and (4.9) give ν { F α : F α ∩ B ( x, r ) = ∅ } ≤ K C r d ≤ K C M / p ′ µ ( B ( x, r )) / p ′ = Cµ ( B ( x, r )) / p ′ . (4.12)Since the choice of B ( x, r ) was arbitrary, (4.12) holds for all such balls, so wehave (4.2). Therefore dim C ∂ X U ≥ p = ss − d for all s > D. Letting s → D givesdim C ∂ X U ≥ DD − d and hence dim tC X ≥ DD − d . (cid:3) Proof of Corollary 4.6.
The family of surfaces Q = {{ x } × [0 , : x ∈ C } satifiesconditions (i) and (iii) of Theorem 4.5. It remains to show that condition (ii) holds.Let d = dim H C and let π : X → C be the projection of X onto C. For F ⊂ Q define(4.13) ν ( F ) = ν C [ Q ∈ F π ( Q ) . By Ahlfors regularity, there is a constant N such that ν C ( π ( B ( x, r ))) ≤ N r d forall z ∈ X and 0 < r ≤ diam X. It now follows from (4.13) that ν { Q ∈ Q : Q ∩ B ( z, r ) = ∅ } = ν C [ E ∩ B ( z,r ) = ∅ π ( Q ) ≤ ν C ( π ( B ( z, r ))) ≤ N r d , (4.14)which is condition (ii). Since X is Ahlfors D − regular with D = 2 + d we have(4.15) dim A ( C × [0 , ) = dim H ( C × [0 , ) = 2 + d = D. Equation (4.15) and Theorem 4.5 yield2 + d / ≤ dim tC X ≤ d. (cid:3) Corollary 4.7.
Topological conformal dimension is not additive under products.Moreover, this quantity can be fractional.
Proof.
Letting C be the middle-thirds Cantor set, the hypotheses of Corollary 4.6are satisfied with d = ln(2)ln(3) . Since C is totally disconnected, dim tC C = 0 , so(4.16) dim tC C + dim tC ([0 , ) = 2 < d / ≤ dim tC ( C × [0 , ) . Also dim tC ( C × [0 , ) ≤ d < , so (4.16) shows that dim tC ( C × [0 , ) is notan integer. (cid:3) Theorem 4.5 provides hope for an affirmative answer to the following conjecture.
Conjecture.
For every d ∈ [2 , ∞ ] there is a metric space X with dim tC X = d. OPOLOGICAL CONFORMAL DIMENSION 11 Comparison with Other Dimensions
In this section, examples are given to demonstrate that topological conformaldimension is different from topological dimension, conformal dimension, topologicalHausdorff dimension, and Hausdorff dimension. It is also shown that conformaldimension and tH-dimension are not comparable. We compute the tC-dimensions ofsome classical fractals. In particular, this type of dimension classifies the Sierpinskicarpet as dimension 1, unlike the Hausdorff, topological Hausdorff, and conformaldimensions.Due to the additivity of Hausdorff dimension under products and to the factthat topological dimension only assumes integer values, the following fact is readilyseen by considering Corollary 4.7.
Fact 5.1.
Topological conformal dimension is different from topological dimensionand Hausdorff dimension.
Write SG for the Sierpinski gasket and SC for the Sierpinski carpet. Laaksoproved that dim C SG = 1 [16], while the value dim C SC remains unknown [11]. Example 5.2. dim tC SG = dim tC SC = 1 . Proof.
Since SG contains a line segment, dim tC SG ≥ , and we have dim tC SG ≤ dim H SG = ln(3)ln(2) < . Since dim tC SG / ∈ (1 , , it follows that dim tC SG = 1 . Similarly, since SC contains a line segment, dim tC SC ≥
1. On the other hand,dim tC SC ≤ dim H SC = ln(8)ln(3) < tC SC = 1 . (cid:3) In light of Example 5.2, we observe the following fact.
Fact 5.3.
Topological conformal dimension is different from conformal dimensionand topological Hausdorff dimension.Proof. If C is the middle-thirds Cantor set, monotonicity of conformal dimensionimplies dim C SC ≥ dim C ( C × [0 , . By Proposition 4.1.11 in [11], dim C ( C × [0 , ln(2)ln(3) , so dim C SC ≥ ln(2)ln(3) . By Theorem 5.4 in [1], dim tH SC =1 + ln(2)ln(3) . (cid:3) Fact 5.4.
Conformal dimension and topological Hausdorff dimension are not com-parable.Proof.
