Diophantine approximations,substituions, and fractals
Shunji ITO (伊藤 俊次)(Tsuda College) (津田塾大学)

at Tsuda College

1996.7.29. –1996.8.1.

古門 麻貴 記Contents
1 Substitution

1

2 Endomorphism

7

3 Dynamical system

21

4 Renormalization

32

5 Fractal boundaries

35

6 Diophantine algorithm and substitutions

38

7 Applications
7.1 Quasi-periodic tiling related to the stepped surface
7.2 Markov partition of group automorphisms on
7.3 Diophantine approximation:

43$T^{3}$

43
44
45

Substitution

1
Let

be the set of words with the alphabet {1, 2, 3}, that is,$W^{*}$$W^{*}:=\bigcup_{n=0}^{\infty}\{1,2,3\}^{n}$

For any words $W=w_{1}\cdots w_{k}$ . $V=v_{1}\cdots v_{l}\in W^{*}$ , let us define the product$W\cdot V$ by the concatination:$W\cdot V=w_{1}\cdots w_{k}v_{1}\cdots v_{1}$Thenis a semi-group. Let :let us assume the following property:$W^{*}$$\sigma$$W^{*}\rightarrow W^{*}$$\sigma(W)=\sigma(w_{1})\sigma(w_{2})\cdots\sigma(w_{k})$for.

be an endomorphism, that is,$W=w_{1}\cdots w_{k}\in W^{*}$

We call satisfying above property by a substitution.
Let :be the canonical homomorphism, that is,$\sigma$$f$$W^{*}\rightarrow Z^{3}$$f(i)$$;=e_{i},$$f(\phi)$$;=0$ ,

$f(W)$$i=1,2,3$ ,for$;=\sum_{i=1}^{k}f(w_{i})$$W=w_{1}\cdots w_{k}\in W^{*}$

where $\{e_{i}|i=1,2,3\}$ the canonical basis of:
let us define thematrix$\sigma$$W^{*}\rightarrow W$ “

For each substitution$Z^{3}\subset R^{3}$$3\times 3$$L_{\sigma}$by$L_{\sigma}=[f(\sigma(1)), f(\sigma(2)), f(\sigma(3))]$,then the following lemma holds.
Lemma 1.1 The commutative relation holds:$W^{*}$$f$$L_{\sigma}$$W^{*}$$\downarrow$$Z^{3}$The matrix$\rightarrow^{\sigma}$$\downarrow$$\rightarrow^{L_{\sigma}}$

is called the matrix of1$f$$Z^{3}$$\sigma$or abeliarization of .
$\sigma$Example 1 (Rauzy substitution [R]) Let us define the substitution

follows:$\sigma$then the matrix of$\sigma$1: 2
3$\rightarrow$$\rightarrow$$\rightarrow$$\sigma$

as

12
13,
1

is given by$L_{\sigma}=\left\{\begin{array}{lll}l & l & 1\\1 & 0 & 0\\0 & l & 0\end{array}\right\}$

The substitution

is called Rauzy substitution.

$\sigma$

Example 2 (Modified Jacobi-Perron substitution [I-O 93]) For each$a\in N$ andas follows:
let us define the substitution$\epsilon\in\{0,1\}$$\sigma\left(\begin{array}{l}a\\\epsilon\end{array}\right)$$\rightarrow\hat{11\cdots 1}2atlmes$:$\sigma\left(\begin{array}{l}a\\0\end{array}\right)$

1
2
3

then the $mat\dot{m}$ of$\rightarrow$$3$$\rightarrow$1$\rightarrow\hat{11\cdots 1}3atimes$:$\sigma\left(\begin{array}{l}a\\1\end{array}\right)$$\rightarrow$

1
2$\sigma\left(\begin{array}{l}a\\1\end{array}\right)$

and$L_{\sigma}\left(\begin{array}{l}a\\0\end{array}\right)=\left\{\begin{array}{lll}a & 0 & l\\1 & 0 & 0\\0 & l & 0\end{array}\right\}$

These substitutions$\rightarrow$are given byand$\sigma\left(\begin{array}{l}a\\0\end{array}\right)$

1
2
3$L_{\sigma}\left(\begin{array}{l}a\\1\end{array}\right)=\left\{\begin{array}{lll}a & l & 0\\0 & 0 & 1\\1 & 0 & 0\end{array}\right\}$$\epsilon\in\{0,1\}$$\sigma\left(\begin{array}{l}a\\\epsilon\end{array}\right)’ a\in N,$

are called $1VIodified$ Jacobi-Perronsubstituions (The definition of Modified Jacobi-Perron algorithm is given in
the section 6).
Definition 1.2 The substitution$\sigma$:$W^{*}\rightarrow W^{*}is$said to be Pisot substitutionif(1) $detL_{\sigma}=\pm 1$ ,(2) there exist $N$ such that
$L_{\sigma}^{N}>O$(Aperiodic condition),2

