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Home , Hausdorff dimension

A DROP PHENOMENON OF

LIANG-YI HUANG AND HUI RAO∗

Abstract. Let E be a metric . We introduce a notion of connectedness index of E, which is the Hausdorff dimension of the union of non-trivial connected components of E. We show that the connectedness index of a fractal E is strictly less than the Hausdorff dimension of E provided that E possesses a trivial connected component. Hence the connectedness index is a new Lipschitz invariant. Moreover, we investigate the relation between the connectedness index and topological Hausdorff dimension.

1. Introduction N d An (IFS) is a family of contractions {ϕj}j=1 on R , and the of the IFS is the unique nonempty compact set K satisfying SN K = j=1 ϕj(K), and it is called a self-similar set [5]. Let n ≥ 2 and let D = d {d1, ··· , dN } ⊂ {0, 1, . . . , n−1} , which we call a digit set. Denote by #D := N the 1 N of D. Then n and D determine an IFS {ϕj(z) = n (z + dj)}j=1, whose attractor E = E(n, D) satisfies the set equation 1 (1.1) E = (E + D). n We call E a fractal cube [12], especially, when d = 2, we call E a fractal [6]. There are some works on topological and metric properties of fractal cubes. Why- burn [11] studied the classification, Bonk and Merenkov [2] studied the quasi-symmetric classification. Lau, Luo and Rao [6] studied when a fractal square is totally disconnected. Xi and Xiong [12] gave a complete classification of Lipschitz equivalence of fractal cubes which are totally disconnected. Recently, the arXiv:2010.12890v1 [math.GN] 24 Oct 2020 studies of [10,13] focus on the the Lipschitz equivalence of fractal which are not totally disconnected. Topological Hausdorff dimension is a new introduced by Buc- zolich and Elekes [1]. It is shown in [1] that for any set K we always have dimtH K ≤ dimH K, where dimtH and dimH denote the topological Hausdorff dimension and Hausdorff dimension respectively. Ma and Zhang [8] calculated topological Haus- dorff of a class of fractal squares.

Date: October 12, 2020. The work is supported by NSFS Nos. 11971195 and 11601172. 2000 Subject Classification: 28A80,26A16 Key words and phrases: fractal cube, connected component, topological Hausdorff dimension . * The correspondence author. 1 Let K be a . A 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) Ic(K) := dimH K \ Λ(K), and we call it the connectedness index of K. It is obvious that Ic(K) ≤ dimH K. Clearly, the connectedness index is a Lipschitz invariant. The main results of the present paper are as follows. Theorem 1.1. Let E = E(n, D) be a d-dimensional fractal cube. If E has a trivial point, then Ic(E) < dimH E. However, Theorem 1.1 is not valid for general self-similar sets, even if the self- similar sets satisfy the open set condition.  ∞  S 1 1 Q 1 Example 1.1. Let Q = {0} ∪ [ 22k+1 , 22k ] . Observe that Q = 4 ∪ [ 2 , 1] and k=0 Q 2 ∪ Q = [0, 1]. Then Q is a self-similar set satisfying the equation Q Q 1 Q 1 Q = ∪ + ∪ + . 4 4 2 2 2

The set Q has only one trivial point, that is 0. Therefore, Ic(Q) = dimH Q = 1. Figure1 illustrates Q0, a two dimensional generalization of Q. Similarly, Q0 is a self-similar set, and the unique trivial point of Q0 is 0.

Figure 1. The self-similar set Q0.

Using Theorem 3.7 of [1] we show the following.

Theorem 1.2. For a non-empty σ-compact metric space K, we have dimtH K ≤ Ic(K).

Zhang [14] asked when dimtH E = dimH E, where E is a fractal square. According to [14], a digit set D is called a Latin digit set, if every row and every column has the same number of elements (see Figure2). For a fractal square E = E(n, D), Zhang showed that if dimtH E = dimH E, then either E = [0, 1] × C, or E = C × [0, 1] for some C ⊂ [0, 1], or 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 dimH E = dimtH E. 2 (a) The digit set of L. (b) The Latin fractal square L.

Figure 2. It is shown in [14] that log 12/ log 6 = dimtH L < dimH L = log 24/ log 6. While by Theorem 1.1 and Theorem 1.2, we directly have dimtH L < dimH L.

