Quasisymmetric Modification of Metrics on Self-Similar Sets

Abstract

Using the notions of scales and their gauge functions associated with self-similar sets, we give a necessary and sufficient condition for two metrics on a self-similar set being quasisymmetric to each other. As an application, we construct metrics on the Sierpinski carpet which is quasisymmetric with respect to the Euclidean metrics and obtain an upper estimate of the conformal dimension of the Sierpinski carpet.

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1 Introduction

The main purpose of this paper is to give a characterization of quasisymmetry for self-similar sets in terms of scales and related notions introduced in [Kig09]. As an application, we will construct a series of metrics on the Sierpinski carpet which are quasisymmetric to the restriction of the Euclidean metric and give an upper estimate of the quasiconformal dimension of the Sierpinski carpet (Fig. 1).

Quasisymmetric maps have been introduced by Tukia and Väisälä in [TV80] as a generalization of quasiconformal mappings in the complex plane.

Definition 1.1

(Quasisymmetry).

• (1) Let $(X, d)$ and $(X, \rho )$ be metric spaces. $\rho$ is said to be quasisymmetric, or QS for short, with respect to $d$ if and only if there exists a homeomorphism $h$ from $[0, +\infty )$ to itself such that $h(0) = 0$ and, for any $t > 0$, $\rho (x, z) < h(t)\rho (x, y)$ whenever $d(x, z) < td(x, y)$. We write $\rho \underset{\mathrm{QS}}{\sim }d$ if $\rho$ is quasisymmetric with respect to $d$.

• (2) Let $(X, d)$ be a metric space. A homeomorphism $f : X \rightarrow X$ is called quasisymmetric if and only if $d \underset{\mathrm{QS}}{\sim }d_f$, where $d_f(x, y)$ is defined by $d_f(x, y) = d(f(x), f(y))$.

The above definition immediately implies the following facts.

Proposition 1.2

Let $(X, d)$ and $(X, \rho )$ be metric spaces.

• (1) If $\rho \underset{\mathrm{QS}}{\sim }d$, then the identity may of $X$ is a homeomorphism from $(X, d)$ to $(X, \rho )$.

• (2) The relation $\underset{\mathrm{QS}}{\sim }$ is an equivalence relation among metrics on $X$. In particular, $\rho \underset{\mathrm{QS}}{\sim }d$ if and only if $d \underset{\mathrm{QS}}{\sim }\rho$.

Associated with the notion of quasisymmetry, the quasiconformal dimension of a metric space has been introduced by Pansu in [Pan89] as an invariant under quasisymmetric modification of a metric.

Definition 1.3

(Quasiconformal dimension) Let $(X, d)$ be a metric space. We define the conformal dimension of $(X, d)$, $\dim _{\mathcal {C}}(X, d)$, by

$$ \dim _{\mathcal {C}}(X, d) = \inf \{\dim _H(X, \rho )|\rho \;\text {is a metric on}\;X\text {and}\;d \underset{\mathrm{QS}}{\sim }\rho \}, $$

where $\dim _H(X,\rho )$ is the Hausdorff dimension of $(X, \rho )$.

Fig. 1

The Sierpinski carpet

Quasisymmetric maps on self-similar sets have been paid much attentions in recent years as well as their conformal dimensions. For example, Bonk and Merenkov have shown that any quasisymmetric homeomorphism from the Sierpinski carpet to itself is a composition of rotations and reflections in [BM00]. About the conformal dimensions, Tyson and Wu have proven that the conformal dimension of the Sierpinski gasket is one in [TW06]. For the Sierpinski carpet, it is known that

$$\begin{aligned} 1 + \frac{\log 3}{\log 2} \le \dim _{\mathcal {C}}(\mathrm{SC}, d_E) < \dim _{H}(\mathrm{SC}, d_E) = \frac{\log 8}{\log 3}, \end{aligned}$$

(1.1)

where SC is the Sierpinski carpet and $d_E$ is the restriction of the Euclidean metric. The strict inequality between the Hausdorff and the quasiconformal dimensions in (1.1) has shown by Keith and Laakso [KL04]. See [MT10] for details.

The first problem we are going to study is to obtain a verifiable characterization of quasisymmetric metrics. It will turn out that scales and related notions introduced in [Kig09] are useful in dealing with such a problem. Let $K$ be a connected self-similar set associated with the family of contractions $\{F_1, \ldots , F_N\}$, i. e. $K = F_1(K) \cup \ldots \cup F_N(K)$. Define $F_{{w}_1\ldots {w}_{m}} = F_{w_1}\circ \ldots \circ {F_{w_m}}$ and $K_{{w}_1\ldots {w}_{m}} = F_{{w}_1\ldots {w}_{m}}(K)$ for any $w_1, \ldots , w_m \in \{1, \ldots , N\}$. The notion of scales has been introduced in order to study how to find a metric under which the contraction mappings $\{F_1, \ldots , F_N\}$ have prescribed values of contraction ratios. A scale essentially gives “diameters” of $K_{{w}_1\ldots {w}_{m}}$’s and induces a family of assumed “balls” $U_s(x)$ around $x \in K$ with radius $s > 0$. See Sect. 2 for precise definitions. In the language of scales, we are going to present an equivalent condition in Theorem 3.4 for metrics being quasisymmetric to each other which is easy to verify for concrete examples, in particular, in the case of “self-similar” metrics.

As an application, we will present a systematic way of constructing a self-similar metric on the Sierpinski carpet which is quasisymmetric to $d_E$ and Ahlfors regular. The main idea is to find an “invisible” set introduced in Sect. 4. Roughly speaking, an invisible set is a collection of places where the shortest paths between two separated boundary points will not visit. (We define the “boundary” of the Sierpinski carpet by the union of four line segments, namely, the most upper, lower, right and left line segments of the square which is the convex hull of the Sierpinski carpet.) Putting an arbitrary weight on an invisible set, we will obtain a self-similar metric having the desired properties mentioned above with an explicit formula for its Hausdorff dimension in Theorem 5.3. Constructing series of invisible sets and taking advantage of the associated metrics, we will show that

$$ \dim _{\mathcal {C}}(\mathrm{SC}, d_E) \le \frac{\log {(\frac{9 + \sqrt{41}}{2})}}{\log 3} = 1.858183... < \frac{\log 8}{\log 3} = 1.892789.... $$

in Sect. 6.¹

Note that the conformal dimension in the above inequality can be replaced by the Ahlfors regular conformal dimension since our metrics are Ahlfors regular. See [MT10] for the definition of the Ahlfors regular conformal dimension.

The following is a convention in notations in this paper.

Let $f$ and $g$ be functions with variables $x_1, \ldots , x_n$. We use “$f \asymp g$ for any $(x_1, \ldots , x_n) \in A$” if and only if there exist positive constants $c_1$ and $c_2$ such that

$$ c_1f(x_1, \ldots , x_n) \le g(x_1, \ldots , x_n) \le c_2f(x_1, \ldots , x_n) $$

for any $(x_1, \ldots , x_n) \in A$.

2 Basic Notions

This section is devoted to introducing fundamental notions and results regarding scales and self-similar sets and scales.

The following is the standard definitions on (finite and infinite) sequences of finite symbols.

Definition 2.1

Let $S$ be a finite set. For $m \ge 0$, define $W_m(S) = S^m = \{w| w = {w}_1\ldots {w}_{m}, w_i \in S\}$, where $W_0(S) = \{\emptyset \}$. Define $W_* = \cup _{m \ge 0} W_m$. Also $\Sigma (S) = S^{\mathbb {N}} = \{\omega | \omega = \omega _1\omega _2\ldots , \omega _i \in S\}$. For $w = w_1\ldots {w_m} \in W_*(S)$, the length $|w|$ of $w$ is defined by $|w| = m$. For $w = w_1\ldots {w_m}$ and $v = v_1\ldots {v_n} \in W_*(S)$, we define $w\cdot {v}$ (or $wv$ for short) by $w{\cdot }v = w_1\ldots {w_m}{v_1}\ldots {v_n}$. For a subseteq $A, B \in W_*(S)$, $A\cdot {B}$ (or $AB$ for short) is defined by $A{\cdot }B = \{wv|w \in A, v \in B\}$.

Remark

The notion of “gauge function” given in the above definition is not related to the notion of “conformal gauge” which is commonly used in literatures concerning the conformal dimension, for example, [MT10].

With the product topology, $\Sigma (S)$ is compact, perfect and totally disconnected. In other words, $\Sigma (S)$ is a Cantor set. A scale is defined by a gauge function which assign a “diameter” to every $w \in W_*(S)$.

Definition 2.2

(Scale). Let $S$ be a finite set.

• (1) A function $g: W_*(S) \rightarrow (0, 1]$ is called a gauge function if and only if $g(\emptyset ) = 1, g({w}_1\ldots {w}_{m}) \le g({w}_1\ldots {w}_{m - 1})$ and $\max _{w \in W_m(S)} g(w) \rightarrow 0$ as $m \rightarrow 0$. A gauge function $g$ is said to be elliptic if and only if there exists $c \in (0, 1)$ and $n$ such that $g_{wi} \ge cg(w)$ for any $i \in S$ and any $w \in W_*(S)$ and $g_{wv} \le cg(w)$ for any $w \in W_*(S)$ and $v \in W_n$.

• (2) Let $g$ be a gauge function. Define

  $$ \Lambda _s^{g} = \{w = {w}_1\ldots {w}_{m} | g(w_1\ldots {w_{m - 1}}) \ge s > g({w}_1\ldots {w}_{m})\} $$

We call $\mathcal {S}^g = \{\Lambda ^g_s\}_{s \in (0, 1]}$ a scale on $\Sigma$ associated with the gauge function $g$.

If no confusion may occur, we omit $S$ in $W_m(S), W_*(S)$ and $\Sigma (S)$ and simply write $W_m, W_*$ and $\Sigma$ respectively.

The notion of self-similar structure describes topological feature of self-similar sets.

Definition 2.3

$(K, S, \{F_i\}_{i \in S})$ is called a self-similar structure if the following four conditions (S1), (S2), (S3) and (S4) are satisfied:

• (S1) $K$ is a compact metrizable set.

• (S2) $S$ is a finite set.

• (S3) $F_s: K \rightarrow K$ is continuous for any $s \in S$.

• (S4) There exists a continuous surjection $\pi : \Sigma (S) \rightarrow K$ such that $F_s\circ {\pi } = \pi \circ {\sigma _s}$ for any $s \in K$, where $\sigma _s: \Sigma (S) \rightarrow \Sigma (S)$ is defined by $\sigma _s(\omega _1\omega _2\ldots ) = s\omega _1\omega _2 \ldots$.

Hereafter in this paper, $(K, S, \{F_s\}_{s \in S})$ is always a self-similar structure.

Notation

Define $F_{{w}_1\ldots {w}_{m}} = F_{w_1}\circ \cdots \circ {F_{w_m}}$ and $K_w = F_w(K)$. Moreover, define $K(A) = \cup _{w \in A} K_w$ for a subset $A \subseteq W_*$.

A scale $\mathcal {S}$ on $\Sigma (S)$ induces a family of “balls” $U^{(n)}(x, s)$ around $x \in X$ with “radius” $s$. One of the main concerns is the existence of a metric under which those “balls” are really balls, in other words, the existence of adapted metric according to the following definition.

Definition 2.4

Let $\mathcal {S}= \{\Lambda _s\}_{s \in (0, 1]}$ be a scale on $\Sigma$ associated with a gauge function $g$.

• (1) For $x \in K$, define $(\Lambda _s)_x^{(n)}$ and $U^{(n)}_{\mathcal {S}}(x, s)$ inductively by

  $$\begin{aligned} (\Lambda _s)_x^{(0)}&= \{w| w \in \Lambda _s, x \in K_w\}\\ U^{(n)}_{\mathcal {S}}(x, s)&= K\left( (\Lambda _s)_x^{(n)}\right) \\ (\Lambda _s)_x^{(n)}&= \{w| w \in \Lambda _s, K_w \cap U^{(n - 1)}_{\mathcal {S}}(x, s) \ne \emptyset \} \end{aligned}$$

• (2) A metric $d$ on $K$ is said to be adapted to the scale $\mathcal {S}$ if and only if there exist $\alpha , \beta > 0$ and $n \ge 1$ such that

  $$ B_d(x, {\alpha }s) \subseteq U^{(n)}_{\mathcal {S}}(x, s) \subseteq B_d(x, {\beta }s) $$

  for any $x \in K$ and any $s$.

The notion of gentleness between scales is introduced in [Kig09] as a part of the equivalence condition for a measure being volume doubling with respect to a scale. Roughly, if two scales are gentle with respect to each other, then the transition to one scale to the other is “smooth”.

Definition 2.5

Let $\mathcal {S}^g$ and $\mathcal {S}^l$ be scales on $\Sigma$ associated with gauge functions $g$ and $l$ respectively. We say $\mathcal {S}^l$ is gentle with respect to $\mathcal {S}^g$ if and only if there exists $c > 0$ such that $l(w) \le cl(v)$ whenever $w, v \in \Lambda _s$ for some $s > 0$ and $K_w \cap K_v \ne \emptyset$. We write $\mathcal {S}^g \underset{\mathrm{GE}}{\sim }\mathcal {S}^l$ if $\mathcal {S}^l$ is gentle with respect to $\mathcal {S}^g$.

Proposition 2.6

Among elliptic scales, i.e. scales whose gauge functions are elliptic, $\underset{\mathrm{GE}}{\sim }$ is an equivalent relation. In particular, if $g$ and $l$ are elliptic, then $\mathcal {S}^g \underset{\mathrm{GE}}{\sim }\mathcal {S}^l$ implies $\mathcal {S}^l \underset{\mathrm{GE}}{\sim }\mathcal {S}^g$.

There exists a natural “pseudo”metric associated with a scale which is defined by the infimum of the “length” of paths between two points.

