AA DIMENSION DROP PHENOMENON OF FRACTAL CUBES
LIANG-YI HUANG AND HUI RAO ∗ Abstract.
Let E be a metric space. We introduce a notion of connectednessindex of E , which is the Hausdorff dimension of the union of non-trivial connectedcomponents of E . We show that the connectedness index of a fractal cube E is strictly less than the Hausdorff dimension of E provided that E possesses atrivial connected component. Hence the connectedness index is a new Lipschitzinvariant. Moreover, we investigate the relation between the connectedness indexand topological Hausdorff dimension. Introduction An iterated function system (IFS) is a family of contractions { ϕ j } Nj =1 on R d ,and the attractor of the IFS is the unique nonempty compact set K satisfying K = (cid:83) Nj =1 ϕ j ( K ), and it is called a self-similar set [5]. Let n ≥ D = { d , · · · , d N } ⊂ { , , . . . , n − } d , which we call a digit set. Denote by D := N the cardinality of D . Then n and D determine an IFS { ϕ j ( z ) = n ( z + d j ) } Nj =1 , whoseattractor E = E ( n, D ) satisfies the set equation(1.1) E = 1 n ( E + D ) . We call E a fractal cube [12], especially, when d = 2, we call E a fractal square [6].There are some works on topological and metric properties of fractal cubes. Why-burn [11] studied the homeomorphism classification, Bonk and Merenkov [2] studiedthe quasi-symmetric classification. Lau, Luo and Rao [6] studied when a fractalsquare is totally disconnected. Xi and Xiong [12] gave a complete classification ofLipschitz equivalence of fractal cubes which are totally disconnected. Recently, thestudies of [10, 13] focus on the the Lipschitz equivalence of fractal squares which arenot totally disconnected.Topological Hausdorff dimension is a new fractal dimension introduced by Buc-zolich and Elekes [1]. It is shown in [1] that for any set K we always have dim tH K ≤ dim H K , where dim tH and dim H denote the topological Hausdorff dimension andHausdorff dimension respectively. Ma and Zhang [8] calculated topological Haus-dorff dimensions of a class of fractal squares. Date : October 12, 2020.The work is supported by NSFS Nos. 11971195 and 11601172.
Key words and phrases: fractal cube, connected component, topological Hausdorff dimension. * The correspondence author. a r X i v : . [ m a t h . GN ] O c t et K be a metric space. A point x ∈ K is called a trivial point of K if { x } is a connected component of K . Let Λ( K ) be the collection of trivial points in K .Denote(1.2) I c ( K ) := dim H K \ Λ( K ) , and we call it the connectedness index of K . It is obvious that I c ( K ) ≤ dim H K .Clearly, the connectedness index is a Lipschitz invariant. The main results of thepresent paper are as follows. Theorem 1.1.
Let E = E ( n, D ) be a d -dimensional fractal cube. If E has a trivialpoint, then I c ( E ) < dim H E . However, Theorem 1.1 is not valid for general self-similar sets, even if the self-similar sets satisfy the open set condition.
Example 1.1.
Let Q = { } ∪ (cid:18) ∞ (cid:83) k =0 [ k +1 , k ] (cid:19) . Observe that Q = Q ∪ [ ,
1] and Q ∪ Q = [0 , Q is a self-similar set satisfying the equation Q = Q ∪ (cid:18) Q (cid:19) ∪ (cid:18) Q (cid:19) . The set Q has only one trivial point, that is 0. Therefore, I c ( Q ) = dim H Q = 1.Figure 1 illustrates Q (cid:48) , a two dimensional generalization of Q . Similarly, Q (cid:48) is aself-similar set, and the unique trivial point of Q (cid:48) is . Figure 1.
The self-similar set Q (cid:48) .Using Theorem 3.7 of [1] we show the following. Theorem 1.2.
For a non-empty σ -compact metric space K , we have dim tH K ≤I c ( K ) . Zhang [14] asked when dim tH E = dim H E , where E is a fractal square. Accordingto [14], a digit set D is called a Latin digit set , if every row and every column has thesame number of elements (see Figure 2). For a fractal square E = E ( n, D ), Zhangshowed that if dim tH E = dim H E , then either E = [0 , × C , or E = C × [0 ,
1] forsome C ⊂ [0 , D is a Latin digit set.As a corollary of Theorem 1.1 and Theorem 1.2, we obtain a new necessary con-dition for dim H E = dim tH E . a) The digit set of L . (b) The Latin fractal square L . Figure 2.