Corollary 1.1. Let E be a d-dimensional fractal cube. If dimH E = dimtH E, then E has no trivial point. Remark 1.1. Another application of Theorem 1.1 is on the gap sequences of fractal cubes, a Lipschitz equivalent invariant introduced by Rao, Ruan and Yang [9]. For a fractal cube K, let {gm(K)}m≥1 be the gap sequence. Using Theorem 3.1 of the present paper, it is proved in [4] that if K has trivial point, then {gm(K)}m≥1 is −1/γ equivalent to {m }m≥1, where γ = dimH K. Finally, we calculate the connectedness indexes of two fractal squares in Figure3, and illustrate the application to Lipschitz classification. Example 1.2. Let K and K0 be two fractal squares indicated by Figure3. It is 0 log 14 seen that dimH K = dimH K = log 5 . By Theorem 1.3 of [8], one can obtain that 0 log 2 dimtH K = dimtH K = 1 + log 5 . We will show in section 5 that √ log(8 + 132/2) log 13 I (K) = and I (K0) = . c log 5 c log 5 So K and K0 are not Lipschitz equivalent.

(a) The digit set of K. (b) The digit set of K0. Figure 3.

3 This article is organized as follows. In section 2, we recall some basic facts of r-face of the [0, 1]d. In section 3, we prove Theorem 1.1. In section 4, we prove Theorem 1.2. In section 5, we give the details of Example 1.2.

2. Preliminaries on r-faces of [0, 1]d We recall some notions about convex , see [15]. Let C ⊂ Rd be a convex polytope, 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 closed 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 . The dimension of a face F , denoted by dim F , is the dimension 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 interior point of F . d Let e1,..., ed be the canonical basis of R . The following facts about the r-faces of [0, 1]d are obvious, see Chapter 2 of [15]. Lemma 2.1. (i) Let A ∪ B = {1, . . . , d} be a partition with #A = r. Then the set ( ) X (2.1) F = cjej; cj ∈ [0, 1] + b j∈A is an r-face of [0, 1]d if and only if b ∈ T , where ( ) X (2.2) T := εjej; εj ∈ {0, 1} ; j∈B (ii) For any r-face F of [0, 1]d, there exists a partition A ∪ B = {1, . . . , d} with #A = r such that F can be written as (2.1). P We will call F0 = { cjej; cj ∈ [0, 1]} a basic r-face related to the partition j∈A A ∪ B. We give a partition B = B0 ∪ B1 according to b by setting B = {j ∈ B; the j-th coordinate of b is 0}, (2.3) 0 B1 = {j ∈ B; the j-th coordinate of b is 1}. P P d Let x = αjej + βiei ∈ [0, 1] , we define two projection maps as follows: j∈A i∈B X X (2.4) πA(x) = αjej, πB(x) = βiei. j∈A i∈B If F is an r-face of [0, 1]d, we denote by F˚ the relative interior of F .

Lemma 2.2 (Chapter 2 of [15]). Let C ⊂ Rd 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. 4 The following lemma will be needed in section 3.

d d Lemma 2.3. Let F = F0 +b be an r-face of [0, 1] given by (2.1). Let u ∈ Z . Then F˚ ∩ (u + [0, 1]d) 6= ∅ if and only if u = b − b0 for some b0 ∈ T , where T is defined in (2.2).

0 0 0 d Proof. “⇐”: Suppose b ∈ T , then F − (b − b ) = F0 + b , and it is an r-face of [0, 1] by Lemma 2.1 (i). Applying a translation b − b0 we see that F ⊂ (b − b0) + [0, 1]d, which completes the proof of the sufficiency. ˚ d “⇒”: Suppose F ∩ (u + [0, 1] ) 6= ∅. Let z0 be a point in the intersection and let F 0 = [0, 1]d ∩(u+[0, 1]d). Then F 0 is a face of both [0, 1]d and u+[0, 1]d. So we have 0 0 F ⊂ F since F contains z0, a relative interior point of F . Hence F is an r-face of F 0 by Lemma 2.2 (i). It follows that F − u is an r-face of F 0 − u. Notice that F 0 is a face of u + [0, 1]d, then F 0 − u is a face of [0, 1]d. By Lemma d 2.2 (ii), F − u = F0 + (b − u) is an r-face of [0, 1] . By Lemma 2.1 (i) we have b − u ∈ T .  3. Trivial points of fractal cubes Let Σ = {1, 2,...,N}. Denote by Σ∞ and Σk the sets of infinite words and words ∗ S k of length k over Σ respectively. Let Σ = k≥0 Σ be the set of all finite words. For k any σ = σ1 . . . σk ∈ Σ , let ϕσ = ϕσ1 ◦ · · · ◦ ϕσk . In this section, we always assume that E = E(n, D) is a d-dimensional fractal cube defined in (1.1) with IFS {ϕj}j∈Σ. In the following, we always assume that For a point z ∈ E, we say F is the containing face of z means that F is a face of the polytope [0, 1]d and it is the containing face of z.