Definition 2.7

• (1) A sequence $(w(1), \ldots , w(n))$ is called a path in $K$ if and only if $w(1), \ldots , w(n) \in W_*$, $K_{w(i)} \cap K_{w(i + 1)} \ne \emptyset$ for any $i = 1, \ldots , n - 1$. The collection of all the paths is denoted by $\mathcal {C}\mathcal {H}$. For $U,V\subseteq K$, a path $(w(1), \ldots , w(n))$ is called a path between $U$ and $V$ if and only if $K_{w(1)} \cap U \ne \emptyset$ and $K_{w(n)} \cap V \ne \emptyset$. We use $\mathcal {C}\mathcal {H}(U, V)$ to denote the collection of paths between $U$ and $V$. For two paths $\mathbf p_1 = (w(1), \ldots , w(n))$ and $\mathbf p_2 = (v(1), \ldots , v(m))$, if $K_{w(n)} \cap K_{v(1)} \ne \emptyset$, we define $\mathbf p_1 \vee \mathbf p_2 \in \mathcal {C}\mathcal {H}$ by $\mathbf p_1 \vee \mathbf p_2 = (w(1), \ldots , w(n), v(1), \ldots , v(m))$.

• (2) Let $\mathcal {S}$ be a scale on $\Sigma$ associated with a gauge function $g$. For any $x, y \in K$, we define

  $$ D_{\mathcal {S}}(x, y) = \inf \{\sum _{i = 1}^ng(w(i))|(w(1), \ldots , w(n)) \in \mathcal {C}\mathcal {H}(x, y)\}. $$

Remark

We identify a point $x \in X$ and a set $\{x\}$ if no confusion may occur.

Remark

We often use $D_{g}$ instead of $D_{\mathcal {S}}$ if $\mathcal {S}$ is the scale associated with a gauge function $g$.

Proposition 2.8

$D_{\mathcal {S}}$ is a pseudometric, i.e. $D_{\mathcal {S}}(x, y) = D_{\mathcal {S}}(y, x)$, $D_{\mathcal {S}}(x, y) \ge 0$, $D_{\mathcal {S}}(x, x) = 0$ and $D_{\mathcal {S}}(x, y) \le D_{\mathcal {S}}(x, z) + D_{\mathcal {S}}(z, y)$.

By [Kig09, Lemma 2.3.10], we have the following theorem, which says that a metric adapted to a scale $\mathcal {S}$, if such a metric exists at all, is essentially $D_{\mathcal {S}}$.

Theorem 2.9

Let $\mathcal {S}$ be a scale. There exists a metric $d$ on $K$ such that $d$ is adapted to $\mathcal {S}$ if and only if $D_{\mathcal {S}}$ is a metric on $K$ which is adapted to $\mathcal {S}$.

3 Quasisymmetric Metrics and Scales

In this section, we give an equivalent condition for two metrics on a self-similar set being quasisymmetric in terms of scales and related notions introduced in Sect. 2.

Let $(K, S, \{F_i\}_{i \in S})$ be a self-similar structure. Assume that $K \ne \overline{V_0}$. Hereafter in this section, every metric on $K$ is assumed to satisfy the following two properties:

1. It produces the same topology as the original topology of $K$.

2. The diameter of $K$ under it equals one.

The next lemma can be verified immediately by the definitions in the previous section.

Lemma 3.1

Let $\mathcal {S}_1 = \{\Lambda ^1_s\}$ and $\mathcal {S}_2 = \{\Lambda ^2_s\}$ be scales. If $\mathcal {S}_1 \underset{\mathrm{GE}}{\sim }\mathcal {S}_2$, then for any $n \ge 1$, there exists $c_n \in (0, 1)$ such that

$$ U^{(n)}_1(x, c_nt) \subseteq U^{(n)}_2(x, s) \subseteq U^{(n)}_1(x, t/c_n) $$

for any $x \in K$, any $(s, t)$ with $\Lambda _{t, x}^1 \cap \Lambda _{s, y}^2 \ne \emptyset$, where $U^{(n)}_i(x, s) = U^{(n)}_{S_i}(x, s)$ and $\Lambda _{t, x}^i = (\Lambda _s^i)_x^{(0)}$ for $i = 1, 2$.

First we define a scale associated with a metric.

Definition 3.2

Let $d$ be a metric on $K$ with $\mathrm{diam}(K, d) = 1$. Define $\mathcal {S}^d = \{\Lambda _s^d\}$ be the scale with the gauge function $d_w = \mathrm{diam}(K_w, d)$.

Lemma 3.3

Let $\mathcal {S}= \{\Lambda _s\}$ be an elliptic scale and let $d$ be a metric on $K$ which is adapted to $\mathcal {S}$. Let $l(w)$ be the gauge function of $\mathcal {S}$. Then

• (1) $d_w \asymp l(w)$ for any $w \in W_*$.

• (2) The pseudometric $D_{\mathcal {S}}$ associated with $\mathcal {S}$ is a metric and $\mathcal {D}_{\mathcal {S}}(x, y) \asymp d(x, y)$ for any $x, y \in K$.

• (3) $\mathcal {S}^d$ is elliptic and $d$ is adapted to $\mathcal {S}^d$.

Proof

Write $U^{(n)}(x, r) = U^{(n)}_{\mathcal {S}}(x, r)$. Since $d$ is adapted to $\mathcal {S}$, we have

$$\begin{aligned} U^{(n)}(x, {\beta }s) \subseteq B_d(x, s) \subseteq U^{(n)}(x, {\alpha }r) \end{aligned}$$

(3.2)

• (1) For $w \in W_*$, $U^{(n)}(x, l(w)) \subseteq B_d(x, {\alpha }l(w))$. Hence $d_w \le {\alpha }l(w)$. Now by [Kig09, Lemma 1.3.12], there exists $y \in K_w$ and $\gamma \in (0, 1)$ such that $U^{(n)}(y, {\gamma }l(w)) \subseteq K_w$. Hence $B_d(x, {\beta }{\gamma }l(w)) \subseteq K_w$. Since $K$ is connected, we have ${\beta }{\gamma }l(w) \le d_w$.

• (2) This is immediate from Theorem 2.9.

• (3) These claims are immediate from (1) and Lemma 3.1. $\square$

Now we present one of the main results of this paper. The following theorem gives an equivalent condition for certain metrics on a self-similar set being quasisymmetic. It plays a crucial role in the proof of Theorem 5.3.

Theorem 3.4

Let $d$ be a metric on $K$ and let $\mathcal {S}= \{\Lambda _s\}$ be an elliptic scale. Assume that $d$ is adapted to $\mathcal {S}$. Let $\rho$ be a metric on $K$. Then $d \underset{\mathrm{QS}}{\sim }\rho$ if and only if $\mathcal {S}^{\rho }$ is elliptic, $\mathcal {S}\underset{\mathrm{GE}}{\sim }\mathcal {S}^{\rho }$ and $\rho$ is adapted to $\mathcal {S}^{\rho }$.

The rest of this section is devoted to the proof of Theorem 3.4.

Proof

First we show $\Rightarrow$. Assume $d \underset{\mathrm{QS}}{\sim }\rho$. By Lemma 3.3, we may regard the gauge function of $\mathcal {S}$ is $d_w$ and hence $\mathcal {S}= \mathcal {S}^d$.

By the results in [Kig00, Part 2], $d \underset{\mathrm{QS}}{\sim }\rho$ is equivalent to the facts that there exists $\delta \in (0, 1)$ such that

$$\begin{aligned} \begin{array}{ll} B_d(x, r) &{}\supseteq B_{\rho }\left( x, \delta \overline{\rho }_d(x, s)\right) \\ B_{\rho }(x, r) &{}\supseteq B_d\left( x, \delta \overline{d}_{\rho }(x, r)\right) \end{array} \end{aligned}$$

(3.3)

and

$$\begin{aligned} \overline{\rho }_d(x, r/2)&\ge \delta \overline{\rho }_d(x, r) \\ \overline{d}_{\rho }(x, r/2)&\ge \delta \overline{d}_{\rho }(x, r), \end{aligned}$$

(3.4)

where $\overline{\rho }_d(x, r) = \sup _{y \in B_d(x, r)} \rho (x, y)$ and $\overline{d}_d(x, r) = \sup _{y \in B_{\rho }(x, r)}d(x, y)$.

First we show the following claim.

Claim 1 Let $w \in \Lambda _s^d$. Then there exists $z \in K_w$ such that $\rho _w \ge c\overline{\rho }_d(z, s)$, where c is a constant which is independent of w and s.

Proof of Claim 1 By [Kig09, Lemma 1.3.12] and (3.3), we may find $z \in K_w$ such that

$$ K_w \supseteq U^{(n)}_d(z, {\gamma }s) \supseteq B_d(z, {\gamma }s/\alpha ) \supseteq B_{\rho }\left( z, \delta \overline{\rho }_d(z, {\gamma }s/\alpha )\right) $$

Hence by (3.4)

$$ \rho _w \ge \delta \overline{\rho }_d(z, {\gamma }s/\alpha ) \ge c\overline{\rho }_d(z, s). $$

$\square$

Step 1: $\mathcal {S}^{\rho }$ is elliptic.

Proof of “$\rho _{wi} \ge c\rho _w$ for any $w \in W_*$ and any $i \in S$:

By Claim 1, it follows that

$$\begin{aligned} \rho _{wi} \ge c'\overline{\rho }_d(z, d_{wi}) \ge c'\overline{\rho }_d(z, 2d_{wi}). \end{aligned}$$

(3.5)

for some $z \in K_{wi}$”. On the other hand,

$$ K_w \subseteq B_d(z, 2d_w) \subseteq B_{\rho }(z, \rho _d(x, 2d_w)). $$

Hence

$$ \rho _w \le \overline{\rho }_d(z, 2d_w). $$

This with (3.5) suffices.

Proof of “there exists $c \in (0, 1)$ and $m$ such that $\rho _{wv} \le c\rho _w$ for any $w \in W_*$ and any $v \in W_m$”.

Since $K_{wv} \subseteq B_{\rho }(x, \overline{\rho }_d(x, 2d_{wv}))$, we have

$$\begin{aligned} \rho _{wv} \le \overline{\rho }_d(x, 2d_{wv}) \le \delta \overline{\rho }_d(x, d_{wv}), \end{aligned}$$

(3.6)

where $x \in K_{wv}$. On the other hand, by [Kig09, Lemma 1.3.12], there exists $z \in K_w$ such that

$$ K_w \supseteq U^{(n)}_d(x, {\gamma }d_w) \supseteq B_{\rho }\left( x, \delta \overline{\rho }_d(x, {\gamma }d_w)\right) . $$

Hence

$$\begin{aligned} \rho _w \ge \delta \overline{\rho }_d(x, \gamma {d_w}) \ge \delta '\overline{\rho }_d(x, d_w) \end{aligned}$$

(3.7)

Now, since $\mathcal {S}^d$ is elliptic, there exists $a \in (0, 1)$ such that

$$ d_{wv} \le ca^{|v|}d_w $$

for any $w$ and $v$. Hence by (3.6) and (3.7), the uniform decay of $\rho$ with respect to $d$, (See [Kig00, Proposition 10.7]),

$$ \rho _{wv} \le \delta \overline{\rho }_d(x, d_{wv}) \le \delta \overline{\rho }_d(x, ca^{|v|}d_w) \le cb^{|v|}\overline{\rho }_d(x, d_w) \le c'b^{|v|}\rho _w, $$

where $b \in (0, 1)$. Hence choosing sufficiently large $m = |v|$, we obtain the desired inequality.

Thus we have shown that $\mathcal {S}^{\rho }$ is elliptic.

Step 2: $\mathcal {S}\underset{\mathrm{GE}}{\sim }\mathcal {S}^{\rho }$

Let $w, v \in \Lambda _s^d$ with $K_w \cap K_v \ne \emptyset$. Choose $x \in K_w$ and $y \in K_v$. Then $d(x, y) \le 2s$ and hence $B_d(x, 3s) \supseteq B_d(y, s)$. This implies $\overline{\rho }_d(x, 3s) \ge \overline{\rho }_d(y, s)$. By (3.4),

$$ \overline{\rho }_d(x, s) \asymp \overline{\rho }_d(y, s). $$

By Claim 1, choosing $y \in K_v$ properly, we see that $\rho _v \ge c\overline{\rho }_d(y, s)$. Since $\overline{\rho }_d(x, 2s) \ge \rho _w$, (3.4) shows that $\mathcal {S}^d \underset{\mathrm{GE}}{\sim }\mathcal {S}$.

Step 3: $\rho$ is adapted to $\mathcal {S}^{\rho }$.

Let $x \in K$ and let $w \in \Lambda _{r, x}^d \cap \Lambda _{s, x}^{\rho }$. Then by Lemma 3.1, (3.3) and (3.4),

$$\begin{aligned} U^{(n)}_{\rho }(x, cs) \supseteq U^{(n)}_{d}(x, r)\,&\supseteq B_d(x, r/\alpha ) \supseteq B_{\rho }\left( x, \delta \overline{\rho }_d(x, r/\alpha )\right) \\&\supseteq B_{\rho }\left( x, \delta '\overline{\rho }_d(x, 2r)\right) \supseteq B_{\rho }(x, \delta '\rho _w) \supseteq B_{\rho }(x, \delta ''s). \end{aligned}$$

On the other hand, let $x \in K$ and let $w \in \Lambda _s^{\rho } \cap \Lambda _t^{d}$. Then

$$\begin{aligned} B_{\rho }(x, s) \supseteq B_d\left( x, \delta \overline{d}_{\rho }(x, s)\right) \supseteq U^{(n)}_d\left( x, \beta \delta \overline{d}_{\rho }(x, s)\right) \supseteq U^{(n)}_{\rho }(x, c'r), \end{aligned}$$

(3.8)

where $wv \in \Lambda _{\beta \delta \overline{d}_{\rho }(x, s), x}^d \cap \Lambda _{r, x}^{\rho }$. Since $B_{\rho }(x, 2s) \supseteq K_w$, we see that $\overline{d}_{\rho }(x, 2s) \ge d_w$. Hence $\overline{d}_{\rho }(x, s) \ge c_1d_w$. Consequently, $d_{wv} \ge c_2d_w$, where $c_2$ is independent of $w$ and $v$. This implies that $|v|$ is uniformly bounded. Since $\mathcal {S}^{\rho }$ is elliptic, $\rho _{wv} \ge c_3\rho _w$. This implies $U^{(n)}_{\rho }(x, c'r) \supseteq U^{(n)}_{\rho }(x, c_4s)$. By (3.8), it follows that $B_{\rho }(x, s) \supseteq U^{(n)}_{\rho }(x, c_5s)$. Thus we have shown that $\rho$ is adapted to $\mathcal {S}^{\rho }$.