We produce spaces X and Y such that dim C X < dim tH X and dim C Y > dim tH Y. Let 0 < α < , n ) α be the metric space that results fromsnowflaking [0 , n , as in Fact 6.1 below. By Fact 6.1,dim tH (([0 , n ) α ) = n − α + 1 > n = dim C (([0 , n ) α ) . On the other hand, if K ⊂ R is a uniformly perfect Cantor set with dim H K = 1 , then dim C K = 1 > dim tH K = 0 by Corollary 3.3 in [6]. (cid:3) Proposition 5.5. If K is homeomorphic to [0 , , then dim tC K = 1 . Remark . In particular, the topological conformal dimension of the von Kochsnowflake is 1 . Proof of Proposition 5.5.
Inequality (2.2) implies dim tC K ≥ . On the other hand,if K is homeomorphic to [0 ,
1] then the usual basis of [0 ,
1] gives a basis U for K such that | ∂U | = 2 for all U ∈ U . Then dim C ∂U = 0 for all U ∈ U and hencedim tC K ≤ . (cid:3) There are some well known fractals for which the tC-dimension remains un-known, such as Rickman’s rug (defined in [11]) and the Heisenberg group. We can,however, compute both the tC and tH-dimensions of the Menger sponge (defined in[11], section 3.5). The following theorem is useful for computing the tH-dimensionof the sponge.
Theorem 5.7. If X is nonempty, locally compact, Ahlfors d -regular, and totallydisconnected, and if Y is any separable metric space with dim tH Y ≥ , then dim tH ( X × Y ) = dim H X + dim tH Y. Proof.
If dim tH Y = ∞ then the statement is trivial, so assume dim tH Y < ∞ . Wewill first show(5.1) dim tH ( X × Y ) ≤ dim H X + dim tH Y. Since X is locally compact, dim t X = 0 by Theorem 29.7 in [18], so X has abasis U consisting of clopen sets. Let ε > V of Y suchthat dim H ∂V ≤ dim tH V − ε for all V ∈ V . For U × V ∈ U × V we have ∂ ( U × V ) = U × ∂V. Since X is Ahlfors regular, Theorem 5.7 in [13] implies thatits upper Minkowski dimension dim M X is equal to dim H X . Corollary 8.10 in [13]gives dim H ( U × ∂V ) ≤ dim H ( X × ∂V ) = dim H X + dim H ∂V ≤ dim H X + dim tH Y − ε. (5.2)Since ε > S ⊂ ( X × Y ) where ( X × Y ) \ S is totally disconnected,(5.3) dim H S ≥ dim H X + dim tH Y − . Once (5.3) is established for all such S , Theorem 3.6 in [1] will give the result. Forexample, if X is the middle-thirds Cantor set and if Y = [0 , , one such subset is S = X × ( Y \ ( C × C )) . To this end, suppose there is S such that (5.3) fails. Choose β such that(5.4) dim H ( S ) < β < dim H X + dim tH Y − X is proper since it is complete and Ahlfors regular. Put dim H X = d. By virtue of the coarea inequality (Theorem 2.10.25 in [4])(5.5) Z ∗ X H β − d ( S ∩ ( { x } × Y )) d H d ( x ) ≤ C H β ( S ) . Then H d ( S ) = 0 since β > dim H S, so there exists x ∈ X such that(5.6) dim H ( S ∩ ( { x } × Y )) ≤ β − dim H X. On the other hand, since Y is separable,(5.7) dim H ( S ∩ ( { x } × Y )) ≥ dim tH Y − OPOLOGICAL CONFORMAL DIMENSION 13 by Theorem 3.6 in [1]. Inequalities (5.6) and (5.7) yielddim tH Y − ≤ β − dim H X, which contradicts (5.4). (cid:3) Let M denote the Menger sponge. We compute dim tH M followed by dim tC M. Example 5.8. dim tH M = 1 + ln(4)ln(3) . Proof.
Consider the sets of intervals W ′ = { [0 , b ) : 0 < b < b is a dyadic rational } ,W ′′ = { ( a,
1] : 0 < a < a is a dyadic rational } ,W ′′′ = { ( a, b ) : 0 < a < b < a and b are dyadic rationals } . Put W = W ′ ∪ W ′′ ∪ W ′′′ and let U = M ∩ ( W × W × W ) . Note that U is a basisof M such that for all U ∈ U , ∂U is a union of at most six sides of a cube in M. Each side of ∂U is a finite union of sets geometrically similar to C × C, where C isthe middle-thirds Cantor set. Then dim H ∂U = ln(4)ln(3) so that dim tH M ≤ ln(4)ln(3) . On the other hand, M contains C × SC , so dim tH M ≥ dim tH ( C × SC ) bymonotonicity of the topological Hausdorff dimension. Note that Theorem 5.4 in [1]gives dim tH SC = 1 + ln(2)ln(3) . Since C × SC satisfies the hypotheses of Theorem 5.7,dim tH ( C × SC ) = dim H C + dim tH SC = 1 + ln(4)ln(3) . Thus dim tH M ≥ ln(4)ln(3) . (cid:3) Example 5.9. dim tC M = 1 . Proof.