It is shown in [14] that log 12 / log 6 = dim tH L < dim H L = log 24 / log 6. While by Theorem 1.1 and Theorem 1.2,we directly have dim tH L < dim H L . Corollary 1.1.
Let E be a d -dimensional fractal cube. If dim H E = dim tH E , then E has no trivial point. Remark 1.1.
Another application of Theorem 1.1 is on the gap sequences of fractalcubes, a Lipschitz equivalent invariant introduced by Rao, Ruan and Yang [9]. Fora fractal cube K , let { g m ( K ) } m ≥ be the gap sequence. Using Theorem 3.1 of thepresent paper, it is proved in [4] that if K has trivial point, then { g m ( K ) } m ≥ isequivalent to { m − /γ } m ≥ , where γ = dim H K .Finally, we calculate the connectedness indexes of two fractal squares in Figure 3,and illustrate the application to Lipschitz classification. Example 1.2.
Let K and K (cid:48) be two fractal squares indicated by Figure 3. It isseen that dim H K = dim H K (cid:48) = log 14log 5 . By Theorem 1.3 of [8], one can obtain thatdim tH K = dim tH K (cid:48) = 1 + log 2log 5 . We will show in section 5 that I c ( K ) = log(8 + √ / I c ( K (cid:48) ) = log 13log 5 . So K and K (cid:48) are not Lipschitz equivalent. (a) The digit set of K . (b) The digit set of K (cid:48) . Figure 3. his article is organized as follows. In section 2, we recall some basic facts of r -face of the polytope [0 , d . In section 3, we prove Theorem 1.1. In section 4, weprove Theorem 1.2. In section 5, we give the details of Example 1.2.2. Preliminaries on r -faces of [0 , d We recall some notions about convex polytopes, see [15]. Let C ⊂ R d be a convexpolytope, let F be a convex subset of C . The affine hull of F , denoted by aff( F ),is the smallest affine subspace containing F . We say F is a face of C , if any closedline segment in C with a relative interior in F has both endpoints in F .The dimension of an affine subspace is defined to be the dimension of the corre-sponding linear vector space. The dimension of a face F , denoted by dim F , is thedimension of its affine hull. Moreover, F is called an r -face of C , if F is a face of C with dimension r . We take it by convention that C is a d -face of itself if dim C = d .For z ∈ C , a face F of C is called the containing face of z if z is a relative interiorpoint of F .Let e , . . . , e d be the canonical basis of R d . The following facts about the r -facesof [0 , d are obvious, see Chapter 2 of [15]. Lemma 2.1. (i)
Let A ∪ B = { , . . . , d } be a partition with A = r . Then the set (2.1) F = (cid:40)(cid:88) j ∈ A c j e j ; c j ∈ [0 , (cid:41) + b is an r -face of [0 , d if and only if b ∈ T , where (2.2) T := (cid:40)(cid:88) j ∈ B ε j e j ; ε j ∈ { , } (cid:41) ;(ii) For any r -face F of [0 , d , there exists a partition A ∪ B = { , . . . , d } with A = r such that F can be written as (2.1) . We will call F = { (cid:80) j ∈ A c j e j ; c j ∈ [0 , } a basic r -face related to the partition A ∪ B . We give a partition B = B ∪ B according to b by setting(2.3) B = { j ∈ B ; the j -th coordinate of b is 0 } ,B = { j ∈ B ; the j -th coordinate of b is 1 } . Let x = (cid:80) j ∈ A α j e j + (cid:80) i ∈ B β i e i ∈ [0 , d , we define two projection maps as follows:(2.4) π A ( x ) = (cid:88) j ∈ A α j e j , π B ( x ) = (cid:88) i ∈ B β i e i . If F is an r -face of [0 , d , we denote by ˚ F the relative interior of F . Lemma 2.2 (Chapter 2 of [15]) . Let C ⊂ R d be a polytope. (i) If G and F are faces of C and F ⊂ G , then F is a face of G . (ii) If G is a face of C , then any face of G is also a face of C . he following lemma will be needed in section 3. Lemma 2.3.