This concludes the proof of $\Rightarrow$. $\square$

To show the converse direction of Theorem 3.4, we need the following lemma.

Lemma 3.5

Assume that $d$ is adapted to $\mathcal {S}^d$. Then, for any $n$ and $k$, there exists $\lambda \in (0, 1)$ such that

$$ U^{(n)}_d(x, r) \supseteq U^{(n + k)}_d(x, \lambda {r}) $$

for any $x \in K$ and any $r$.

Proof

Since $d$ is adapted to $\mathcal {S}^d$, there exists $c > 0$ such that $U^{(n)}_d(x, r) \supseteq B_d(x, cr)$. Then $B_d(x, cr) \supseteq U^{(n + k)}_d(x, cr/(n + k + 2))$. $\square$

Proof

(of $\Leftarrow$ of Theorem 3.4) Since $d$ and $\rho$ are adapted to $\mathcal {S}^d$ and $\mathcal {S}^{\rho }$ respectively,

$$\begin{aligned} U^{(n)}_d(x, \beta _1r)\,&\subseteq B_d(x, r) \subseteq U^{(n)}_d(x, \alpha _1r)\\ U^{(m)}_{\rho }(x, \beta _2r)\,&\subseteq B_{\rho }(x, r) \subseteq U^{(m)}_{\rho }(x, \alpha _2r). \end{aligned}$$

First we show (3.3). By Lemma 3.1,

$$\begin{aligned} B_d(x, r) \subseteq U^{(n)}_d(x, {\alpha _1}r) \subseteq U^{(n)}_{\rho }(x, c\rho _w), \end{aligned}$$

(3.9)

where $w \in \Lambda _{{\alpha _1}r, x}^d$. Using Lemma 3.5 if necessary, we obtain

$$ B_d(x, r) \subseteq U^{(m)}_{\rho }(x, c_1\rho _w) \subseteq B_{\rho }(x, c_2\rho _w). $$

Hence $\overline{\rho }_d(x, r) \le c_2\rho _w$. Now by Lemma 3.1,

$$\begin{aligned} B_d(x, r) \supseteq U^{(n)}_d(x, \beta _1r) \supseteq U^{(n)}_{\rho }(x, c'\rho _{wv}), \end{aligned}$$

(3.10)

where $wv \in \Lambda _{\beta _1r, x}^d$. By making use of Lemma 3.5 if necessary, we have

$$ B_d(x, r) \supseteq U^{(m)}_{\rho }(x, c''\rho _{wv}) \supseteq B_{\rho }(x, c''\beta _2\rho _{wv}). $$

Since $\mathcal {S}^d$ is elliptic, the fact that $w \in \Lambda _{\alpha _1r, x}^d$ and $wv \in \Lambda _{\beta _1r, x}^d$ implies that $|v|$ is uniformly bounded with respect to $x$ and $r$. Since $\mathcal {S}^{\rho }$ is also elliptic, we see that $\rho _wv \ge c_3\rho _w \ge c_4\overline{\rho }_d(x, r)$. Hence (3.3) holds. (By exchanging $\rho$ and $d$, we also obtain the other one.)

Next we show (3.4). By (3.9),

$$ \overline{\rho }_d(x, r) \le c(n + 1)\rho _w, $$

where $w \in \Lambda _{\alpha _1r}^d$. Replacing $r$ by $\lambda {r}$ for $\lambda \in (0, 1)$ in (3.10), we have

$$ B_d(x, \lambda {r}) \supseteq U^{(n)}(x, c'\rho _{wv}), $$

where $wv \in \Lambda _{\lambda \beta _1r, x}^d$. This implies $\overline{\rho }_d(x, \lambda {r}) \ge c'\rho _{wv}$. The same arguments as above show that $|v|$ is uniformly bounded and $\rho _{wv} \ge c\rho _w$. Combining all these, we obtain

$$ \overline{\rho }_d(x, \lambda {r}) \ge c'\rho _{wv} \ge c''\rho _w \ge c'''\overline{\rho }_d(x, r). $$

Again the other one is obtained by exchanging $d$ and $\rho$. Thus we have obtained (3.4). $\square$

4 Sierpinski Carpet and Its Invisible Sets

In this and the following sections, we are going to apply Theorem 3.4 to the Sierpinski carpet. First we give the definition of the Sierpinski carpet.

Definition 4.1

Let $S = \{\swarrow , \downarrow , \searrow , \rightarrow , \nearrow , \uparrow , \nwarrow , \leftarrow \}$. Define $p_{\swarrow } = -1 - \sqrt{-1}$, $p_{\downarrow } = -\sqrt{-1}$, $p_{\searrow } = 1 - \sqrt{-1}$, $p_{\rightarrow } = 1$, $p_{\nearrow } = 1 + \sqrt{-1}$, $p_{\uparrow } = \sqrt{-1}$, $p_{\nwarrow } = -1 + \sqrt{-1}$ and $p_{\leftarrow } = -1$. Moreover, define $F_s : \mathbb {C}\rightarrow \mathbb {C}$ for $s \in S$ by

$$ F_s(z) = \frac{(z - p_s)}{3} + p_s. $$

The Sierpinski carpet $K$ is the unique non-empty compact set which satisfies

$$ K = \bigcup _{s \in S} F_s(K). $$

Let $d_E$ be the restriction of the Euclidean metric on the Sierpinski carpet $K$.

We consider $d_E$ as the standard metric on $K$ and are going to construct metrics which is quasisymmetric with respect to $d_E$. Obviously, the scale $\mathcal {S}_{d_E}$ associated with $d_E$ is elliptic and $d_E$ is adapted to the scale $\mathcal {S}_{d_E}$. In fact, the gauge function associated with $d_E$ is given by $3^{-|w|}$ for any $w \in W_*$.

Next we introduce notions and notations regarding the boundary of the Sierpinski carpet (Fig. 2).

Definition 4.2

• (1) Define $L = K \cap \{z| \mathrm{Re}\,z = -1\}$, $R = K \cap \{z| \mathrm{Re}\,z = 1\}$, $T = K \cap \{z| \mathrm{Im}\,z = 1\}$ and $B = K \cap \{z| \mathrm{Im}\,z = -1\}$. Let $H_w = F_w(H)$ for any $w \in W_*$ and any $H \in \{L, R, T, B\}$. Moreover define $\partial _m = \{L_w, R_w, T_w, B_w| w \in W_m\}$.

• (2) Define $L^m = \{\swarrow , \leftarrow , \nwarrow \}^m, R^m = \{\searrow , \rightarrow , \nearrow \}^m, T^m = \{\nwarrow , \uparrow , \nearrow \}^m$, $B^m = \{\swarrow , \downarrow , \searrow \}^m$ and $\delta _m = L^m \cup R^m \cup T^m \cup B^m$.

Remark

Recall that $K(A) = \cup _{w \in A} K_w$ for a subset $A \subseteq W_*$. The map $A \rightarrow K(A)$ can be regarded as a map from the subsets of $W_*$ to the subsets of $K$. In the case of the Sierpinski carpet, this map is injective, i.e. if $A \ne B$, then $K(A) \ne K(B)$. Therefore, if no confusion may occur, we identify $A \subseteq W_*$ with $K(A) \subseteq K$.

Fig. 2

Generation of the Sierpinski carpet

Note that $D_{d_E}(x, y) \ge 1$ for any $(x, y) \in (L \times R) \cup (T \times B)$. This fact may remain true even if you put $0$ as weights (length) of some pieces of $w$’s. Such a collection of $w$’s is called an invisible set, whose precise definition is given below.

Definition 4.3

• (1) Let

  $$ \mathcal {C}\mathcal {H}_m = \{(w(1), \ldots , w(n))|(w(1), \ldots , w(n)) \in \mathcal {C}\mathcal {H}, w(i) \in W_m\} $$

  and let $\mathcal {C}\mathcal {H}_m(U, V) = \mathcal {C}\mathcal {H}(U, V) \cap \mathcal {C}\mathcal {H}_m$ for $U, V \subseteq K$.

• (2) Let $A \subseteq W_m$. For $\mathbf p= (w(1), \ldots , w(n)) \in \mathcal {C}\mathcal {H}_m$, define

  $$ \ell _{A}(\mathbf p) = \frac{\#\{i| i = 1, \ldots , n, w(i) \notin A\}}{3^m} $$

• (3) Let $A \subseteq W_m$. $A$ is said to be an invisible set if and only if

  $$ \inf _{\mathbf p\in \mathcal {C}\mathcal {H}_m(L, R) \cup \mathcal {C}\mathcal {H}_m(T, B)} \ell _A(\mathbf p) \ge 1 $$

• (4) Let $A \subseteq W_m$. $A$ is said to be $+$-invariant if and only if $K(A)$ is symmetric with respect to both the real and imaginary axes.

Since $L^m$, $R^m$, $T^m$ and $B_m$ are the shortest paths, we have the following proposition.

Proposition 4.4

Let $A \subseteq W_m$. If $A$ is invisible, then $A \cap \delta _m = \emptyset$.

The next theorem is one of the fundamental property of an invisible set. It will play a key role in constructing a metric associated with an invisible set in the next section.

Theorem 4.5

Let $A \subseteq W_m$ be an invisible set and let $X \subseteq W_n$ be an invisible and $+$-invariant set. Then $AW_n \cup W_mX$ is an invisible set.

The rest of this section is devoted to the proof of Theorem 4.5.

Definition 4.6

• (1) Let $A \subseteq W_m$. Define $\partial _mA = \{F| F \in \partial _m, F \subseteq K(A) \cap \overline{K\backslash K(A)}\}$.

• (2) Define $f_{m, \rightarrow }(z) = z + 3^{-m}$, $f_{m, \leftarrow }(z) = z - 3^{-m}$, $f_{m, \uparrow }(z) = z + 3^{-m}\sqrt{-1}$ and $f_{m, \downarrow }(z) = z - 3^{-m}\sqrt{-1}$. Moreover, let $f_{m, \swarrow } = f_{m, \downarrow }\circ {f_{m, \leftarrow }}$, $f_{m, \searrow } = f_{m, \downarrow }{\circ }f_{m, \rightarrow }$, $f_{m, \nwarrow } = f_{m, \uparrow }{\circ }f_{m, \leftarrow }$ and $f_{m, \nearrow } = f_{m, \uparrow }{\circ }f_{m, \rightarrow }$.

• (3) Let $w \in W_m$. For $s \in S$, if there exists $w' \in W_m$ such that $f_{m, s}(K_w) = K_{w'}$, then define $(w)_s = w'$. Otherwise define $(w)_s = \%$, where $\%$ is used as the symbol which represents non-existence (Fig. 3).

Lemma 4.7

Let $F \in \partial _m$ and let $G \in \partial _m(W_m(F))$. If $X \subseteq W_n$ is invisible and $+$-invariant, then $\ell _{W_mX}(\mathbf p) \ge 3^{-m}$ for any $\mathbf p\in \mathcal {C}\mathcal {H}_{m + n}(F, G)$.

Proof

Note that $\#(W_m(F)) \le 6$. Up to parallel translations, the reflections in the real and the imaginary axes and the $\pi /2$-rotation, we may assume that $F = B_w$ for some $w \in W_m$. Then $W_m(F) \subseteq \{w, (w)_{\leftarrow }, (w)_{\swarrow }, (w)_{\downarrow }, (w)_{\searrow }, (w)_{\rightarrow }\}$, where some of them may be $\%$. In fact there are 7 cases. (See Fig. 4.)

 

Case 1:

$\#(W_m(F)) = 6$.

Case 2:

$\#(W_m(F)) = 5$ and $(w)_{\searrow } = \%$.

Case 3:

$\#(W_m(F)) = 5$ and $(w)_{\downarrow } = \%$.

Case 4:

$\#(W_m(F)) = 4$ and $(w)_{\downarrow } = (w)_{\searrow } = \%$.

Case 5:

$\#(W_m(F)) = 3$ and $(w)_{\downarrow } = (w)_{\searrow } = (w)_{\swarrow } = \%$.

Case 6:

$\#(W_m(F)) = 3$ and $(w)_{\leftarrow } = (w)_{\swarrow } = (w)_{\searrow } = \%$.

Case 7:

$\#(W_m(F)) = 2$ and $(w)_{\downarrow } = (w)_{\swarrow } = (w)_{\searrow } = (w)_{\leftarrow } = \%$.

 

Fig. 3

Structures of $W_m(F)$

Fig. 4

Construction of $\Updownarrow _n$, $\Uparrow _n$ and $\swarrow \!\!\!\!\swarrow \!\!\!_n$

We consider the first case. The other cases can be treated by the similar discussion. If $D = \cup _{U \in \partial _m(W_m(F))} U$, then $D = {\partial }K(W_m(F))$. Let $\mathbf p= (w(1), \ldots , w(k)) \in \mathcal {C}\mathcal {H}_{m + n}(F, G)$. The reflection in the line containing $F$ induces a natural bijection from $W_m(F)\cdot {W_n}$ to itself, which is denoted by $\eta$. Define $\theta :W_m(F)\cdot {W_n} \rightarrow \{(w)_{\leftarrow }, w, (w)_{\rightarrow }\}\cdot {W_n}$ by

$$ \theta (uv) = {\left\{ \begin{array}{ll} uv &{}\text {if}\; u \in \{(w)_{\leftarrow }, w, (w)_{\rightarrow }\}\;\text {and}\;v \in W_n,\\ \eta (uv) &{}\text {if}\; u \in \{(w)_{\swarrow }, (w)_{\downarrow }, (w)_{\searrow }\}\;\text {and}\;v \in W_n . \end{array}\right. }$$

Define $v(i) = \theta (w(i))$ and $\widetilde{\mathbf p} = (v(1), \ldots , v(k))$. Then the $+$-invariant property of $X$ implies that $\widetilde{\mathbf p} \in \mathcal {C}\mathcal {H}_{m + n}(F, D_1)$, where $D_1 = L_{(w)_{\leftarrow }} \cup T_{(w)_{\leftarrow }} \cup T_w \cup T_{(w)_{\rightarrow }} \cup R_{(w)_{\rightarrow }}$, and

$$ \ell _{W_mX}(\mathbf p) = \ell _{W_mX}(\widetilde{\mathbf p}), $$

If $v(k) \cap L_{(w)_{\leftarrow }} \ne \emptyset$, then there exists $j$ such that $(v(j), v(j + 1), \ldots , v(k)) \in \mathcal {C}\mathcal {H}_m(R_{(w)_{\leftarrow }}, L_{(w)_{\leftarrow }})$ and $K_{v(i)} \subseteq (w)_{\leftarrow }\cdot {W_n}$ for any $i \in \{j, j + 1, \ldots , k\}$. Since $X$ is invisible, it follows that

$$ \ell _{W_mX}(\mathbf p) \ge \ell _{W_mX}((v(j), \ldots , v(k))) \ge 3^{-m}. $$

The same discussion shows that $\ell _{W_mX}(\mathbf p) \ge 3^{-m}$ if $K_{v(k)} \cap R_{(w)_{\rightarrow }} \ne \emptyset$.