Let U be as in Example 5.8. A similar argument shows that dim tC M ≤ C ( C × C ) . Since C × C is uniformly disconnected, dim C ( C × C ) = 0 ([11],section 1.3], so dim tC M ≤ . By monotonicity dim tC M ≥ . (cid:3) Quasisymmetric Distortion of Topological Hausdorff Dimension
Topological Hausdorff dimension is not quasisymmetrically invariant. For exam-ple, it increases under the snowflake transformation:
Fact 6.1.
Let ( X, d ) be a metric space and let < α < . If X α = ( X, d α ) isthe snowflaked metric space defined by the transformation d ( x, y ) d ( x, y ) α , then dim tH ( X α ) = α (dim tH X −
1) + 1 . Proof.
For any metric space Z, snowflaking has the following effect on Hasdorffdimension: dim H ( Z α ) = α dim H Z ([11], Corollary 1.4.18). Applying this fact toopen subsets of X α givesdim tH X α = inf { c : X α has basis U α , dim H ( ∂U α ) ≤ c − ∀ U α ∈ U α } = inf { c : X has basis U , α dim H ∂U ≤ c − ∀ U ∈ U} = inf { c : X has basis U , dim H ∂U ≤ ( αc − α + 1) − ∀ U ∈ U} (6.1)The desired equality now follows from the definition of tH-dimension. (cid:3) More interestingly, there exist spaces which are minimal for conformal dimension,yet their topological Hausdorff dimension can be lowered by quasisymmetric maps.This is the content of Theorem 6.2. Note that if C is a compact Ahlfors regularmetric space, then C × [0 ,
1] is minimal for conformal dimension [15], i.e. dim C ( C × [0 , H ( C × [0 , . Theorem 6.2.
Let C be an Ahlfors d − regular metric measure space with
Let D = { a i : a, i ∈ N , a ≤ i } and write D = { y , y , . . . } . For each n let µ n be the restriction of H d to C × { y n } . We define a measure ν on X by ν ( E ) = ∞ X n =1 µ n ( E )2 n Let α > . By Theorem 4.1 in [9] there is a quasisymmetric map f : X → Y such that for all z = ( x, y ) ∈ X and for all r > f ( B ( z, r )) ⊂ B ( f ( z ) , R ) , where R = min { C ( α ) r / α ν ( B ( z, r )) − / α , r } . (6.2)Since C is Ahlfors d − regular, there is a constant K such that for every x ∈ C andfor 0 < r ≤ diam C ν ( B (( x, y j ) , r )) = ∞ X n =1 µ n ( B (( x, y j ) , r )2 n ≥ µ j ( B (( x, y j ) , r )2 j ≥ K − j r d = K j r d . (6.3)Let x , x ∈ C and put z = ( x , y j ) , z = ( x , y j ) , r = | x − x | = d X ( z , z ) = | z − z | . By (6.3) R ≤ C ( α ) r / α ν ( B ( z , r )) / α ≤ C ( α ) r / α K / α j ( r d ) / α = C ( j, α ) r − dα = C ( j, α ) | z − z | − dα (6.4)where C ( j, α ) is a constant that depends only on j and α. Continuity of f and (6.2)imply f ( B ( z , r )) ⊂ B ( f ( z ) , R ) . Then | f ( z ) − f ( z ) | ≤ R , so by (6.4)(6.5) | f ( z ) − f ( z ) | ≤ R ≤ C ( j, α ) | z − z | − dα . OPOLOGICAL CONFORMAL DIMENSION 15
By (6.5) we see that for all j dim H f ( C × { y j } ) ≤ α − d dim H ( C × { y j } ) = αd − d . Since α was arbitrary, for every ε > f ε : X → Y such that dim H f ε ( C × { y j } ) ≤ ε for all j. Notice that since f ε is a homeomorphism, f ε ( X ) \ f ε ( C × D ) is totally disconnected. Also dim H f ε ( C × D ) = sup j dim H f ε ( C ×{ y j } ) ≤ ε so that Theorem 3.7 in [1] gives dim tH f ε ( X ) ≤ ε. To see that dim tC X = 1 , first note that dim tC X ≥ ε > , dim tC X = dim tC f ( X ) ≤ dim tH f ( X ) ≤ ε. (cid:3) As stated in the introduction, topological conformal dimension gives a lowerbound on the topological Hausdorff dimensions of quasisymmetric images of a givenmetric space.(6.6) dim tC X ≤ inf { dim tH f ( X ) : f is quasisymmetric } . Question 6.4.
Does equality hold in (6.6) for every metric space?
Acknowledgement.
This paper is based on a part of a PhD thesis written by theauthor under the supervision of Leonid Kovalev. The author thanks the anonymousreferee for many useful suggestions in revising this paper.
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