Let F = F + b be an r -face of [0 , d given by (2.1) . Let u ∈ Z d . Then ˚ F ∩ ( u + [0 , d ) (cid:54) = ∅ if and only if u = b − b (cid:48) for some b (cid:48) ∈ T , where T is defined in (2.2) .Proof. “ ⇐ ”: Suppose b (cid:48) ∈ T , then F − ( b − b (cid:48) ) = F + b (cid:48) , and it is an r -face of [0 , d by Lemma 2.1 (i). Applying a translation b − b (cid:48) we see that F ⊂ ( b − b (cid:48) ) + [0 , d ,which completes the proof of the sufficiency.“ ⇒ ”: Suppose ˚ F ∩ ( u + [0 , d ) (cid:54) = ∅ . Let z be a point in the intersection and let F (cid:48) = [0 , d ∩ ( u + [0 , d ). Then F (cid:48) is a face of both [0 , d and u + [0 , d . So we have F ⊂ F (cid:48) since F (cid:48) contains z , a relative interior point of F . Hence F is an r -face of F (cid:48) by Lemma 2.2 (i). It follows that F − u is an r -face of F (cid:48) − u .Notice that F (cid:48) is a face of u + [0 , d , then F (cid:48) − u is a face of [0 , d . By Lemma2.2 (ii), F − u = F + ( b − u ) is an r -face of [0 , d . By Lemma 2.1 (i) we have b − u ∈ T . (cid:3) Trivial points of fractal cubes
Let Σ = { , , . . . , N } . Denote by Σ ∞ and Σ k the sets of infinite words and wordsof length k over Σ respectively. Let Σ ∗ = (cid:83) k ≥ Σ k be the set of all finite words. Forany σ = σ . . . σ k ∈ Σ k , let ϕ σ = ϕ σ ◦ · · · ◦ ϕ σ k .In this section, we always assume that E = E ( n, D ) is a d -dimensional fractalcube defined in (1.1) with IFS { ϕ j } j ∈ Σ . In the following, we always assume thatFor a point z ∈ E , we say F is the containing face of z means that F is a face ofthe polytope [0 , d and it is the containing face of z . Lemma 3.1.
Let z ∈ E and σ ∈ Σ k for some k > . Let F be the containingface of z , let F (cid:48) be the containing face of ϕ σ ( z ) . Then either ϕ σ ( z ) ∈ F or dim F (cid:48) ≥ dim F + 1 .Proof. Let A ∪ B be the partition in Lemma 2.1 (i) which defines F . By the definitionof containing face, we have z ∈ ˚ F . Suppose that ϕ σ ( z ) / ∈ F .Take any point x ∈ F \ { z } and let I be the closed line segment in F suchthat x is an endpoint of I and z is a relative interior point of I . It is clear that ϕ σ ( I ) ⊂ ϕ σ ([0 , d ) ⊂ [0 , d . Since ϕ σ ( z ) ∈ F (cid:48) , we have ϕ σ ( I ) ⊂ F (cid:48) . By thearbitrary of x we deduce that ϕ σ ( F ) ⊂ F (cid:48) , hence(3.1) dim F (cid:48) ≥ dim ϕ σ ( F ) = dim F. We claim that F (cid:48) is not an r -face of [0 , d . This claim together with (3.1) implydim F (cid:48) ≥ dim F + 1.Suppose on the contrary that F (cid:48) is an r -face of [0 , d . Then there exists a partition A (cid:48) ∪ B (cid:48) = { , . . . , d } such that F (cid:48) = F (cid:48) + b (cid:48) , where F (cid:48) = { (cid:80) j ∈ A (cid:48) c j e j ; c j ∈ [0 , } and b (cid:48) ∈ { (cid:80) j ∈ B (cid:48) ε j e j ; ε j ∈ { , }} . Since F n k + bn k + ϕ σ ( ) = ϕ σ ( F ) ⊂ F (cid:48) = F (cid:48) + b (cid:48) , e have F (cid:48) = F . Hence A (cid:48) = A and B (cid:48) = B . It follows that(3.2) b (cid:48) = π B ( ϕ σ ( z )) = bn k + π B ( ϕ σ ( )) ∈ T. Notice that(3.3) π B ( ϕ ω ( )) ∈ (cid:40)(cid:88) j ∈ B c j e j ; c j ∈ [0 , n k − n k ] (cid:41) for any ω ∈ Σ k , which together with (3.2) imply that π B ( ϕ σ ( )) = ( n k − n k b . Hence b (cid:48) = b and it follows that ϕ σ ( z ) ∈ F , a contradiction. The claim is confirmed andthe lemma is proven. (cid:3) For each σ = σ . . . σ k ∈ Σ k , we call ϕ σ ([0 , d ) ⊂ E k a k -th cell of E k . Denote(3.4) Σ σ = { ω ∈ Σ k ; π A ( ϕ ω ( )) = π A ( ϕ σ ( )) } and set(3.5) H σ = (cid:91) ω ∈ Σ σ ϕ ω ([0 , d ) . Indeed, H σ is the union of all k -th cells having the same projection with ϕ σ ([0 , d )under π A . From now on, we always assume that(3.6) z is a trivial point of E and F is the containing face of z . Lemma 3.2.