Next suppose $v(K) \cap \left( T_{(w)_{\leftarrow }} \cup T_w \cup T_{(w)_{\rightarrow }}\right) \ne \emptyset$. Then using the reflections in the lines containing $L_w$ and $R_w$, we may construct $(u(1), \ldots , u(k)) \in \mathcal {C}\mathcal {H}_{m + n}(B_w, T_w)$ which satisfies $u(i) \in w\cdots {W_n}$ for any $i$ and $\ell _{W_mX}(\mathbf p) = \ell _{W_mX}((u(1), \ldots , u(k)))$. Since $X$ is invisible, it follows that $\ell _{W_mX}(\mathbf p) \ge 3^{-m}$. $\square$

Lemma 4.8

Let $F, G \in \partial _m$ with $F \cap G = \emptyset$ and let $\mathbf p= (w(1), \ldots , w(k)) \in \mathcal {C}\mathcal {H}_{m + n}(F, G)$. If $\{w(i)\}_{i = 1}^k \cap AW_n = \emptyset$, then there exists $\mathbf p_* \in \mathcal {C}\mathcal {H}_m(F, G)$ such that $\ell _A(\mathbf p_*) \le \ell _{AW_n \cup W_mX}(\mathbf p)$.

Proof

Let $k_1 = \max \{j| \{w(i)\}_{i = 1}^j \subseteq W_{m + n}(F)\}$. Define $v(1) = [w(k_1)]_m$. Note that $v(1) \notin A$. There exists a unique $F_1 \in \{L_{v(1)}, R_{v(1)}, T_{v(1)}, B_{v(1)}\} \cap \partial _m(W_m(F))$ such that $K_{w(k_1)} \subseteq F_1$. By Lemma 4.7,

$$ \ell _{AW_n \cup W_mX}((w(1), \ldots , w(k_1))) \ge 3^{-m} = \ell _A((v(1))). $$

Now, if $F_1 \cap G \ne \emptyset$, $(v(1)) \in \mathcal {C}\mathcal {H}_m(F, G)$ and $\ell _{AW_n \cup W_mX}(\mathbf p) \ge \ell _A((v(1))$. Hence we have constructed $\mathbf p_* = (v(1))$. Otherwise, replacing $(w(1), \ldots , w(k))$ and $F$ by $(w(k_1), \ldots , w(k))$ and $F_1$ respectively, we repeat the same procedure as above and obtain $k_2$, $v(2)$ and $F_2$. Inductively, we have $\mathbf p_* = (v(1), \ldots , v(l))$ with the desired properties. $\square$

Lemma 4.9

Let $F, G \in \partial _m$ with $F \cap G = \emptyset$. Then for any $\mathbf p\in \mathcal {C}\mathcal {H}_{m + n}(F, G)$, there exists $\mathbf p_* \in \mathcal {C}\mathcal {H}_m(F, G)$ such that $\ell _A(\mathbf p_*) \le \ell _{AW_n \cup W_mX}(\mathbf p)$.

Proof

Let $\mathbf p= (w(1), \ldots , w(k))$. If $w(i) \notin AW_n$ for any $i$, then Lemma 4.8 suffices. Hence we assume that there exists $i$ such that $w(i) \in AW_n$.

Claim 1 Without loss of generality, we may assume that there exists $p_1 \ge 1$ and $G_1 \in \partial _m$ such that $w(1), \ldots , w(p_1) \in W_{m + n}\backslash AW_n$, $w(p_1 + 1) \in AW_n$, $G_1 \cap F = \emptyset$, $G_1 \subseteq K_{[w(p_1 + 1)]_m}$ and $(w(1), \ldots , w(p_1)) \in \mathcal {C}\mathcal {H}_{m + n}(F, G_1)$.

Proof of Claim 1 Case 1: $F \cap K(A) = \emptyset$

In this case, define

$$ p_1 = \min \{i|w(i) \in AW_n\} - 1 $$

and choose $G_1 \in \partial _m$ so that $G_1 \cap K_{w(p_1)} \cap K_{w(p_1 + 1)} \ne \emptyset$ and $G_1 \subseteq K_{[w(p_1)]_m}$.

Case 2: $F \cap K(A) \ne \emptyset$

In this case, $F$ intersects at most two connected components of $K(A)$. Let $C_1$ and $C_2$ be those connected components of $K(A)$ (It is possible that $C_1 = C_2$).

Case 2.1: $\{i| K_{w(i)} \subseteq C_1 \cup C_2\} = \emptyset$.

Define $p_1$ and choose $G_1$ as in Case 1. Then $p_1$ and $G_1$ satisfies the desired property.

Case 2.2: $\{i | K_{w(i)} \subseteq C_1 \cup C_2\} \ne \emptyset$.

Define

$$ q = \max \{i| K_{w(i)} \in C_1 \cup C_2\}. $$

We may choose $F_0 \in \partial _m$ so that $F_0 \cap K_{w(q)} \cap K_{w(q + 1)} \ne \emptyset$ and $F_0 \subseteq K_{[w(q)]_m}$. Moreover, we may choose $\mathbf p^0 = (v(1), \ldots , v(k_0)) \in \mathcal {C}\mathcal {H}_m(F, F_0)$ so that $v(i) \in AW_n$ for any $i = 1, \ldots , k_0$ and $v(k_0) = [w(q)]_m$. Note that $\ell _A(\mathbf p^0) = 0$. If $F_0 \cap G \ne \emptyset$, then $K_{v(k_0)} \cap G \ne \emptyset$ and $\mathbf p^0 \in \mathcal {C}\mathcal {H}_m(F, G)$. Hence letting $\mathbf p_* = \mathbf p^0$, we have constructed $\mathbf p_*$ which satisfies all the conditions. Assume that $F_0 \cap G = \emptyset$. Since $(w(1), \ldots , w(q)) \in \mathcal {C}\mathcal {H}_{m + n}(F, F_0)$ corresponds $\mathbf p^0 \in \mathcal {C}\mathcal {H}_m(F, F_0)$, it is enough to show the statement of the lemma in the case where $F$ and $\mathbf p$ are replaced by $F_0$ and $(w(q + 1), \ldots , w(k))$ respectively. In this situation, the counterpart of Case 2.1 holds and so does Claim 1 (End of Proof of Claim 1). $\square$

Claim 2 Without loss of generality, we may assume that there exists $k_*$ and $F_* \in \partial _m$ such that $w(k_*), \ldots , w(k) \in W_{m + n}\backslash AW_n$, $w(k_* - 1) \in AW_n$, $F_* \cap G = \emptyset$, $F_* \subseteq K_{w(k_*)}$ and $(w(k_*), \ldots , w(k)) \in \mathcal {C}\mathcal {H}_{m + n}(F_*, G)$.

Proof of Claim 2 By considering the chain $(w(k), w(k - 1), \ldots , w(1)) \in \mathcal {C}\mathcal {H}_{m + n} (G, F)$, the same argument as in the proof of Claim 1 yields this claim (End of Proof of Claim 2).

Now under Claim 1 and Claim 2, we may choose $p_1, \ldots , p_{j + 1}$ and $q_0, q_1, \ldots , q_j$ which satisfy the following conditions (A), (B), (C) and (D):

• (A) $q_0 = 0$, $p_{j + 1} = k$, $q_i < p_{i + 1} < q_{i + 1}$ for any $i$.

• (B) $\{(w(q_{i - 1} + 1), \ldots , w(p_i)\} \cap AW_n = \emptyset$ for any $i = 1, \ldots , j + 1$

• (C) $K_{w(p_i + 1)}$ and $K_{w(q_i)}$ belong to the same connected component of $K(A)$ for any $i = 1, \ldots , j$.

• (D) $K_{w(q_i)}$ and $K_{w(p_{i + 1} + 1)}$ belong to the different connected components of $K(A)$ for any $i = 1, 2, \ldots , j - 1$

Let $\mathbf p_i = (w(q_{i - 1} + 1), \ldots , w(p_i))$ for $i = 1, \ldots , j + 1$. Define $F_1 = F$. For $i \ge 2$, we may choose $F_i \in \partial _m$ so that $F_i \cap K_{w(q_{i - 1})} \cap K_{w(q_{i - 1} + 1)} \ne \emptyset$ and $F \subseteq K_{[w(q_{i - 1})]_m}$. Moreover, for $i = 1, \ldots , j$, we may choose $G_i \in \partial _m$ so that $G_i \cap K_{w(p_i)} \cap K_{w(p_i + 1)} \ne \emptyset$ and $G_i \subseteq K_{[w(p_i + 1)]_m}$. Also let $F_{j + 1} = G$. By the condition (D), $F_i \cap G_i = \emptyset$ for any $i = 1, \ldots , j + 1$. Hence letting $F = F_i, G = G_i$ and $\mathbf p= \mathbf p_i$ and applying Lemma 4.8, we obtain $\mathbf p_{*, i} = (v(i, 1), \ldots , v(i, k_i)) \in \mathcal {C}\mathcal {H}_m(F_i, G_i)$ which satisfies $\ell _A(\mathbf p_{*, i}) \le \ell _{AW_n \cup W_mX}(\mathbf p_i)$.

Note that $G_i$ and $F_i$ belong to the same connected component of $K(A)$ by the condition (C). Hence there exists $\mathbf p^1_i = (u(i, 1), \ldots , u(i, l_i)) \in \mathcal {C}\mathcal {H}_m(G_i, F_i)$ such that $u(i, 1), \ldots , u(i, l_i) \in A$.

Finally let $\mathbf p_* = (\mathbf p_{*, 1}, \mathbf p^1_1, \mathbf p_{*, 2}, \ldots , \mathbf p^1_j, \mathbf p_{*, j + 1})$. Then $\mathbf p_* \in \mathcal {C}\mathcal {H}_m(F, G)$ and $\ell _A(\mathbf p_*) \le \ell _{AW_n \cup W_mX}(\mathbf p)$. $\square$

Proof

of Theorem 4.5 Let $\mathbf p= (w(1), \ldots , w(k)) \in \mathcal {C}\mathcal {H}_{m + n}(L, R)$. Set $F = L_{[w(1)]_m}$ and $G = R_{[w(k)]_m}$. By Lemma 4.9, there exists $\mathbf p_* \in \mathcal {C}\mathcal {H}_m(F, G)$ such that $\ell _A(\mathbf p_*) \le \ell _{AW_n \cup W_mX}(\mathbf p)$. Since $A$ is invisible, we have $\ell _A(\mathbf p_*) \ge 1$. Hence $\ell _{AW_n \cup W_mX}(\mathbf p) \ge 1$. In the same way, if $\mathbf p' \in \mathcal {C}\mathcal {H}_{m + n}(T, B)$, it follows that $\ell _{AW_n \cup W_mX}(\mathbf p') \ge 1$. Thus $AW_n \cup W_mX$ is invisible. $\square$

5 Metric Associated with Invisible Set

In this section, we construct a metric associated with a $+$-invariant invisible set and characterize the Hausdorff dimension and the Hausdorff measure with respect to the metric.

Throughout this section, we fix a $+$-invariant invisible set $A \subseteq W_m$.

Notation We write $W_{m, n} = (W_m)^n = W_{mn}$, $W_{m, *} = \cup _{n \ge 0} W_{m, n}$ and $\Sigma ^{(m)} = (W_m)^{\mathbb {N}}$.

Naturally $W_{m, *}$ is regarded as a subset of $W_*$ and $\Sigma ^{(m)}$ is identified with $\Sigma$.

Definition 5.1

• (1) Let $\epsilon > 0$. Define $D_{\epsilon }^A(w)$ for $w \in W_m$ by

  $$ D_{\epsilon }^A(w) = {\left\{ \begin{array}{ll} 3^{-m} &{}\text {if}\,w \notin A ,\\ \epsilon &{}\text {if}\,w \in A. \end{array}\right. } $$

  and $D_{\epsilon }^A(\emptyset ) = 1$ for $\emptyset \in W_0$. For any $w = w^{(1)}\cdots {w^{(n)}} \in W_{m, n}$, where $w^{(i)} \in W_m$, define

  $$ D_{\epsilon }^A(w) = D_{\epsilon }^A(w^{(1)})D_{\epsilon }^A(w^{(2)}){\cdots }D_{\epsilon }^A(w^{(n)}). $$

• (2) Define

  $$\begin{aligned} \mathcal {C}\mathcal {H}^{(m)} = \{(w(1),&\ldots , w(k))| \\&(w(1), \ldots , w(k)) \in \mathcal {C}\mathcal {H}, w(i) \in W_{m, *} \,\,\,\text {for any}\,i = 1, \ldots , k\}. \end{aligned}$$

  and $\mathcal {C}\mathcal {H}^{(m)}(U, V) = \mathcal {C}\mathcal {H}(U, V) \cap \mathcal {C}\mathcal {H}^{(m)}$ for $U, V \subseteq K$. Moreover, define $\ell ^{A, \epsilon }(\mathbf p) = \sum _{i = 1}^k D_{\epsilon }^A(w(i))$ for any $\mathbf p= (w(1), \ldots , w(k)) \in \mathcal {C}\mathcal {H}^{(m)}$ and, for $x, y \in K$,

  $$ d_{\epsilon }^A(x, y) = \inf \{\ell ^{A, \epsilon }(\mathbf p)| \mathbf p\in \mathcal {C}\mathcal {H}^{(m)}(x, y)\}. $$

$D_{\epsilon }^A(\cdot )$ is a gauge function on $\Sigma ^{(m)}$ and $d_{\epsilon }^A$ is the associated pseudometric.