k Lemma 3.1. Let z0 ∈ E and σ ∈ Σ for some k > 0. Let F be the containing 0 face of z0, let F be the containing face of ϕσ(z0). Then either ϕσ(z0) ∈ F or dim F 0 ≥ dim F + 1. Proof. Let A∪B be the partition in Lemma 2.1 (i) which defines F . By the definition ˚ of containing face, we have z0 ∈ F . Suppose that ϕσ(z0) ∈/ F . Take any point x ∈ F \{z0} and let I be the closed in F such that x is an endpoint of I and z0 is a relative interior point of I. It is clear that d d 0 0 ϕσ(I) ⊂ ϕσ([0, 1] ) ⊂ [0, 1] . Since ϕσ(z0) ∈ F , we have ϕσ(I) ⊂ F . By the 0 arbitrary of x we deduce that ϕσ(F ) ⊂ F , hence 0 (3.1) dim F ≥ dim ϕσ(F ) = dim F. We claim that F 0 is not an r-face of [0, 1]d. This claim together with (3.1) imply dim F 0 ≥ dim F + 1. Suppose on the contrary that F 0 is an r-face of [0, 1]d. Then there exists a partition 0 0 0 0 0 0 P A ∪ B = {1, . . . , d} such that F = F0 + b , where F0 = { cjej; cj ∈ [0, 1]} and j∈A0 0 P b ∈ { εjej; εj ∈ {0, 1}}. Since j∈B0 F b 0 + + ϕ (0) = ϕ (F ) ⊂ F 0 = F 0 + b0, nk nk σ σ 0 5 0 0 0 we have F0 = F0. Hence A = A and B = B. It follows that b (3.2) b0 = π (ϕ (z )) = + π (ϕ (0)) ∈ T. B σ 0 nk B σ Notice that ( ) X nk − 1 (3.3) π (ϕ (0)) ∈ c e ; c ∈ [0, ] B ω j j j nk j∈B

k (nk−1) for any ω ∈ Σ , which together with (3.2) imply that πB(ϕσ(0)) = nk b. Hence 0 b = b and it follows that ϕσ(z0) ∈ F , a contradiction. The claim is confirmed and the lemma is proven.  k d For each σ = σ1 . . . σk ∈ Σ , we call ϕσ([0, 1] ) ⊂ Ek a k-th cell of Ek. Denote k (3.4) Σσ = {ω ∈ Σ ; πA(ϕω(0)) = πA(ϕσ(0))} and set [ d (3.5) Hσ = ϕω([0, 1] ).

ω∈Σσ d Indeed, Hσ is the union of all k-th cells having the same projection with ϕσ([0, 1] ) under πA. From now on, we always assume that

(3.6) z0 is a trivial point of E and F is the containing face of z0. k Lemma 3.2. Let k > 0, fix σ ∈ Σ . If Hσ is not connected or Hσ ∩ F = ∅, then ∗ there exists ω ∈ Σσ such that ϕω∗ (z0) ∈/ 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 Hσ ∩F 6= ∅, then there is only one k-th cell in Hσ which intersects nk−1 d k d F . Actually, since ϕω(0) ∈ [0, nk ] for any ω ∈ Σ , if ϕω([0, 1] ) ∩ F 6= ∅ for some (nk−1) ω ∈ Σσ, then similar to the proof of Lemma 3.1 we must have πB(ϕω(0)) = nk b. On the other hand, πA(ϕω(0)) = πA(ϕσ(0)), so ω is unique in Σσ. Furthermore, ϕω(z0) ∈ F in this scenario. By the assumption of the lemma and the claim above, there is a connected com- d ponent U of Hσ such that U ∩ F = ∅. Let W = {ω ∈ Σσ; ϕω([0, 1] ) ⊂ U}. For each ω ∈ W , write X X πB(ϕω(0)) = αj(ω)ej + βj(ω)ej

j∈B0 j∈B1 First, we take the subset W 0 ⊂ W by ( ) 0 X W = ω ∈ W ; αj(ω) attains the minimum .