Let k > , fix σ ∈ Σ k . If H σ is not connected or H σ ∩ F = ∅ , thenthere exists ω ∗ ∈ Σ σ such that ϕ ω ∗ ( z ) / ∈ F and it is a trivial point of E .Proof. Let dim F = r and let A ∪ B be the partition in Lemma 2.1 (i) which defines F .We claim that if H σ ∩ F (cid:54) = ∅ , then there is only one k -th cell in H σ which intersects F . Actually, since ϕ ω ( ) ∈ [0 , n k − n k ] d for any ω ∈ Σ k , if ϕ ω ([0 , d ) ∩ F (cid:54) = ∅ for some ω ∈ Σ σ , then similar to the proof of Lemma 3.1 we must have π B ( ϕ ω ( )) = ( n k − n k b .On the other hand, π A ( ϕ ω ( )) = π A ( ϕ σ ( )), so ω is unique in Σ σ . Furthermore, ϕ ω ( z ) ∈ F in this scenario.By the assumption of the lemma and the claim above, there is a connected com-ponent U of H σ such that U ∩ F = ∅ . Let W = { ω ∈ Σ σ ; ϕ ω ([0 , d ) ⊂ U } . For each ω ∈ W , write π B ( ϕ ω ( )) = (cid:88) j ∈ B α j ( ω ) e j + (cid:88) j ∈ B β j ( ω ) e j First, we take the subset W (cid:48) ⊂ W by W (cid:48) = (cid:40) ω ∈ W ; (cid:88) j ∈ B α j ( ω ) attains the minimum (cid:41) . Then we take ω ∗ ∈ W (cid:48) such that (cid:88) j ∈ B β j ( ω ∗ ) = max { (cid:88) j ∈ B β j ( ω ); ω ∈ W (cid:48) } . ince U ∩ F = ∅ , we have ϕ ω ∗ ( z ) / ∈ F .Let us check that ϕ ω ∗ ( z ) is a trivial point of E . To this end, we only need toshow that(3.7) ϕ ω ∗ ( z ) / ∈ ϕ ω ([0 , d ) , where ω ∈ Σ k \ { ω ∗ } . Notice that ϕ ω ∗ ( z ) ∈ ϕ ω ∗ ( ˚ F ), it is clear that (3.7) holds forany ω / ∈ Σ σ . Since U is a connected component of H σ , we see that (3.7) holds forany ω / ∈ W .Now suppose ϕ ω ∗ ( z ) ∈ ϕ ω ([0 , d ) for some ω ∈ W , then ϕ ω ∗ ( z ) ∈ ϕ ω ∗ ([0 , d ) ∩ ϕ ω ([0 , d ). By Lemma 2.3 we have π B ( ϕ ω ( z )) − π B ( ϕ ω ∗ ( z )) = π B ( ϕ ω ( )) − π B ( ϕ ω ∗ ( )) ∈ b − b (cid:48) n k , where b (cid:48) ∈ T . By the definition of B and B in (2.3), we know that the j -thcoordinate of b − b (cid:48) is 0 or − j ∈ B and is 0 or 1 if j ∈ B . According tothe choosing process of ω ∗ , on one hand, we have (cid:80) j ∈ B ( α j ( ω ) − α j ( ω ∗ )) ≥
0. So α j ( ω ) = α j ( ω ∗ ) for j ∈ B , that is to say, ω ∈ W (cid:48) . On the other hand, since ω ∈ W (cid:48) , we have (cid:80) j ∈ B ( β j ( ω ) − β j ( ω ∗ )) ≤
0, which forces that β j ( ω ) = β j ( ω ∗ ) for j ∈ B . Therefore, b = b (cid:48) and hence ω = ω ∗ . This finishes the proof. (cid:3) For k >
0, denote D k = D + n D + · · · + n k − D . We call E k = ([0 , d + D k ) /n k the k -th approximation of E . Clearly, E k ⊂ E k − for all k ≥ E = (cid:84) ∞ k =0 E k .For σ = ( σ (cid:96) ) (cid:96) ≥ ∈ Σ ∞ , we denote σ | k = σ . . . σ k for k >