The next fact is obvious from the definition.

Proposition 5.2

$d_0^A(x, y) \le d_{\epsilon }^A(x, y)$ for any $x, y \in K$ and any $\epsilon > 0$.

The next theorem shows that $d_{\epsilon }^a$ is really a metric and $d_{\epsilon }^A \underset{\mathrm{QS}}{\sim }d_E$.

Theorem 5.3

For any $\epsilon > 0$, $d_{\epsilon }^A$ is a metric on $K$ which is quasisymmetric with respect to $d_E$. The Hausdorff dimension of $K$ with respect to the metric $d_{\epsilon }^A$, $\dim _H(K, d_{\epsilon }^A)$ is given by the unique $\alpha$ which satisfies

$$\begin{aligned} (8^m - \#(A))3^{-m{\alpha }} + \#(A)\epsilon ^{\alpha } = 1. \end{aligned}$$

(5.1)

Furthermore, let $\mathcal {H}^{\alpha }$ be the $\alpha$-dimensional Hausdorff measure on $(X, d_{\epsilon }^A)$. Then the metric measure space $(X, d_{\epsilon }^A, \mathcal {H}^{\alpha })$ is Ahlfors $\alpha$-regular, i.e.

$$ \mathcal {H}^{\alpha }(B_d(x, r)) \asymp r^{\alpha } $$

for any $x \in K$ and $r \in [0, \mathrm{diam}(X, d_{\epsilon }^A))$.

Letting $\epsilon \downarrow 0$ in (5.1), we obtain the following corollary.

Corollary 5.4

$$ \dim _{\mathcal {C}} (K, d_E) \le \frac{\log 8}{\log 3} + \frac{1}{m\log 3}\log {\Big (1 - \frac{\#(A)}{8^m}\Big )}. $$

In the rest of this section, we are going to prove the above theorem. Hereafter, we omit $A$ in the notations $D_{\epsilon }^A(w)$, $\ell ^{A, \epsilon }(\mathbf p)$ and $d_{\epsilon }^A(x, y)$ and write $D_{\epsilon }(w)$, $\ell ^{\epsilon }(\mathbf p)$ and $d_{\epsilon }(x, y)$ respectively.

Lemma 5.5

Define $A_n \subseteq W_{mn}$ inductively by $A_1 = A$ and

$$ A_{n + 1} = AW_{mn} \cup W_mA_n. $$

Then $A_n$ is $+$-invariant and invisible.

Proof

Letting $X = A_n$ and applying Theorem 4.5, we see inductively that $A_{n + 1}$ is $+$-invariant and invisible. $\square$

Lemma 5.6

$d_0^A(x, y) \ge 1$ for any $(x, y) \in (L \times R) \cup (T \times B)$.

Proof

Define $I(\mathbf p) = \max _{i = 1, \ldots , k} |w(i)|/m$ for any $\mathbf p= (w(1), \ldots , w(k)) \in \mathcal {C}\mathcal {H}^{(m)}(L, R)$. We are going to show that $\ell ^0(\mathbf p) \ge 1$ by an induction in $I(\mathbf p)$. If $I(\mathbf p) = 0$, then $\mathbf p= (\emptyset )$ and $\ell ^0(\mathbf p) = D_0(\emptyset ) = 1$. Let $J = \{i| i = 1, \ldots , k, |w(i)| = I(\mathbf p)m\}$. Then there exists $k_1, \ldots , k_l$ and $j_1, \ldots ,j_l$ such that $k_i \le j_i < k_{i + 1}$ and $J = \cup _{i = 1, \ldots , l} \{j| k_i \le j \le j_i\}$. Let $\mathbf p^i = (w(k_i), \ldots , w(j_i))$. Since $|w(k_i - 1)| \le (I(\mathbf p) - 1)m$ and $|w(j_i + 1) \le (I(\mathbf p) - 1)m$, there exist $F, G \in \partial _{M}$, where $M = (I(\mathbf p) - 1)m$, such that $F \subseteq K_{w(k_i - 1)}$, $F \cap K_{w(k_i)} \ne \emptyset$, $G \subseteq K_{w(j_i + 1)}$ and $G \cap K_{w(j_i)} \ne \emptyset$. If $F \cap G = \emptyset$, then $K_{w(k_i - 1)} \cap K_{w(j_i + 1)} \ne \emptyset$. Hence if $\mathbf p' = (w(1), \ldots , w(k_i - 1), w(j_i + 1), \ldots , w(k)) \in \mathcal {C}\mathcal {H}^{(m)}(L, R)$, then we define $\mathbf p_*^i$ as the empty sequence. Note that $\ell ^0(\mathbf p) \ge \ell ^0(\mathbf p')$. Now assume that $F \cap G = \emptyset$. Set $X = A_M$. Lemma 5.5 shows that $X$ is $+$-invariant and invisible. Then by Lemma 4.9, there exists $\mathbf p_*^i = (v(1), \ldots , v(l)) \in \mathcal {C}\mathcal {H}_{M}(F, G)$ such that $\ell _{A_M}(\mathbf p_*^i) \le \ell _{A_MW_m \cup W_{M}A}(\mathbf p^i)$. Note that $A_MW_m \cup W_MA = A_{I(\mathbf p)m}$, that $\ell _{A_M}(\mathbf p_*^i) = \ell ^0(\mathbf p_*^i)$ and that $\ell _{A_MW_m \cup W_MA}(\mathbf p^i) = \ell ^0(\mathbf p^i)$. Let $\mathbf p_*$ be the chain where $\mathbf p^i$ is replaced by $\mathbf p_*^i$ for all $i$. Then $\mathbf p_* \in \mathcal {C}\mathcal {H}^{(m)}(L, R)$, $I(\mathbf p_*) < I(\mathbf p)$ and $\ell ^0(\mathbf p) \ge \ell ^0(\mathbf p_*)$. Now we have $\ell ^0(\mathbf p) \ge \ell ^0(\mathbf p_*) \ge 1$ by induction hypothesis.

Now, $d_0^A(x, y) \ge \inf \{\ell ^0(\mathbf p)|\mathbf p\in \mathcal {C}\mathcal {H}^{(m)}(x, y)\} \ge 1$ for any $x \in L$ and any $y \in R$. In the same manner, it follows that $d_0^A(x, y) \ge 1$ for any $x \in T$ and any $y \in B$ as well. $\square$

Lemma 5.7

$d^A_{\epsilon }(\cdot , \cdot )$ is a metric on $K$ for any $\epsilon > 0$.

Proof

Let $x, y \in K$ with $x \ne y$. Then $\mathrm{Re}\,x \ne \mathrm{Re}\,y$ or $\mathrm{Im}\,x \ne \mathrm{Im}\,y$. Suppose $\mathrm{Re}\,x < \mathrm{Re}\,y$. Then there exist $n$ and $i \in \{0, 1, \ldots , 3^{mn} - 1\}$ such that $\mathrm{Re}\,x \le (2i - 3^{mn})3^{-mn} < (2i + 2 - 3^{mn})3^{-mn}\le \mathrm{Re}\,y$.

Claim $d_{\epsilon }^A(x, y) \ge \min \{D_{\epsilon }^A(w)| w \in W_{m, k}, k = 0, 1, \ldots , n\}$.

Proof of Claim Define $W_{m, n}^i = \{w|w \in W_{m, n}, K_w \subseteq \{z | (2i - 3^{mn})3^{-mn} \le \mathrm{Re}\,z \le (2i + 2 - 3^{mn})3^{-mn}\}$. Let $D_{mn, i} = \min \{D_{\epsilon }(w)| w \in W_{m, n}^i\}$. We also define $L_{mn, i} = \cup _{w \in W^i_{m, n}} L_w$ and $R_{mn, i} = \cup _{w \in W^i_{m, n}} R_w$. Let $\mathbf p= (w(1), \ldots , w(k)) \in \mathcal {C}\mathcal {H}^{(m)}(x, y)$. If $|w(i)| \le mn$ for some $i$, then the claim is trivial. Hence assume that $|w(i)| < mn$ for any $i = 1, \ldots , k$. Then $\mathbf p$ contains $(w(p), w(p + 1), \ldots , w(q)) \in \mathcal {C}\mathcal {H}(L_{mn, i}, R_{mn, i})$ which satisfies $w(i) \in \cup _{w \in W^i_{m, n}} wW_{m, *}$. Let $w(i) = u(i)v(i)$ for $i = p, \ldots , q$, where $u(i) \in W^i_{m, n}$ and $v(i) \in W_{m, *}$. It follows that

$$\begin{aligned} \ell ^{A, \epsilon }(\mathbf p) \ge \ell ^{A, \epsilon }((w(p), w(p + 1), \ldots , w(q))) \ge D_{mn, i}\sum _{i = p}^q D_{\epsilon }(v(i)). \end{aligned}$$

(5.2)

Now the reflection $\psi$ in the real axis induces a natural bijection $\varphi _{\leftrightarrow } :W_* \rightarrow W_*$ defined by $\psi (K_w) = K_{\varphi _{\leftrightarrow }(w)}$ which satisfies $\varphi _{\leftrightarrow }(\varphi _{\leftrightarrow }(w)) = w$. Hereafter in this section, we write $\varphi = \varphi _{\leftrightarrow }$. There exist $p_1, p_2, \ldots p_l$ such that $p_1 = p, p_l = q + 1$, $p_i < p_{i + 1}$, $u(p_i) = u(p_i + 1) = \ldots = u(p_{i + 1} - 1)$ and $u(p_i) \ne u(p_{i + 1})$ for any $i$. Let $\bar{v}(j) = \varphi ^i(v(j))$ for $j = p_i, p_i + 1, \ldots , p_{i + 1} - 1$, where $\varphi ^j$ is the $j$-th iteration of $\varphi$. Then $(\bar{v}(p), \bar{v}(p + 1), \ldots , \bar{v}(q)) \in \mathcal {C}\mathcal {H}^{(m)}(L, R)$. Since $A$ is $+$-invariant, $\sum _{i = p}^q D_{\epsilon }(v(i)) = \sum _{i = p}^q D_{\epsilon }(\bar{v}(i))$. Hence Lemma 5.6 implies that

$$ \sum _{i = p}^q D_{\epsilon }(v(i)) = \sum _{i = p}^q D_{\epsilon }(\bar{v}(i)) \ge \sum _{i = p}^q D_0(\bar{v}(i)) \ge 1. $$

Combining this with (5.2), we have $\ell ^{A, \epsilon }(\mathbf p) \ge D_{mn, i}$. Hence the claim holds (End of Proof of Claim).

The claim shows that $d^A_{\epsilon }(x, y) > 0$ if $\mathrm{Re}\,x \ne \mathrm{Re}\,y$. Similar discussion implies $d^A_{\epsilon }(x, y) > 0$ if $\mathrm{Im}\,x \ne \mathrm{Im}\,y$. $\square$

Proof

of Theorem 5.3 Let $\mathcal {S}^{(m)}(A, \epsilon ) = \{\Lambda ^{(m)}_s(A, \epsilon )\}_{s \in (0, 1]}$ be the scale on $\Sigma ^{(m)}$ whose gauge function is $D_{\epsilon }^A$ and let $\mathcal {S}^{(m)}$ by the scale on $\Sigma ^{(m)}$ whose gauge function $g$ is given by $g(w(1)\ldots {w(k)}) = 3^{-mk}$ for $w(1)\ldots {w(k)} \in W_{m, *}$ with $w(1), \ldots , w(k) \in W_m$. Obviously $\Sigma ^{(m)}$ is adapted to the Euclidean metric on $K$. Also since $\mathcal {S}^{(m)}(A, \epsilon )$ and $\mathcal {S}^{(m)}$ are self-similar, they are elliptic.

Note that $(K, W_m, \{F_w\}_{w \in W_m})$ is a rationally ramified self-similar structure. (See [Kig09, Sect. 1.5] for the definition of rationally ramified self-similar structures.) In fact, define $h:L^1 \rightarrow R^1$ by $h(\nwarrow ) = \nearrow , h(\leftarrow ) = \rightarrow , h(\swarrow ) = \searrow$ and $g: T^1 \rightarrow B^1$ by $g(\nwarrow ) = \swarrow , g(\uparrow ) = \downarrow , g(\nearrow ) = \searrow$. Then define $h_m : L^m \rightarrow R^m$ by $h_m(w_1\ldots {w_m}) = h(w_1)\ldots {h(w_m)}$ for $w_1\ldots {w_m} \in L^m$ and $g_m: T^m \rightarrow B^m$ by $g_m(w_1{\ldots }w_m) = g(w_1){\ldots }g(w_m)$ for $w_1\ldots {w_m} \in T^m$. Then a relation set $\mathcal {R}_m$ of $(K, W_m, \{F_w\}_{w \in W_m})$ is given by

$$\begin{aligned} \mathcal {R}_m = \{(L^m, R^m, h_m, w, v)| w, v \in&W_m, L_w = R_v\} \cup \\&\{(T^m, B^m, g_m, w, v)| w, v \in W_m, T_w = B_v\} \end{aligned}$$

By Proposition 4.4, $D_{\epsilon }^A(w) = 3^{-m}$ for any $w \in L^m \cup R^m \cup T^m \cup B^m$. Using [Kig09, Theorem 1.6.6], we see that $\mathcal {S}^{(m)}(A, \epsilon ) \underset{\mathrm{GE}}{\sim }\mathcal {S}^{(m)}$.

Theorems 1.6.1 and 2.2.7 in [Kig09] imply that $\mathcal {S}^{(m)}(A, \epsilon )$ is intersection type finite. Since $d_{\epsilon }^A$ is a metric on $K$ by Lemma 5.7, we may apply [Kig09, Theorem 2.3.16] and show that $d_{\epsilon }^A$ is adapted to the scale $\mathcal {S}^{(m)}(A, \epsilon )$. Thus we have obtained all the conditions in Theorem 3.4 and hence shown that $d_{\epsilon }^A$ is quasisymmetric with respect to the Euclidean metric.

The Hausdorff dimension and Ahlfors regularity of the Hausdorff measure of $(K, d_{\epsilon }^A)$ are immediately obtained by [Kig01, Theorem 1.5.7]. $\square$

6 Construction of Invisible Sets

Under the existence of an invisible set, we have constructed a corresponding metric which is quasisymmetric with respect to $d_E$ and characterized the associated Hausdorff dimension in the previous two sections. In this section, it is shown that invisible sets do exist. In fact, we construct a series of invisible sets inductively.