j∈B0 Then we take ω∗ ∈ W 0 such that X ∗ X 0 βj(ω ) = max{ βj(ω); ω ∈ W }.

j∈B1 j∈B1 6 Since U ∩ F = ∅, we have ϕω∗ (z0) ∈/ F . Let us check that ϕω∗ (z0) is a trivial point of E. To this end, we only need to show that d (3.7) ϕω∗ (z0) ∈/ ϕω([0, 1] ), k ∗ ˚ where ω ∈ Σ \{ω }. Notice that ϕω∗ (z0) ∈ ϕω∗ (F ), it is clear that (3.7) holds for any ω∈ / Σσ. Since U is a connected component of Hσ, we see that (3.7) holds for any ω∈ / W . d d Now suppose ϕω∗ (z0) ∈ ϕω([0, 1] ) for some ω ∈ W , then ϕω∗ (z0) ∈ ϕω∗ ([0, 1] ) ∩ d ϕω([0, 1] ). By Lemma 2.3 we have b − b0 π (ϕ (z )) − π (ϕ ∗ (z )) = π (ϕ (0)) − π (ϕ ∗ (0)) ∈ , B ω 0 B ω 0 B ω B ω nk 0 where b ∈ T . By the definition of B0 and B1 in (2.3), we know that the j-th 0 coordinate of b − b is 0 or −1 if j ∈ B0 and is 0 or 1 if j ∈ B1. According to ∗ P ∗ the choosing process of ω , on one hand, we have (αj(ω) − αj(ω )) ≥ 0. So j∈B0 ∗ 0 αj(ω) = αj(ω ) for j ∈ B0, that is to say, ω ∈ W . On the other hand, since 0 P ∗ ∗ ω ∈ W , we have (βj(ω) − βj(ω )) ≤ 0, which forces that βj(ω) = βj(ω ) for j∈B1 0 ∗ j ∈ B1. Therefore, b = b and hence ω = ω . This finishes the proof.  k−1 d k For k > 0, denote Dk = D + nD + ··· + n D. We call Ek = ([0, 1] + Dk)/n T∞ the k-th approximation of E. Clearly, Ek ⊂ Ek−1 for all k ≥ 1 and E = k=0 Ek. ∞ For σ = (σ`)`≥1 ∈ Σ , we denote σ|k = σ1 . . . σk for k > 0. We say σ is a coding of T a point x ∈ E if {x} = k≥1 ϕσ1...σk (E).

Definition 3.1. Let U be a connected component of Ek, we call U a k-th island if U ∩ ∂[0, 1]d = ∅.

Lemma 3.3. If Ek contains a k-th island for some k > 0, then E has a trivial point.

k Proof. Since we can regard E(n, D) as E(n , Dk), without loss of generality, we S d assume that E1 has an island and denote it by U. Write U = j∈J ϕj([0, 1] ), where ∞ J ⊂ Σ. We call a letter j ∈ J a special letter. A sequence σ = (σi)i≥1 ∈ Σ is called 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 = (σi)i≥1 be a coding of z such that σ is a special sequence. Suppose σk is a special letter, it is easy to see that z ∈ ϕ ([0, 1]d) ⊂ ϕ (U) and ϕ (U) is a connected σ1...σk σ1...σk−√1 σ1...σk−1 k−2 component of Ek with diam(ϕσ1...σk−1 (U)) ≤ d/n . Notice that special letters occur infinitely often in σ, we conclude that z is a trivial point.  Theorem 3.1. Let E be a fractal cube with dim aff(E) = d. Then E has a trivial point if and only if Ek contains a k-th island for some k ≥ 1. 7 Proof. Let z0 ∈ E be a trivial point. We claim that there exists another trivial point z∗ ∈ E ∩ (0, 1)d, that is, the dimension of the containing face of z∗ is d. Suppose F is the containing face of z0 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`)`≥1 ∈ Σ be a coding of z0. Then for each k > 0, z0 ∈ Hσ|k ∩ F , where Hσ|k is defined in (3.5). We will show by two cases that E contains another trivial point of the form ∗ ϕω(z0), ω ∈ Σ , and it is not in F .