0. We say σ is a coding ofa point x ∈ E if { x } = (cid:84) k ≥ ϕ σ ...σ k ( E ). Definition 3.1.
Let U be a connected component of E k , we call U a k -th island if U ∩ ∂ [0 , d = ∅ . Lemma 3.3. If E k contains a k -th island for some k > , then E has a trivialpoint.Proof. Since we can regard E ( n, D ) as E ( n k , D k ), without loss of generality, weassume that E has an island and denote it by U . Write U = (cid:83) j ∈ J ϕ j ([0 , d ), where J ⊂ Σ. We call a letter j ∈ J a special letter. A sequence σ = ( σ i ) i ≥ ∈ Σ ∞ iscalled a special sequence, if special letters occur infinitely many times in σ .Let(3.8) P = { x ∈ E ; at least one coding of x is a special sequence } . We claim that every point in P is a trivial point. Let z ∈ P and let σ = ( σ i ) i ≥ bea coding of z such that σ is a special sequence. Suppose σ k is a special letter, it iseasy to see that z ∈ ϕ σ ...σ k ([0 , d ) ⊂ ϕ σ ...σ k − ( U ) and ϕ σ ...σ k − ( U ) is a connectedcomponent of E k with diam ( ϕ σ ...σ k − ( U )) ≤ √ d/n k − . Notice that special lettersoccur infinitely often in σ , we conclude that z is a trivial point. (cid:3) Theorem 3.1.
Let E be a fractal cube with dim aff( E ) = d . Then E has a trivialpoint if and only if E k contains a k -th island for some k ≥ . roof. Let z ∈ E be a trivial point. We claim that there exists another trivial point z ∗ ∈ E ∩ (0 , d , that is, the dimension of the containing face of z ∗ is d .Suppose F is the containing face of z with dim F = r , where 0 ≤ r ≤ d −
1. Let A ∪ B be the partition in Lemma 2.1 (i) which defines F . Let σ = ( σ (cid:96) ) (cid:96) ≥ ∈ Σ ∞ be a coding of z . Then for each k > z ∈ H σ | k ∩ F , where H σ | k is defined in(3.5). We will show by two cases that E contains another trivial point of the form ϕ ω ( z ) , ω ∈ Σ ∗ , and it is not in F . Case 1. H σ | k is not connected for some k > ω ∗ ∈ Σ σ | k such that z = ϕ ω ∗ ( z ) / ∈ F is a trivialpoint of E . Case 2. H σ | k is connected for all k > p > C p is the connected component of E p containing z and diam ( C p ) < . It is clear that H σ | p ⊂ C p , so we have diam ( H σ | p ) < . Sincedim aff( E ) = d , there exist j ∈ Σ such that(3.9) ϕ j ([0 , d ) ∩ F = ∅ . We consider the set H jσ ...σ p . Let Σ j = { i ∈ Σ; π A ( ϕ i ( )) = π A ( ϕ j ( )) } . It is easyto see that H jσ ...σ p = (cid:83) i ∈ Σ j ϕ i ( H σ | p ).If j = 1, then H jσ ...σ p ∩ F = ϕ j ( H σ | p ) ∩ F = ∅ . If j >
1, we have ϕ i ( H σ | p ) ∩ ϕ i (cid:48) ( H σ | p ) = ∅ for any i, i (cid:48) ∈ Σ j since diam ( H σ | p ) < . Hence H jσ ...σ p isnot connected. So by Lemma 3.2, there exists ω ∗ ∈ Σ jσ ...σ p such that ϕ ω ∗ ( z ) / ∈ F and it is a trivial point of E .Then by Lemma 3.1, the containing face of this trivial point has dimension noless than r + 1. Inductively, we can finally obtain a trivial point z ∗ whose containingface is [0 , d . The claim is proved.Now suppose on the contrary that E k contains no k -th island for all k ≥