Definition 6.1

Let $\psi _{\updownarrow }$ and $\psi _{\leftrightarrow }$ be the reflections in the real and complex axes respectively. Then $\psi _{\updownarrow }$ induces a natural bijection $\varphi _{\updownarrow }$ from $W_*$ to itself defined by $\psi _{\updownarrow }(K_w) = K_{\varphi _{\updownarrow (w)}}$. In the same way, we define a bijection $\varphi _{\leftrightarrow }$ from $W_*$ to itself by $\psi _{\leftrightarrow }(K_w) = K_{\varphi _{\leftrightarrow }(w)}$. Moreover, let $R$ be the $\pi /2$-rotation around the origin $0$ and let $\rho : W_* \rightarrow W_*$ be the bijection defined by $R(K_w) = K_{\rho (w)}$.

The idea to have invisible sets is to divide the notion of a invisible set into a vertically invisible set and a horizontally invisible set. The final existence of invisible sets are established by taking intersections of vertically invisible set and horizontally invisible set in Theorem 6.4.

Definition 6.2

Define $\Updownarrow _n, \Uparrow _n$ and $\swarrow \!\!\!\!\swarrow \!\!\!_n$ as subsets of $W_n$ inductively by

$$\begin{aligned} \Updownarrow _{n + 1}&= \{\nwarrow , \leftarrow , \swarrow , \nearrow , \rightarrow , \searrow \}\cdot \Updownarrow _n \cup \uparrow \cdot \Uparrow _n \cup \downarrow \cdot \Downarrow _n \end{aligned}$$

(6.1)

$$\begin{aligned} \Uparrow _{n + 1}&= \{\nearrow , \leftarrow , \nwarrow , \rightarrow \}\cdot \Updownarrow _n \cup \uparrow \cdot \Uparrow _n \cup \swarrow \cdot \swarrow \!\!\!\!\swarrow \!\!\!_n \cup \downarrow \cdot {W_n} \cup \searrow \cdot \searrow \!\!\!\!\searrow _n \end{aligned}$$

(6.2)

$$\begin{aligned} \swarrow \!\!\!\!\swarrow \!\!\!_{n + 1}&= \{\nwarrow , \leftarrow \}\cdot \Updownarrow _n \cup \{\downarrow , \searrow \}\cdot \Leftrightarrow _n \cup \{\uparrow , \rightarrow , \swarrow \}\cdot \swarrow \!\!\!\!\swarrow \!\!\!_n \cup \nearrow \cdot {W_n} \end{aligned}$$

(6.3)

and $\Updownarrow _0 = \Uparrow _0 = \swarrow \!\!\!\!\swarrow \!\!\!_0 = \emptyset$, where $\Downarrow _n = \varphi _{\updownarrow }(\Uparrow _n)$, $\searrow \!\!\!\!\searrow _n = \varphi _{\leftrightarrow }(\swarrow \!\!\!\!\swarrow \!\!\!_n)$ and $\Leftrightarrow _n = \rho (\Updownarrow _n)$.

Lemma 6.13 will show that $\Updownarrow _n$ and $\Leftrightarrow _n$ are vertically and horizontally invisible sets respectively.

Lemma 6.3

$$ \#(\Updownarrow _n) = 8^n - \frac{7 + \sqrt{41}}{2\sqrt{41}}\Bigg (\frac{9 + \sqrt{41}}{2}\Bigg )^n + \frac{7 - \sqrt{41}}{2\sqrt{41}}\Bigg (\frac{9 - \sqrt{41}}{2}\Bigg )^n $$

Proof

Write $a_n = \#(\Updownarrow _n)$, $b_n = \#(\Uparrow _n)$ and $c_n = \#(\swarrow \!\!\!\!\swarrow \!\!\!_n)$. By (6.1), (6.2) and (6.3), it follows that

$$\begin{aligned} a_{n + 1}&= 6a_n + 2b_n\\ b_{n + 1}&= 4a_n + b_n + 2c_n + 8^n\\ c_{n + 1}&= 4a_n + 3c_n + 8^n. \end{aligned}$$

Solving these with $a_0 = b_0 = c_0 = 0$, we obtain $a_n$ as in the statement of the lemma. $\square$

Now we have the main theorem of this section.

Theorem 6.4

Let $A_n =\,\Updownarrow _n \cap \Leftrightarrow _n$. Then $A_n$ is a $+$-invariant invisible set and

$$ \alpha _n \le 8^n - \#(A_n) \le 2\alpha _n, $$

where

$$ \alpha _n = \frac{7 + \sqrt{41}}{2\sqrt{41}}\Bigg (\frac{9 + \sqrt{41}}{2}\Bigg )^n - \frac{7 - \sqrt{41}}{2\sqrt{41}}\Bigg (\frac{9 - \sqrt{41}}{2}\Bigg )^n. $$

Example 6.5

$A_0 = A_1 = A_2 = A_3 = \emptyset$.

$$\begin{aligned} A_4&= \{\uparrow \downarrow \leftarrow \rightarrow , \uparrow \downarrow \rightarrow \leftarrow , \downarrow \uparrow \leftarrow \rightarrow , \downarrow \uparrow \leftarrow \rightarrow , \leftarrow \rightarrow \uparrow \downarrow , \leftarrow \rightarrow \downarrow \uparrow , \rightarrow \leftarrow \uparrow \downarrow , \\&\qquad \rightarrow \leftarrow \downarrow \uparrow \}. \end{aligned}$$

Applying Corollary 5.4 and letting $n \rightarrow \infty$, we obtain the following upper estimate of the conformal dimension of the Sierpinski carpet.

Corollary 6.6

$$ \dim _{\mathcal {C}} (K, d_E) \le \frac{\log {(\frac{9 + \sqrt{41}}{2})}}{\log 3} = 1.858183... < \frac{\log 8}{\log 3} = 1.892789.... $$

Remark

The known lower bound of $\dim _{\mathcal {C}}(K, d_E)$ given in (1.1) is $\frac{\log 6}{\log 3} = 1.630929...$.

The rest of this section is devoted to proving Theorem 6.4.

Lemma 6.7

• (1) $\varphi _{\leftrightarrow }(\Updownarrow _n) =\,\, \Updownarrow _n$ and $\varphi _{\updownarrow }(\Updownarrow _n) = \,\, \Updownarrow _n$.

• (2) $\varphi _{\leftrightarrow }(\Uparrow _n) =\,\, \Uparrow _n$.

• (3) $\varphi _{\leftrightarrow }\circ \rho (\swarrow \!\!\!\!\swarrow \!\!\!_n) =\,\, \swarrow \!\!\!\!\swarrow \!\!\!_n$.

Definition 6.8

Define the vertical index $I_{\updownarrow }^n: W_n \rightarrow \{1, \ldots , 3^n\}$ by

$$ I_{\updownarrow }^n(w) = \frac{3^n(\mathrm{Im}\,(F_w(\sqrt{-1})) + 1)}{2} $$

For $H \in \{L, R\}$, define $w_H^n(i)$ for $i = 1, \ldots , 3^n$ as the unique $w \in H^n$ which satisfies $I_{\updownarrow }^n(w) = i$. Moreover, for $w, v \in W_n$, define $\mathbf p_H^n(w, v) \in \mathcal {C}\mathcal {H}_n$ by

$$ \mathbf p_H^n(w, v) = {\left\{ \begin{array}{ll} (w_H^n\left( I_{\updownarrow }^n(w)\right) , w_H^n\left( I_{\updownarrow }^n(w) + 1\right) , \ldots , w_H^n\left( I_{\updownarrow }^n(v)\right) &{}\text {if}\,I_{\updownarrow }^n(w) \le I_{\updownarrow }^n(v),\\ (w_H^n\left( I_{\updownarrow }^n(w)\right) , w_H^n\left( I_{\updownarrow }^n(w) - 1\right) , \ldots , w_H^n\left( I_{\updownarrow }^n(v)\right) &{}\text {if}\,I_{\updownarrow }^n(v) \le I_{\updownarrow }^n(w). \end{array}\right. } $$

In the same way, we define the horizontal index $I_{\leftrightarrow }^n: W_n \rightarrow \{1, \ldots , 3^n\}$, $w_T^n(i)$, $w_B^n(i)$, $\mathbf p_T^n(w, v)$ and $\mathbf p_B^n(w, v)$.

Lemma 6.9

Let $A \subseteq W_n$. Assume that

$$\begin{aligned} \inf \{\ell _{A}(\mathbf p_{*})| \mathbf p_* \in \mathcal {C}\mathcal {H}_n(T, p_{\swarrow })\} \ge 1 \end{aligned}$$

(6.4)

Let $\mathbf p= (w(1), \ldots , w(k)) \in \mathcal {C}\mathcal {H}_n$. If $(w(1), w(k)) \in (T^n \cup L^n) \times L^n$, then

$$\begin{aligned} \ell _{A}(\mathbf p) \ge \frac{|I_{\updownarrow }^n(w(1)) - I_{\updownarrow }^n(w(k))| + 1}{3^n} = \ell _{A}(\mathbf p_L^n(w(1), w(k))). \end{aligned}$$

(6.5)

Remark

Using the symmetries, we may exchange $(T, L, p_{\swarrow })$ in the statement of Lemma 6.9 by $(T, R, p_{\searrow })$, $(B, L, p_{\nwarrow })$ and $(B, R, p_{\nearrow })$.

Proof

Since $\ell _{A}((w_L^n(i))_{i = 1, \ldots , 3^n}) \ge 1$, it follows that $\{w_L^n(i)|i = 1, \ldots , 3^n\} \cap A = \emptyset$. Let $\mathbf p= (w(1), \ldots , w(k)) \in \mathcal {C}\mathcal {H}_n$.

Suppose that $(w(1), w(k)) \in T^n \times L^n$. Note that $w(k) = w_L^n(i)$ for some $i$. Then $\mathbf p_L^n(w(1), w(k)) = (w_L^n(3^n), w_L^n(3^n - 1), \ldots , w_L^n(i))$ and $\ell _{A}(\mathbf p_L^n(w(1), w(k)))$ $= 1 - (i - 1)/3^n$. Since $\mathbf p\vee \mathbf p_L^n(w_L^n(i - 1), w_L^n(1)) \in \mathcal {C}\mathcal {H}_n(T, p_{\swarrow })$, (6.4) implies

$$ \ell _{A}(\mathbf p) + \ell _{A}(\mathbf p_L^n(w_L^n(i - 1), w_L^n(1))) = \ell _{A}(\mathbf p\vee \mathbf p_L^n(w_L^n(i - 1), w_L^n(1))) \ge 1. $$

This shows (6.5) in this case.

Suppose that $(w(1), w(k)) \in L^n \times L^n$. Set $w(1) = w_L^n(j)$ and $w(k) = w_L^n(i)$. If $j < i$, then we consider $(w(k), \ldots , w(1))$ in place of $(w(1), \ldots , w(k))$. In this way, we may assume that $j \ge i$ without loss of generality. Let $\widetilde{\mathbf p} = \mathbf p_L^n\left( w_L^n\left( 3^n\right) , w_L^n(j + 1)\right) \vee \mathbf p\vee \mathbf p_L^n\left( w_L^n(j - 1), w_L^n(1)\right)$. Since $\widetilde{\mathbf p} \in \mathcal {C}\mathcal {H}_n(T, p_{\swarrow })$, we have

$$ \frac{3^n - j}{3^n} + \ell _{A}(\mathbf p) + \frac{i - 1}{3^n} = \ell _{A}(\widetilde{\mathbf p}) \ge 1 $$

This immediately implies (6.5) in this case. $\square$

Lemma 6.10

Let $X, Y \subseteq W_n$. Assume that

$$ \inf \{\ell _X(\mathbf p)|\mathbf p\in \mathcal {C}\mathcal {H}_n(T, B)\} \ge 1 $$

and that

$$ \inf \{\ell _Y(\mathbf p)|\mathbf p\in \mathcal {C}\mathcal {H}_n(T, p_{\swarrow })\} \ge 1. $$

Define $A =\,\, \nwarrow \!\cdot {X}\,\, \cup \uparrow \!\!{\cdot }Y$. If $\mathbf p= (w(1), \ldots , w(k)) \in \mathcal {C}\mathcal {H}_{n + 1}(T, B_{\nwarrow })$ and $\{w(1), \ldots , w(k)\} \subseteq \{\nwarrow , \uparrow \}{\cdot }W_n$, then

$$ \ell _A(\mathbf p) \ge \frac{1}{3}. $$

Proof

Let $w(i) = s(i)v(i)$, where $s(i) \in \{\nwarrow , \uparrow \}$ and $v(i) \in W_n$.

First assume that $s(1) = \nwarrow$. Then there exist $j_1, j_2, \ldots , j_{2p + 2}$ which satisfies the following three conditions (C1), (C2) and (C3):

• (C1) $j_1 = 1$, $j_{2p + 2} = k + 1$ and $j_1 < j_2 < \ldots < j_{2p + 2}$

• (C2) $s(i) = \,\,\nwarrow$ for $i = j_{2q - 1}, \ldots , j_{2q} - 1$ and $q = 1, \ldots , p + 1$

• (C3) $s(i) =\,\,\, \uparrow \,\,$ for $i = j_{2q}, \ldots , j_{2q + 1} - 1$ and $q = 1, \ldots , p$.