Case 1. Hσ|k is not connected for some k > 0. ∗ ∗ By Lemma 3.2, there exists ω ∈ Σσ|k such that z1 = ϕω (z0) ∈/ F is a trivial point of E.

Case 2. Hσ|k is connected for all k > 0. Let p > 0 be an integer such that Cp is the connected component of Ep containing 1 1 z0 and diam(Cp) < 3 . It is clear that Hσ|p ⊂ Cp, so we have diam(Hσ|p ) < 3 . Since dim aff(E) = d, there exist j ∈ Σ such that d (3.9) ϕj([0, 1] ) ∩ F = ∅.

We consider the set Hjσ1...σp . Let Σj = {i ∈ Σ; πA(ϕi(0)) = πA(ϕj(0))}. It is easy to see that H = S ϕ (H ). jσ1...σp i∈Σj i σ|p

If #Σj = 1, then Hjσ1...σp ∩ F = ϕj(Hσ|p ) ∩ F = ∅. If #Σj > 1, we have 0 1 0 ϕi(Hσ|p ) ∩ ϕi (Hσ|p ) = ∅ for any i, i ∈ Σj since diam(Hσ|p ) < 3 . Hence Hjσ1...σp is ∗ ∗ not connected. So by Lemma 3.2, there exists ω ∈ Σjσ1...σp such that ϕω (z0) ∈/ F and it is a trivial point of E. Then by Lemma 3.1, the containing face of this trivial point has dimension no less than r +1. Inductively, we can finally obtain a trivial point z∗ whose containing face is [0, 1]d. The claim is proved. Now suppose on the contrary that Ek contains no k-th island for all k ≥ 1. We ∗ d will derive a contradiction. Let z ∈ E ∩ (0, 1) be a trivial point. Let Uk be the ∗ d connected component of Ek containing z , then we have Uk ∩ ∂[0, 1] 6= ∅. By the Weiestrass-Balzano property of the Hausdorff metric, there exists a subsequence kj ∗ ∗ such that Ukj converge. We denote U to be the limit. On one hand, U is connected ∗ ∗ ∗ d since Ukj is connected for each kj. On the other hand, z ∈ U and U ∩∂[0, 1] 6= ∅. 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.  Proof of Theorem 1.1. First, let us assume dim aff(E) = d. Since E contains a trivial point, by Theorem 3.1, there exists k > 0 such that Ek contains a k-th island. S d Without lose of generality, suppose E1 has an island C. Write C = j∈J ϕj([0, 1] ), where J ⊂ Σ. Let P be defined as (3.8). It has been proved in Lemma 3.3 that every point in P is a trivial point. c 0 0 We denote P = E \ P . Let D = D\{dj; j ∈ J} and let E be the fractal cube determined by n and D0. It is easy to see that ∞ c [ [ 0 [ 0 P = ϕσ1...σk (E ) ⊂ ϕσ(E ). k ∗ k=0 σ1...σk∈Σ ,σk∈J σ∈Σ 8 c 0 log #D0 c Consequently, dimH P ≤ dimH E = log n < dimH E. Notice that E \ Λ(E) ⊂ P , we have Ic(E) = dimH E \ Λ(E) < dimH E. d Next, assume that dim aff(E) < d. Then there exist α = (α1, . . . , αd) ∈ R \{0} and c ∈ R such that (3.10) hx, αi = c, ∀x ∈ E. x+h Without loss of generality, we may assume that α1 6= 0. Since n ∈ E for any x ∈ E and any h ∈ D, we deduce that (3.11) hh, αi = (n − 1)c.

d Let x = (x1, . . . , xd) ∈ R , we define a map by π(x) = (x2, . . . , xd). Denote De = {π(h); h ∈ D} and let Ee be the fractal cube determined by n and De. Define g : Rd−1 → Rd by g(x2, . . . , xd) = (c − hπ(x), π(α)i, π(x)).