1. Wewill derive a contradiction. Let z ∗ ∈ E ∩ (0 , d be a trivial point. Let U k be theconnected component of E k containing z ∗ , then we have U k ∩ ∂ [0 , d (cid:54) = ∅ . By theWeiestrass-Balzano property of the Hausdorff metric, there exists a subsequence k j such that U k j converge. We denote U ∗ to be the limit. On one hand, U ∗ is connectedsince U k j is connected for each k j . On the other hand, z ∗ ∈ U ∗ and U ∗ ∩ ∂ [0 , d (cid:54) = ∅ .So U ∗ is a non-trivial connected component of E containing z ∗ , a contradiction.This together with Lemma 3.3 finish the proof of the theorem. (cid:3) Proof of Theorem 1.1 . First, let us assume dim aff( E ) = d . Since E contains atrivial point, by Theorem 3.1, there exists k > E k contains a k -th island.Without lose of generality, suppose E has an island C . Write C = (cid:83) j ∈ J ϕ j ([0 , d ),where J ⊂ Σ. Let P be defined as (3.8). It has been proved in Lemma 3.3 thatevery point in P is a trivial point.We denote P c = E \ P . Let D (cid:48) = D \ { d j ; j ∈ J } and let E (cid:48) be the fractal cubedetermined by n and D (cid:48) . It is easy to see that P c = ∞ (cid:91) k =0 (cid:91) σ ...σ k ∈ Σ k ,σ k ∈ J ϕ σ ...σ k ( E (cid:48) ) ⊂ (cid:91) σ ∈ Σ ∗ ϕ σ ( E (cid:48) ) . onsequently, dim H P c ≤ dim H E (cid:48) = log D (cid:48) log n < dim H E . Notice that E \ Λ( E ) ⊂ P c ,we have I c ( E ) = dim H E \ Λ( E ) < dim H E .Next, assume that dim aff( E ) < d . Then there exist α = ( α , . . . , α d ) ∈ R d \ { } and c ∈ R such that(3.10) (cid:104) x, α (cid:105) = c, ∀ x ∈ E. Without loss of generality, we may assume that α (cid:54) = 0. Since x + hn ∈ E for any x ∈ E and any h ∈ D , we deduce that(3.11) (cid:104) h, α (cid:105) = ( n − c. Let x = ( x , . . . , x d ) ∈ R d , we define a map by π ( x ) = ( x , . . . , x d ). Denote (cid:101) D = { π ( h ); h ∈ D} and let (cid:101) E be the fractal cube determined by n and (cid:101) D . Define g : R d − → R d by g ( x , . . . , x d ) = ( c − (cid:104) π ( x ) , π ( α ) (cid:105) , π ( x )) . According to (3.10) and (3.11), one can show that E = g ( (cid:101) E ). So we have I c ( E ) = I c ( (cid:101) E ) and dim H E = dim H (cid:101) E .Therefore, by the first part of the proof and induction we have I c ( E ) < dim H E .This finish the proof. (cid:3) Application to topological Hausdorff dimension
The topological Hausdorff dimension is defined as follows:
Definition 4.1 ( [1]) . Let X be a metric space. The topological Hausdorff dimensionof X is defined as(4.1) dim tH X = inf U is a basis of X (cid:18) U ∈U dim H ∂U (cid:19) , where dim H ∂U denotes the Hausdorff dimension of the boundary of U and we adoptthe convention that dim tH ∅ = dim H ∅ = − . The following theorem gives an alternative definition of the topological Hausdorffdimension.