Set $\mathbf p_{1, q} = \left( w(j_{2q - 1}), \ldots , w(j_{2q} - 1)\right)$ and $\widetilde{\mathbf p}_{1, q} = \left( v(j_{2q - 1}), \ldots , v(j_{2q} - 1)\right)$. Since $\left( v(j_{2q - 1}), v(j_{2q} - 1)\right) \in (T^n \times R^n) \cup (R^n \times R^n) \cup (R^n \times B^n)$, Lemma 6.9 and its variants explained in the remark imply

$$\begin{aligned} \ell _{A}\left( \mathbf p_{1, q}\right) = \frac{1}{3}\ell _{X}\left( \widetilde{\mathbf p}_{1, q}\right) \ge \frac{1}{3}\ell _{X}\left( \mathbf p_R^n\left( v\left( j_{2q - 1}\right) , v\left( j_{2q} - 1\right) \right) \right) . \end{aligned}$$

(6.6)

Set $\mathbf p_{2, q} = \left( w\left( j_{2q}\right) , \ldots , w\left( j_{2q + 1} - 1\right) \right)$ and $\widetilde{\mathbf p}_{2, q} = \left( v\left( j_{2q}\right) , \ldots , v\left( j_{2q + 1} - 1\right) \right)$. Since $\left( v\left( j_{2q}\right) , v\left( j_{2q + 1} - 1\right) \right) \in L^n \times L^n$, Lemma 6.9 shows that

$$\begin{aligned} \ell _{A}\left( \mathbf p_{2, q}\right) = \frac{1}{3}\ell _{Y}\left( \widetilde{\mathbf p}_{2, q}\right) \ge \frac{1}{3}\ell _{Y}\left( \mathbf p_L^n\left( v\left( j_{2q}\right) , v\left( j_{2q + 1} - 1\right) \right) \right) . \end{aligned}$$

(6.7)

Note that for any $i = 1, \ldots , 3^n$, there exists $l = 1, 2, \ldots , 2q + 1$ such that $I_{\updownarrow }^n\left( v\left( j_l\right) \right) \le i \le I_{\updownarrow }^n\left( v\left( j_{l + 1} - 1\right) \right)$ or $I_{\updownarrow }^n\left( v\left( j_l\right) \right) \ge i \ge I_{\updownarrow }^n\left( v\left( j_{l + 1} - 1\right) \right)$. Hence

$$ \sum _{q = 1}^{p + 1}\ell _{X}\left( \mathbf p_R^n\left( v\left( j_{2q - 1}\right) , v\left( j_{2q} - 1\right) \right) \right) + \sum _{q = 1}^p \ell _{Y}\left( \mathbf p_L^n\left( v\left( j_{2q}\right) , v\left( j_{2q + 1} - 1\right) \right) \right) \ge 1. $$

Combining this with (6.6) and (6.7), we obtain

$$ \ell _{A}(\mathbf p) = \sum _{q = 1}^{p + 1} \ell _{A}\left( \mathbf p_{1, q}\right) + \sum _{q = 1}^p \ell _{A}\left( \mathbf p_{2, q}\right) \ge \frac{1}{3}. $$

Thus we have shown the desired statement in the case when $s(1) = \nwarrow$.

If $s(1) = \,\,\uparrow$, slight modification of the above arguments yields the lemma as well. $\square$

Definition 6.11

Define $\pi : W_* \rightarrow W_*$ by

$$ \pi (w) = {\left\{ \begin{array}{ll} w &{}\text {if} \;\mathrm{Re}\,F_{w}(0) \le 0,\\ \varphi _{\leftrightarrow }(w) &{}\text {if}\; \mathrm{Re}\,F_{w}(0) > 0. \end{array}\right. } $$

For $\mathbf p= (w(1), \ldots , w(k)) \in \mathcal {C}\mathcal {H}_n$, we define $\pi _n(\mathbf p) = (\pi (w(1)), \ldots , \pi (w(k))$. Also define $\xi : W_* \rightarrow W_*$ by

$$ \xi (w) = {\left\{ \begin{array}{ll} w &{}\text {if} \;\mathrm{Re}\,F_{w}(0) \le \mathrm{Im}\,F_{w}(0),\\ \varphi _{\leftrightarrow }(\rho (w)) &{}\text {if} \;\mathrm{Re}\,F_{w}(0) > \mathrm{Im}\,F_{w}(0). \end{array}\right. } $$

For $\mathbf p= (w(1), \ldots , w(k)) \in \mathcal {C}\mathcal {H}_n$, we define $\xi _n(\mathbf p) = (\xi (w(1)), \ldots , \xi (w(k))$.

By the symmetry of $\Updownarrow _n$, $\Uparrow _n$ and $\swarrow \!\!\!\!\swarrow \!\!\!_n$ given in Lemma 6.7, we have the following lemma.

Lemma 6.12

• (1) $\pi _n:\mathcal {C}\mathcal {H}_n \rightarrow \mathcal {C}\mathcal {H}_n$, $\ell _{\Updownarrow _n}\left( \pi _n(\mathbf p)\right) = \ell _{\Updownarrow _n}(\mathbf p)$ and $\ell _{\Uparrow _n}\left( \pi _n(\mathbf p)\right) = \ell _{\Uparrow _n}(\mathbf p)$.

• (2) $\xi _n:\mathcal {C}\mathcal {H}_n \rightarrow \mathcal {C}\mathcal {H}_n$ and $\ell _{\swarrow \!\!\!\!\swarrow \!\!\!_n}(\xi _n(\mathbf p)) = \ell _{\swarrow \!\!\!\!\swarrow \!\!\!_n}(\mathbf p)$.

Lemma 6.13

Suppose that

$$\begin{aligned} \inf \{\ell _{\Updownarrow _n}(\mathbf p)| \mathbf p\in \mathcal {C}\mathcal {H}_n(T, B)\} \ge 1 \end{aligned}$$

(6.8)

and

$$\begin{aligned} \inf \{\ell _{\Uparrow _n}(\mathbf p)|\mathbf p\in \mathcal {C}\mathcal {H}_n(T, \{p_{\swarrow }, p_{\searrow }\}) \ge 1. \end{aligned}$$

(6.9)

If $\mathbf p= (w(1), \ldots , w(k)) \in \mathcal {C}\mathcal {H}_{n + 1}(T, B_{\nwarrow } \cup B_{\nearrow })$ and $\{w(i)\}_{i = 1}^k \subseteq \{\nwarrow , \uparrow , \nearrow \}{\cdot }W_n$, then $\ell _{\Updownarrow _{n + 1}}(\mathbf p) \ge 1/3$.

Proof

Replacing $\mathbf p$ by $\pi _{n + 1}(\mathbf p)$, we may assume that $w(1), \ldots , w(k) \in \{\nwarrow , \uparrow \}{\cdot }W_n$ and $w(k) \in \nwarrow {\cdot }B^n$ without loss of generality. Set $X = \Updownarrow _n$ and $Y = \Uparrow _n$. Then the assumptions (6.8) and (6.9) of Lemma 6.10 follows. Hence $\ell _{\Updownarrow _{n + 1}}(\mathbf p) \ge 1/3$. $\square$

Lemma 6.14

Suppose that (6.8) holds and that

$$\begin{aligned} \inf \{\ell _{\swarrow \!\!\!\!\swarrow \!\!\!_n}(\mathbf p)|\mathbf p\in \mathcal {C}\mathcal {H}_n(T \cup R, p_{\swarrow }) \ge 1. \end{aligned}$$

(6.10)

Let $\mathbf p= (w(1), \ldots , w(k)) \in \mathcal {C}\mathcal {H}_{n + 1}(T, B_{\swarrow })$. If $\{w(1), \ldots , w(k)\} \subseteq {\nwarrow , \uparrow , \nearrow ~}{\cdot }W_n$, then $\ell _{\swarrow \!\!\!\!\swarrow \!\!\!_{n + 1}}(\mathbf p) \ge 1/3$.

Proof

First assume that $\{w(1), \ldots , w(k)\} \subseteq \{\nwarrow , \uparrow \}{\cdot }W_n$. By (6.3), applying Lemma 6.10 with $X = \,\,\Updownarrow _n$ and $Y = \,\,\swarrow \!\!\!\!\swarrow \!\!\!_n$, we have $\ell _{\swarrow \!\!\!\!\swarrow \!\!\!_{n + 1}}(\mathbf p) \ge 1/3$.

Next, suppose that $w(i) \in \nearrow {\cdot }W_n$ for some $i$. Let

$$ i_* = \max \{i|w(i) \in {\nearrow }{\cdot }W_n\} + 1. $$

and let

$$ j_* = \min \{j|w(j) \in {\nwarrow }{\cdot }W_n, j \ge i_*\} - 1. $$

Then, for $i = \{i_*, \ldots , j_*\}$, there exists $v(i) \in W_n$ such that $w(i) = \uparrow \cdot {v(i)}$. Define

$$ \mathbf p_* = \left( \uparrow {\cdot }\xi \left( v\left( i_*\right) \right) , \uparrow {\cdot }\xi \left( v\left( i_* + 1\right) \right) , \ldots , \uparrow {\cdot }\xi \left( v\left( j_*\right) \right) , w\left( j_* + 1\right) , \ldots , w\left( k\right) \right) . $$

By (6.3) and Lemma 6.12,

$$\begin{aligned} \ell _{\swarrow \!\!\!\!\swarrow \!\!\!_{n + 1}} \left( w\left( i_*\right) , \ldots , w\left( j_*\right) \right)&= \frac{1}{3}\ell _{\swarrow \!\!\!\!\swarrow \!\!\!_n}\left( v\left( i_*\right) , \ldots , v\left( j_*\right) \right) \\&= \frac{1}{3}\ell _{\swarrow \!\!\!\!\swarrow \!\!\!_n}\left( \xi \left( v\left( i_*\right) \right) , \ldots , \xi \left( v\left( j_*\right) \right) \right) \\&= \ell _{\swarrow \!\!\!\!\swarrow \!\!\!_{n + 1}}\left( \uparrow {\cdot }\xi \left( v\left( i_*\right) \right) , \ldots , \uparrow {\cdot }\xi \left( v\left( j_*\right) \right) \right) . \end{aligned}$$

Hence $\ell _{\swarrow \!\!\!\!\swarrow \!\!\!_{n + 1}}(\mathbf p) \ge \ell _{\swarrow \!\!\!\!\swarrow \!\!\!_{n + 1}}\left( \mathbf p_*\right)$. Let $\mathbf p_* = \left( w_*(1), w_*(2), \ldots , w_*(l)\right)$. Then $w^*(i) \in \{\nwarrow , \uparrow \}{\cdot }W_n$ for any $i = 1, \ldots , l$. Now replacing $\mathbf p$ by $\mathbf p_*$, we are exactly in the first case and hence the desired inequality is satisfied. $\square$

Lemma 6.15

(6.8), (6.9) and (6.10) hold for any $n \ge 0$.

Proof

We use induction on $n$. Obviously (6.8), (6.9) and (6.10) holds for $n = 0$ since $\Updownarrow _n, \Uparrow _n$ and $\swarrow \!\!\!\!\swarrow \!\!\!_n$ are the empty sets. Assume that (6.8), (6.9) and (6.10) are true for $n = m$.

First we show (6.8) holds for $n = m + 1$. Let $\mathbf p= (w(1), \ldots , w(k)) \in \mathcal {C}\mathcal {H}_{m + 1}(T, B)$. Note that by Lemma 6.7-(1), $\pi _{n + 1}(\mathbf p) \in \mathcal {C}\mathcal {H}_{m + 1}(T, B)$ and $\ell _{\Updownarrow _{m + 1}}(\mathbf p) = \ell _{\Updownarrow _{m + 1}}(\pi _{n + 1}(\mathbf p))$. Hence replacing $\mathbf p$ by $\pi _{n + 1}(\mathbf p)$, we may assume that $w(i) \in \{\nwarrow , \uparrow , \leftarrow , \swarrow , \downarrow \}{\cdot }W_m$ for any $i = 1, \ldots , k$ without loss of generality. Set $w(i) = s(i)v(i)$, where $s(i) \in \{\nwarrow , \uparrow , \leftarrow , \swarrow , \downarrow \}$ and $v(i) \in W_m$. We may choose $i_1, i_2, i_3$ and $i_4$ which satisfies $i_1 < i_2 < i_3 < i_4$ and the following tree conditions (a1), (b1) and (c1):

• (a1) $s(1), \ldots , s(i_1) \in \{\nwarrow , \uparrow \}$, $(v(1), \ldots , v(i_1)) \in \mathcal {C}\mathcal {H}_m(T, B_{\swarrow })$,

• (b1) $s(i) =\,\, \leftarrow$ for $i = i_2, \ldots , i_3$, $(v(i_2), \ldots , v(i_3)) \in \mathcal {C}\mathcal {H}_m(T, B)$,

• (c1) $s(i_4), \ldots , s(k) \in \{\swarrow , \downarrow \}$, $w_*(i_4) \in {\swarrow }{\cdot }T^m$.

Let $\mathbf p_1 = (w(1), \ldots , w(i_1))$. Then by the induction hypothesis, we may apply Lemma 6.13 and see that $\ell _{\Updownarrow _{m + 1}}(\psi _1) \ge 1/3$.

Let $\mathbf p_2 = (w(i_2), \ldots , w(i_3))$. Since $(v(i_2), \ldots , v(i_3)) \in \mathcal {C}\mathcal {H}_m(T, B)$, the induction hypothesis implies

$$ \ell _{\Updownarrow _{m + 1}}(\mathbf p_2) = \frac{1}{3}\ell _{\Updownarrow _m}(v(i_2), \ldots , v(i_3)) \ge \frac{1}{3}. $$

Set $\mathbf p_3 = (w(i_3), \ldots , w(k))$ and $\widetilde{\mathbf p}_3 = \big (\varphi _{\updownarrow }\left( w\left( k\right) \right) , \varphi _{\updownarrow }\left( w\left( k - 1\right) \right) , \ldots ,\varphi _{\updownarrow }\left( w\left( i_3\right) \right) \big )$. Then $\ell _{\Updownarrow _{m + 1}}(\mathbf p_3) = \ell _{\Updownarrow _{m + 1}}(\widetilde{\mathbf p}_3)$. As $\mathbf p_1$, we may apply Lemma 6.13 to $\widetilde{\mathbf p}_3$ and obtain $\ell _{\Updownarrow _{m + 1}}(\widetilde{\mathbf p}_3) \ge 1/3$. Combining all the estimates on $\ell _{\Updownarrow _{m + 1}}(\mathbf p_i)$ for $i = 1, 2, 3$, we have $\ell _{\Updownarrow _{m + 1}}(\mathbf p) \ge 1$.