According to (3.10) and (3.11), one can show that E = g(Ee). So we have Ic(E) = Ic(Ee) and dimH E = dimH Ee. Therefore, by the first part of the proof and induction we have Ic(E) < dimH E. This finish the proof. 

4. 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 dimension of X is defined as   (4.1) dimtH X = inf 1 + sup dimH ∂U , U is a basis of X U∈U where dimH ∂U denotes the Hausdorff dimension of the boundary of U and we adopt the convention that dimtH ∅ = dimH ∅ = −1. The following theorem gives an alternative definition of the topological Hausdorff dimension. Theorem 4.1 (Theorem 3.7 of [1]). For a non-empty σ-compact metric space X, it holds that

dimtH X = min{h; ∃S ⊂ X such that dimH S ≤ h − 1 and X \ S is totally disconnected}. Proof of Theorem 1.2. Let G = X \ Λ(X). Clearly X \ G = Λ(X) is totally disconnected. Let t = dimtH G. By Theorem 4.1, for any δ > 0, there exists S ⊂ G such that G \ S is totally disconnected, and

dimH S + 1 < t + δ. We can see that X \ S = Λ(X) ∪ (G \ S) is also totally disconnected; for otherwise there is a connected component of E connecting a point x ∈ Λ(X) and a point 9 y ∈ G \ S. Again by Theorem 4.1, dimtH X ≤ dimH S + 1 < t + δ. Since δ is arbitrary, we have dimtH X ≤ dimtH G. Therefore,

dimtH X ≤ dimtH G ≤ dimH G = Ic(X). 

5. Calculation of Ic(K) in Example 1.2 2 We identify R with C. Let n = 5. Let D = {d1, . . . , d14} be the digit set illustrated in Figure3 (a), denote Σ = {1,..., 14}. Let K be the fractal square z+dj determined by n and D, and let {ϕj = 5 }j∈Σ be the IFS of K. Denote

JXX = {j ∈ Σ; dj ∈ D \ {i, 2i, 3i}}; J = {j ∈ Σ; d ∈ {i, 2i, 3i}}; (5.1) XY j JYX = {j ∈ Σ; dj ∈ D \ {i, 2i, 3i, 4, 4 + 4i}};

JYY = {j ∈ Σ; dj ∈ {i, 2i, 3i, 4, 4 + 4i}}, see Figure4. Let ! ! ! ! [ [ [ [ X = ϕj(X) ∪ ϕj(Y ) ,Y = ϕj(X) ∪ ϕj(Y ) .

j∈JXX j∈JXY j∈JYX j∈JYY Then X and Y are graph-directed sets (see [7]). The directed graph G is given in Figure5.

XXXXX XXXXY Y Y Y X Y Y Y XXXXX XXXXY

(a) The first iteration of X. (b) The first iteration of Y . Figure 4.

J XY

J J XX X Y YY

JYX Figure 5. The directed graph G. Each d ∈ JXY defined an edge from X to Y , and the corresponding map of this edge is (z + d)/5. The same hold for JXX ,JYX and JYY .

10 (`) For each ` > 0, let JYX be the collection of paths with length ` which start (`) (`) from Y and end at X in the graph G. Similarly, we can define JXX ,JXY and (`) S 2 S 2 JYY . Let K` = ϕσ([0, 1] ) and Y` = ϕσ([0, 1] ) be the be the `-th σ∈Σ` σ∈J(`) ∪J(`) YX YY T T approximations of K and Y respectively. Then K = K` and Y = Y`. `>0 `>0 Lemma 5.1. Let C be the connected component of K containing 0. Then (i) C = Y ; (ii) for any non-trivial connected component C0 6= C of K, there exists ω ∈ Σ∗ 0 such that C = ϕω(C).

Proof. (i) Let C` be the connected component of K` containing 0. We only need to show that C` = Y` for all ` > 0. Now we define a label map h on the cells in C` as ` follows. We set h(σ1 . . . σ`) = X if there exists ω1 . . . ω` ∈ Σ such that 1 (5.2) ϕ ([0, 1]2) = ϕ ([0, 1]2) + ∈ C , ω1...ω` σ1...σ` n` ` otherwise set h(σ1 . . . σ`) = Y . We will prove by induction that ( (`) JYX , if h(σ1 . . . σ`) = X, (5.3) σ1 . . . σ` ∈ (`) JYY , if h(σ1 . . . σ`) = Y. For ` = 1, (5.3) holds by (5.1). Assume that (5.3) holds for `.