Theorem 4.1 (Theorem 3.7 of [1]) . For a non-empty σ -compact metric space X , itholds that dim tH X = min { h ; ∃ S ⊂ X such that dim H S ≤ h − and X \ S is totally disconnected } . Proof of Theorem 1.2 . Let G = X \ Λ( X ). Clearly X \ G = Λ( X ) is totallydisconnected. Let t = dim tH G . By Theorem 4.1, for any δ >
0, there exists S ⊂ G such that G \ S is totally disconnected, anddim H S + 1 < t + δ. We can see that X \ S = Λ( X ) ∪ ( G \ S ) is also totally disconnected; for otherwisethere is a connected component of E connecting a point x ∈ Λ( X ) and a point ∈ G \ S . Again by Theorem 4.1, dim tH X ≤ dim H S + 1 < t + δ . Since δ isarbitrary, we have dim tH X ≤ dim tH G . Therefore,dim tH X ≤ dim tH G ≤ dim H G = I c ( X ) . (cid:3) Calculation of I c ( K ) in Example 1.2 We identify R with C . Let n = 5. Let D = { d , . . . , d } be the digit setillustrated in Figure 3 (a), denote Σ = { , . . . , } . Let K be the fractal squaredetermined by n and D , and let { ϕ j = z + d j } j ∈ Σ be the IFS of K . Denote(5.1) J XX = { j ∈ Σ; d j ∈ D \ { i, i, i }} ; J XY = { j ∈ Σ; d j ∈ { i, i, i }} ; J Y X = { j ∈ Σ; d j ∈ D \ { i, i, i, , i }} ; J Y Y = { j ∈ Σ; d j ∈ { i, i, i, , i }} , see Figure 4. Let X = (cid:32) (cid:91) j ∈ J XX ϕ j ( X ) (cid:33) ∪ (cid:32) (cid:91) j ∈ J XY ϕ j ( Y ) (cid:33) , Y = (cid:32) (cid:91) j ∈ J Y X ϕ j ( X ) (cid:33) ∪ (cid:32) (cid:91) j ∈ J Y Y ϕ j ( Y ) (cid:33) . Then X and Y are graph-directed sets (see [7]). The directed graph G is given inFigure 5. X X X X XYYYX X X X XX (a) The first iteration of X . X X X X YYYYX X X X Y (b) The first iteration of Y . Figure 4.
X Y XX J YX J XY J YY J Figure 5.
The directed graph G . Each d ∈ J XY defined an edgefrom X to Y , and the corresponding map of this edge is ( z + d ) / J XX , J Y X and J Y Y . or each (cid:96) >
0, let J ( (cid:96) ) Y X be the collection of paths with length (cid:96) which startfrom Y and end at X in the graph G . Similarly, we can define J ( (cid:96) ) XX , J ( (cid:96) ) XY and J ( (cid:96) ) Y Y . Let K (cid:96) = (cid:83) σ ∈ Σ (cid:96) ϕ σ ([0 , ) and Y (cid:96) = (cid:83) σ ∈ J ( (cid:96) ) Y X ∪ J ( (cid:96) ) Y Y ϕ σ ([0 , ) be the be the (cid:96) -thapproximations of K and Y respectively. Then K = (cid:84) (cid:96)> K (cid:96) and Y = (cid:84) (cid:96)> Y (cid:96) . Lemma 5.1.
Let C be the connected component of K containing . Then (i) C = Y ; (ii) for any non-trivial connected component C (cid:48) (cid:54) = C of K , there exists ω ∈ Σ ∗ such that C (cid:48) = ϕ ω ( C ) .Proof. (i) Let C (cid:96) be the connected component of K (cid:96) containing . We only need toshow that C (cid:96) = Y (cid:96) for all (cid:96) >
0. Now we define a label map h on the cells in C (cid:96) asfollows. We set h ( σ . . . σ (cid:96) ) = X if there exists ω . . . ω (cid:96) ∈ Σ (cid:96) such that(5.2) ϕ ω ...ω (cid:96) ([0 , ) = ϕ σ ...σ (cid:96) ([0 , ) + 1 n (cid:96) ∈ C (cid:96) , otherwise set h ( σ . . . σ (cid:96) ) = Y . We will prove by induction that(5.3) σ . . . σ (cid:96) ∈ (cid:40) J ( (cid:96) ) Y X , if h ( σ . . . σ (cid:96) ) = X,J ( (cid:96) ) Y Y , if h ( σ . . . σ (cid:96) ) = Y. For (cid:96) = 1, (5.3) holds by (5.1). Assume that (5.3) holds for (cid:96) . Case 1. h ( σ . . . σ (cid:96) ) = X .In this case, (5.2) holds, which means that the right neighbor of ϕ σ ...σ (cid:96) ([0 , )belongs to C (cid:96) . If h ( σ . . . σ (cid:96) σ (cid:96) +1 ) = X , then the right neighbor of ϕ σ ...