Secondly, we show that (6.9) holds for $n = m + 1$. Let $\mathbf p= (w(1), \ldots , w(k)) \in \mathcal {C}\mathcal {H}_{m + 1}(T, \{p_{\swarrow }, p_{\searrow }\})$. As in the first case, we may assume that $w(i) \in \{\nwarrow , \uparrow , \leftarrow , \swarrow , \downarrow \}$ for any $i = 1, \ldots , k$ without loss of generality. Set $w(i) = s(i)v(i)$, where $s(i) \in \{\nwarrow , \uparrow , \leftarrow , \swarrow , \downarrow \}$ and $v(i) \in W_m$. We may choose $i_1, i_2, i_3$ and $i_4$ which satisfies $i_1 < i_2 < i_3 < i_4$ and the following tree conditions (a2), (b2) and (c2):

• (a2) $s(1), \ldots , s(i_1) \in \{\nwarrow , \uparrow \}$, $(v(1), \ldots , v(i_1)) \in \mathcal {C}\mathcal {H}_m(T, B_{\swarrow })$,

• (b2) $s(i) =\,\, \leftarrow$ for $i = i_2, \ldots , i_3$, $(v(i_2), \ldots , v(i_3)) \in \mathcal {C}\mathcal {H}_m(T, B)$,

• (c2) $s(i) = \swarrow$ for $i = i_4, \ldots , k$. $(v(i_4), \ldots , v(k)) \in \mathcal {C}\mathcal {H}_m(T \cup R, p_{\swarrow })$.

Define $\mathbf p_1, \mathbf p_2$ and $\mathbf p_3$ as in the first case. Then using the same discussion as in the first case, we obtain $\ell _{\Uparrow _{m + 1}}(\mathbf p_j) \ge 1/3$ for $j = 1, 2$. Since $(v(i_4), \ldots , v(k)) \in \mathcal {C}\mathcal {H}_m(T \cup R, p_{\swarrow })$, The induction hypothesis and Lemma 6.14 yield that $\ell _{\swarrow \!\!\!\!\swarrow \!\!\!_m}((v(i_4), \ldots , v(k))) \ge 1$. By (6.2), it follows that

$$ \ell _{\Uparrow _{m + 1}}(\mathbf p_3) = \frac{1}{3}\ell _{\swarrow \!\!\!\!\swarrow \!\!\!_m}((v(i_4), \ldots , v(k))) \ge 1/3. $$

Thus, we have shown that $\ell _{\Uparrow _{m + 1}}(\mathbf p) \ge 1$.

Finally we show that (6.10) holds for $n = m + 1$. Let $\mathbf p= (w(1), \ldots , w(k)) \in \mathcal {C}\mathcal {H}_{m + 1}(T \cup R, p_{\swarrow })$. Note that $\xi _{m + 1}(\mathbf p) \in \mathcal {C}\mathcal {H}_{m + 1}(T, p_{\swarrow })$ and $\ell _{\swarrow \!\!\!\!\swarrow \!\!\!_{m + 1}}(\mathbf p) = \ell _{\swarrow \!\!\!\!\swarrow \!\!\!_{m + 1}}(\xi _{m + 1}(\mathbf p))$. Hence replacing $\mathbf p$ by $\xi _{m + 1}(\mathbf p)$, we may assume that $w(i) \in \{\nwarrow , \uparrow , \nearrow , \leftarrow , \swarrow \}{\cdot }W_m$ for any $i = 1, \ldots , k$ without loss of generality (Fig. 5). Set $w(i) = s(i)v(i)$, where $s(i) \in \{\nwarrow , \uparrow , \nearrow , \leftarrow , \swarrow \}$ and $v(i) \in W_m$. We may choose $i_1, i_2, i_3$ and $i_4$ which satisfies $i_1 < i_2 < i_3 < i_4$ and the following tree conditions (a3), (b3) and (c3):

• (a3) $s(1), \ldots , s(i_1) \in \{\nwarrow , \uparrow \}$, $(v(1), \ldots , v(i_1)) \in \mathcal {C}\mathcal {H}_m(T, B_{\swarrow })$,

• (b3) $s(i) =\,\, \leftarrow$ for $i = i_2, \ldots , i_3$, $(v(i_2), \ldots , v(i_3)) \in \mathcal {C}\mathcal {H}_m(T, B)$,

• (c3) $s(i) = \swarrow$ for $i = i_4, \ldots , k$. $(v(i_4), \ldots , v(k)) \in \mathcal {C}\mathcal {H}_m(T \cup R, p_{\swarrow })$.

Define $\mathbf p_1, \mathbf p_2$ and $\mathbf p_3$ as in the above two cases. Then by the induction hypothesis and (6.3), it follows that $\ell _{\swarrow \!\!\!\!\swarrow \!\!\!_{m + 1}}(\mathbf p_j) \ge 1/3$ for $j = 2, 3$. Furthermore, Lemma 6.14 implies $\ell _{\swarrow \!\!\!\!\swarrow \!\!\!_{m + 1}}(\mathbf p_1) \ge 1/3$. Hence we have $\ell _{\swarrow \!\!\!\!\swarrow \!\!\!_{m + 1}}(\mathbf p) \ge 1$.

Thus we have obtained (6.8), (6.9) and (6.10) for $n = m + 1$. $\square$

Proof

of Theorem 6.4 Since $A_n \subseteq \,\,\Updownarrow _n$, $\ell _{A_n}(\mathbf p) \ge 1$ for any $\mathbf p\in \mathcal {C}\mathcal {H}_n(T, B)$. By the fact that $\Leftrightarrow _n = \rho (\Updownarrow _n)$, it follows that $\ell _{\Leftrightarrow _n}(\mathbf p) \ge 1$ for any $\mathbf p\in \mathcal {C}\mathcal {H}_n(L, R)$. Hence $\ell _{A_n}(\mathbf p) \ge 1$ for any $\mathbf p\in \mathcal {C}\mathcal {H}_n(L, R)$. Thus $A_n$ is invisible. By Lemma 6.7-(1), it follows that $A_n$ is $+$-invariant.

Lemma 6.3 shows that $8^n - \#(\Updownarrow _n) = \#(W_n\backslash \!{\Updownarrow _n}) = \alpha _n$. Since $W_n\backslash \!{\Updownarrow _n} \subseteq W_n\backslash A_n \subseteq (W_n\backslash \!{\Updownarrow _n}) \cup (W_n\backslash \!{\Leftrightarrow _n})$, we have $\alpha _n \le 8^n - \#(A_n) \le 2\alpha _n$. $\square$

7 Generalized Sierpinski Carpet

The idea of invisible sets can be exploited for the generalized Sierpinski carpets. We will present results for a special class of the generalized Sierpinski carpet in this section. We fix $N \ge 3$. The complex plane $\mathbb {C}$ is identified with $\mathbb {R}^2$ in the usual manner.

Fig. 5

Construction of $K^{(4)}$

Definition 7.1

• (1) For any $(i, j) \in \{1, \ldots , N\}^2$, we define $J_{(i, j)} = [-1 + 2(i - 1)/N, -1 + 2i/N] \times [-1 + 2(j - 1)/N, -1 + 2j/N]$ and $F_{(i, j)}: \mathbb {R}^2 \rightarrow \mathbb {R}^2$ by $F_{(i, j)}(x, y) = (x/N + a_{(i, j)}, y/N + b_{(i, j)})$, where $a_{(i, j)} = -1 + (2i - 1)/N$ and $b_{(i, j)} = -1 + (2j - 1)/N$.

• (2) Define $S^{(N)} = \{(i, j)| (i, j) \in \{1, \ldots , N\}^2, i \in \{1, N\}\,\,\,\text {or}\,\,\,j \in \{1, N\}\}$. Let $K^{(N)}$ be the unique compact set which satisfies

  $$ K^{(N)} = \bigcup _{(i, j) \in S^{(N)}} F_{(i, j)}\left( K^{(N)}\right) . $$

When $N = 3$, $K^{(3)}$ is the Sierpinski carpet.

Proposition 7.2

$\#(S^{(N)}) = 4N - 4$ and $\displaystyle \dim _H {\left( K^{(N)}, d_E\right) } = \frac{\log {(4N - 4)}}{\log N}$, where $d_E$ is the restriction of the Euclidean metric.

In the following, we occasionally omit $N$ in $S^{(N)}$ and $K^{(N)}$ and write them $S$ and $K$ respectively. Also we use $W_m, W_*$ and $\Sigma$ in place of $W_m\left( S^{(N)}\right) , W_*\left( S^{(N)}\right)$ and $\Sigma \left( S^{(N)}\right)$.

Definition 7.3

Let $A \subseteq W_m$.

• (1) Let $A \subseteq W_m$. For $\mathbf p= (w(1), \ldots , w(k)) \in \mathcal {C}\mathcal {H}_m$, define

  $$ \ell _A(\mathbf p) = \frac{\#(\{i|i = 1, \ldots , k, w(i) \notin A\})}{N^m}. $$

• (2) $A$ is called an invisible set if and only if

  $$ \inf _{\mathbf p\in \mathcal {C}\mathcal {H}_m(T, B) \cup \mathcal {C}\mathcal {H}_m(L, R)} \ell _{A}(\mathbf p) \ge 1, $$

where $T, B, L$ and $R$ are the same as in the last tree sections.

We also define the notion of $+$-invariance exactly same as in the previous sections. Then the analogous results as Theorems 4.5 and 5.3 hold. As a consequence we have the following statement.

Theorem 7.4

Let $A \subset W_m$ be a $+$-invariant invisible set. Then

$$ \dim _{\mathcal {C}} (K^{(N)}, d_E) \le \frac{\log {\left( (4N - 4)^m - \#(A)\right) }}{m\log N}. $$

A procedure which is similar to that in Sect. 6 produces a sequence of invisible sets. We assume $N \ge 4$ hereafter. The maps $\varphi _{\leftrightarrow }$, $\varphi _{\updownarrow }$ and $\rho$ from $W_*$ to itself associated with symmetries can be defined in the same way as in the last section.

Definition 7.5

Define $\Updownarrow _n \subseteq W_n$ and $\searrow \!\!\!\!\searrow _n \subseteq W_n$ inductively by

$$\begin{aligned} \Updownarrow _{n + 1}&= \{(i, j)|(i, j) \in S. i \in \{1, N\}\}\cdot \Updownarrow _n\\&\cup (2, 1){\cdot }\searrow \!\!\!\!\searrow _n \cup (2, N){\cdot }\nearrow \!\!\!\!\nearrow _n \cup (N - 1, 1){\cdot }\swarrow \!\!\!\!\swarrow \!\!\!_n \cup (N - 1, N){\cdot }\nwarrow \!\!\!\!\nwarrow _n\\&\qquad \qquad \qquad \qquad \cup \{(i, j)|(i, j) \in S, j \in \{1, N\}, i \notin \{1, 2, N -1, N\}\}{\cdot }W_n, \end{aligned}$$

$$\begin{aligned} \searrow \!\!\!\!\searrow _{n + 1}&= \{(1, N), (2, 1), (N - 1, N)\}{\cdot }\searrow \!\!\!\!\searrow _n \\&\cup \{(1, j)| j = 1, \ldots , N - 1\}{\cdot }\Updownarrow _n \cup \{(i, N)|i = 2, \ldots , N\}{\cdot }\Leftrightarrow _n\\&\qquad \qquad \quad \cup \{(1, j)|j = 3, \ldots , N\}{\cdot }W_n \cup \{(i, N)|i = 1, \ldots , N - 2\}{\cdot }W_n, \end{aligned}$$

$\Updownarrow _0 = \emptyset$ and $\searrow \!\!\!\!\searrow _0 = \emptyset$, where $\Leftrightarrow _n = \rho (\Updownarrow _n)$, $\swarrow \!\!\!\!\swarrow \!\!\!_n = \varphi _{\leftrightarrow }(\searrow \!\!\!\!\searrow _n)$, $\nearrow \!\!\!\!\nearrow _n = \varphi _{\updownarrow }(\searrow \!\!\!\!\searrow _n)$ and $\nwarrow \!\!\!\!\nwarrow _n = \varphi _{\leftrightarrow }(\nearrow \!\!\!\!\nearrow _n)$ (Fig. 6).

Fig. 6

Construction of $\Updownarrow _n$ and $\searrow \!\!\!\!\searrow _n$ for $N = 5$

By the above definition, it follows that

$$\begin{aligned} x_{n + 1}&= 2Nx_n + 4y_n + 2(N - 4)(4N - 4)^n\\ y_{n + 1}&= 2(N - 1)x_n + 3y_n + (2N - 5)(4N - 4)^n, \end{aligned}$$

where $x_n = \#(\Updownarrow _n)$ and $y_n = \#(\searrow \!\!\!\!\searrow _n)$. Define

$$ \tau _N = \sqrt{4N^2 + 20N - 23}. $$

Then we have

$$\begin{aligned} x_n&= (4N - 4)^n - \Big (\frac{2N + 5}{2\tau _N} + \frac{1}{2}\Big )\Big (\frac{2N + 3 + \tau _N}{2}\Big )^n \\&\quad \;+ \Big (\frac{2N + 5}{2\tau _N} - \frac{1}{2}\Big )\Big (\frac{2N + 3 - \tau _N}{2}\Big )^n \end{aligned}$$

The same discussion as in the last section shows

$$ \inf _{\mathbf p\in \mathcal {C}\mathcal {H}_n(T, B)} \ell _{\Updownarrow _n}(\mathbf p) \ge 1. $$

Hence we obtain the counterpart of Theorem 6.4.

Theorem 7.6

Let $A_n =\,\,\, \Updownarrow _n \cap \Leftrightarrow _n$. Then $A_n$ is $+$-invariant invisible set and there exist $c_1, c_2 > 0$ such that

$$ c_1\left( \frac{2N + 3 + \tau _N}{2}\right) ^n \le (4N - 4)^n - \#(A_n) \le c_2\left( \frac{2N + 3 + \tau _N}{2}\right) ^n $$

for sufficiently large $n$.

As an corollary, we obtain the following estimate of the conformal dimension of $K^{(N)}$. The lower estimate is shown by applying [MT10, Example 4.1.9].

Corollary 7.7

$$\begin{aligned} \frac{\log {(2N)}}{\log N} \le \dim _{\mathcal {C}} \left( K^{(N)}, d_E\right)&\le \frac{\log {\frac{2N + 3 + \tau _N}{2}}}{\log {N}} \\&< \frac{\log {(4N - 4)}}{\log {N}} = \dim _H{\left( K^{(N)}, d_E\right) }. \end{aligned}$$

Remark

$$ 2N + 3 \le \frac{2N + 3 + \tau _n}{2} < 2N + 4. $$