Case 1. h(σ1 . . . σ`) = X. 2 In this case, (5.2) holds, which means that the right neighbor of ϕσ1...σ` ([0, 1] ) 2 belongs to C`. If h(σ1 . . . σ`σ`+1) = X, then the right neighbor of ϕσ1...σ`σ`+1 ([0, 1] ) (`+1) belongs to C`+1 and we have σ`+1 ∈ JXX . Hence σ1 . . . σ`σ`+1 ∈ JYX . Similarly, if (`+1) h(σ1 . . . σ`σ`+1) = Y , then σ`+1 ∈ JXY and σ1 . . . σ`σ`+1 ∈ JYY .

Case 2. h(σ1 . . . σ`) = Y . 2 In this case, the right neighbor of ϕσ1...σ` ([0, 1] ) is not contained in C`. By a (`+1) similar argument as Case 1, we have σ1 . . . σ`σ`+1 ∈ JYX if h(σ1 . . . σ`σ`+1) = X, (`+1) and σ1 . . . σ`σ`+1 ∈ JYY if h(σ1 . . . σ`σ`+1) = Y . Therefore, (5.3) holds for ` + 1. Clearly, (5.3) implies that C` = Y`. Statement (i) is proved. 0 (ii) Notice that ϕi(K) ∩ ϕj(K) ⊂ C for each i, j ∈ Σ with i 6= j. Let C be a non-trivial connected component of K. Let ω be the longest word in Σ∗ such that 0 −1 0 −1 0 C ⊂ ϕω(K). Then ϕω (C ) ⊂ K and there exists i, j ∈ Σ such that ϕω (C ) ∩ −1 0 0 0 ϕi(K) ∩ ϕj(K) 6= ∅. It follows that ϕω (C ) ⊂ C, hence C ⊂ ϕω(C). Since C is a 0 connected component, we have C = ϕω(C). Statement (ii) is proved.  √ log λ 16+ 132 By Lemma 5.1 we have Ic(K) = dimH C = dimH Y = log 5 , where λ = 2 " # 11 8 is the maximal eigenvalue of the matrix . Let K0 be the fractal square in 3 5 0 log 13 Example 1.2. It is obvious that Ic(K ) = log 5 . 11 References [1] R. Balka, Z. Buczolich and M. Elekes, A new fractal dimension: the topological Hausdorff dimension, Adv. Math., 274 (2015), 881-927. [2] M. Bonk and S. Merenkov. Quasisymmetric rigidity of square Sierpinski carpets, Anal. Math., 177.2 (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, and slef-, Indiana, Univ. Math. J. 30 (1981), 713-747. [6] K.S. Lau, J.J. Luo and H. Rao,: Topological structure of fractal squares. Math. Proc. Camb. Phil. Soc, 155 (2013), 73-86. [7] R.D. Mauldin, S.C. Williams, Hausdorff dimension in graph directed constructions, Trans. Amer. Math. Soc. 309 (1988) 811-829. [8] J.H. Ma and Y.F. Zhang, Topological Hausdorff dimension of fractal squares and its application to Lipschitz classification, To appear in Nonlinearlity. [9] H. Rao, H.J. Ruan, Y. M. Yang, Gap sequence, Lipschitz equivalence and box dimension of fractal sets, Nonlinearity, 21 (2008), no. 6, 1339-1347. [10] H. J. Ruan and Y. Wang. Topological invariants and Lipschitz equivalence of fractal squares. J. Math. Anal. Appl., 451.1 (2017), 327-344. [11] G.T. Whyburn, Topological characterization of the Sierpinski , Fund. Math. 45 (1958), 320–324. [12] L.F. Xi and Y. Xiong: Self-similar sets with initial cubic patterns, CR Acad. Sci. Paris, Ser.I, 348 (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, 458.1 (2018), 379-392. [14] Y.F. Zhang, A lower bound of topological Hausdorff domension of fractal squares, To appear in 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]

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