σ (cid:96) σ (cid:96) +1 ([0 , )belongs to C (cid:96) +1 and we have σ (cid:96) +1 ∈ J XX . Hence σ . . . σ (cid:96) σ (cid:96) +1 ∈ J ( (cid:96) +1) Y X . Similarly, if h ( σ . . . σ (cid:96) σ (cid:96) +1 ) = Y , then σ (cid:96) +1 ∈ J XY and σ . . . σ (cid:96) σ (cid:96) +1 ∈ J ( (cid:96) +1) Y Y . Case 2. h ( σ . . . σ (cid:96) ) = Y .In this case, the right neighbor of ϕ σ ...σ (cid:96) ([0 , ) is not contained in C (cid:96) . By asimilar argument as Case 1 , we have σ . . . σ (cid:96) σ (cid:96) +1 ∈ J ( (cid:96) +1) Y X if h ( σ . . . σ (cid:96) σ (cid:96) +1 ) = X ,and σ . . . σ (cid:96) σ (cid:96) +1 ∈ J ( (cid:96) +1) Y Y if h ( σ . . . σ (cid:96) σ (cid:96) +1 ) = Y .Therefore, (5.3) holds for (cid:96) + 1. Clearly, (5.3) implies that C (cid:96) = Y (cid:96) . Statement (i)is proved.(ii) Notice that ϕ i ( K ) ∩ ϕ j ( K ) ⊂ C for each i, j ∈ Σ with i (cid:54) = j . Let C (cid:48) be anon-trivial connected component of K . Let ω be the longest word in Σ ∗ such that C (cid:48) ⊂ ϕ ω ( K ). Then ϕ − ω ( C (cid:48) ) ⊂ K and there exists i, j ∈ Σ such that ϕ − ω ( C (cid:48) ) ∩ ϕ i ( K ) ∩ ϕ j ( K ) (cid:54) = ∅ . It follows that ϕ − ω ( C (cid:48) ) ⊂ C , hence C (cid:48) ⊂ ϕ ω ( C ). Since C (cid:48) is aconnected component, we have C (cid:48) = ϕ ω ( C ). Statement (ii) is proved. (cid:3) By Lemma 5.1 we have I c ( K ) = dim H C = dim H Y = log λ log 5 , where λ = √ is the maximal eigenvalue of the matrix (cid:34)
11 83 5 (cid:35) . Let K (cid:48) be the fractal square inExample 1.2. It is obvious that I c ( K (cid:48) ) = log 13log 5 . eferences [1] R. Balka, Z. Buczolich and M. Elekes, A new fractal dimension: the topological Hausdorffdimension , Adv. Math., (2015), 881-927.[2] M. Bonk and S. Merenkov. Quasisymmetric rigidity of square Sierpinski carpets, Anal. Math., (2013), 591-643.[3] K.J. Falconer,
Fractal geometry: mathematical foundations and applications , John Wiley &Sons, (1990).[4] L.Y. Huang, Y. Zhang,
Gap sequence of high dimensional self-similar sets and self-affine sets ,Preprint.[5] J.E. Hutchinson,
Fractals and slef-similarity , Indiana, Univ. Math. J. (1981), 713-747.[6] K.S. Lau, J.J. Luo and H. Rao,: Topological structure of fractal squares . Math. Proc. Camb.Phil. Soc , (2013), 73-86.[7] R.D. Mauldin, S.C. Williams, Hausdorff dimension in graph directed constructions , Trans.Amer. Math. Soc. (1988) 811-829.[8] J.H. Ma and Y.F. Zhang,
Topological Hausdorff dimension of fractal squares and its applicationto Lipschitz classification , To appear in Nonlinearlity.[9] H. Rao, H.J. Ruan, Y. M. Yang,
Gap sequence, Lipschitz equivalence and box dimension offractal sets , Nonlinearity, (2008), no. 6, 1339-1347.[10] H. J. Ruan and Y. Wang. Topological invariants and Lipschitz equivalence of fractal squares.J. Math. Anal. Appl., (2017), 327-344.[11] G.T. Whyburn, Topological characterization of the Sierpinski curve , Fund. Math. (1958),320–324.[12] L.F. Xi and Y. Xiong: Self-similar sets with initial cubic patterns, CR Acad. Sci. Paris, Ser.I , (2010), 15-20.[13] Y.M. Yang and Y.J. Zhu: Lipschitz equivalence of self-similar sets with two-state automation, J. Math. Anal. Appl , (2018), 379-392.[14] Y.F. Zhang, A lower bound of topological Hausdorff domension of fractal squares , To appearin Fractals, (2020).[15] Ziegler, G¨unter M.,
Lectures on Polytopes , Graduate Texts in Mathematics, 152, Springer,Definition 2.1, p. 51 (1995).
College of Computer, Beijing Institute of Technology, Beijing, 100080, China
Email address : [email protected] Department of Mathematics and Statistics, Central China Normal University,Wuhan, 430079, China
Email address : [email protected]@mail.ccnu.edu.cn