(3) the eigenvalues$\lambda_{j}\lambda’$and$\lambda$“of$L_{\sigma}$satisfy(Pisot condition)$\lambda>1>|\lambda’|,$ $|\lambda’’|$In this lecture, we discuss only on Pisot substitutions.
Let us denote the column and row eigenvectors of
associated with the
maximum eigenva.lue by$L_{\sigma}$$\lambda$,$L_{\sigma}\left\{\begin{array}{l}l\\\alpha\\\beta\end{array}\right\}=\lambda\left\{\begin{array}{l}1\\\alpha\\\beta\end{array}\right\}$Let$\mathcal{P}$${}^{t}L_{\sigma}\left\{\begin{array}{l}l\\\gamma\\\delta\end{array}\right\}=\lambda\left\{\begin{array}{l}1\\\gamma\\\delta\end{array}\right\}$

be the plane given by$\mathcal{P}:=\{x\in R^{3}|\langle x,$$\left\{\begin{array}{l}l\\\gamma\\\delta\end{array}\right\}\rangle=0\}$,

then it is easy to see the following lemma.
Lemma 1.3 The plane, that is,$\mathcal{P}$is the contracting invariant plane with respect to$L_{\sigma}$$L_{\sigma}\mathcal{P}=\mathcal{P}$.

More precisely,(1) In the case of$\lambda’,$$\lambda^{u}\in R$$v’:=\left\{\begin{array}{l}1\\\alpha’\\\beta’\end{array}\right\}$, put the eigenvectors of,$v^{u}:=\left\{\begin{array}{l}1\\\alpha’’\\\beta’\end{array}\right\}\in R^{3}$$\lambda_{j}’\lambda’’$

by

,

then
$v’,$ $v’’\in \mathcal{P}$and$[L_{\sigma}v’, L_{\sigma}v’’]=[v’, v’’]\left\{\begin{array}{ll}\lambda’ & 0\\0 & \lambda’\end{array}\right\}$(2) In the case$of\overline{\lambda’}=\lambda’’\in C_{2}$put the eigenvectors of$v’’:=\left\{\begin{array}{l}1\\\alpha’’\\\beta’\end{array}\right\}\in C^{3}$$v’:=\left\{\begin{array}{l}l\\\alpha’\\\beta’\end{array}\right\}$

3,$\lambda’,$$\lambda’’$

by

and
$u’$

$:=-(v’21+v’’)$ ,$u’’$$:=\frac{1}{2i}(v’-v’’)$,then$u’,$ $u^{u}\in \mathcal{P}$and there exists$\theta$such that$[L_{\sigma}u’, L_{\sigma}u’’]=\frac{1}{\sqrt{\lambda}}[u’, u’’]\left\{\begin{array}{ll}cos\theta & -sin\theta\\sin\theta & cos\theta\end{array}\right\}$, that is, the first alphabet ofcoinLet us assume $\sigma(1)=1$
sides with 1. Then there exists the sequence of {1, 2, 3} given by $u=$and satisfies$\sigma(u)=u$ ,$\sigma(1)$$\nu V_{1}$$u$$\lim_{n\rightarrow\infty}\sigma^{n}(1)$that is, is the fixed point of .
For each Pisot substitution, let, and let us define the set$u$${}^{t}[1,$$\alpha,$$\sigma$$Y_{N}$$:=$$\{\pi\sum_{j=1}^{k}e_{s}j$$k=1,2,$$Y_{N,i}$$;=$$\{\pi\sum_{j_{-}^{-}1}^{k}e_{s}j$$s_{j}=i,$$Y$$Y_{i}$

where:$ 7\iota$$R^{3}\rightarrow \mathcal{P}$be the projection along$\beta]$$s_{j},$$;=\bigcup_{N=1}^{\infty}Y_{N}$$:=$$X_{i}$$:=$$j\in N$

, $N\}$ ,$k=1,2,$$\cdots,$$N,$$i=1,2,3\}$ ,,$;=\bigcup_{N=1}^{\infty}Y_{N,i}$$X$$\cdots$,

the closure of $Y$,
the closure of ,$Y_{i}$are given by$ u=\lim_{n\rightarrow\infty}\sigma^{n}(1)=s_{1}s_{2}\cdots s_{k}\cdots\cdots$

Then we can find the domain $X$ and$X_{i},$

4$i=1,2,3$ with fractal boundaries.

On Example 1 [Rauzy substitution]:

Figure 1: The figure of the domain $X$ with fractal boundaries on Rauzy substitution

On Example 2 [Modified Jacobi-Perron substitution]: in the case of $a=1$.
and$\epsilon=0$

Figure 2: The figure of the domain $X$ with fractal boundaries on Modified Jacobi-Perronsubstitutions

5Note. We introduce the substitution onin this lecture, but it is easy to extend the definition of substitutions on the alphabet$\{1, 2\cdots, N\}$ . In particular, the simplest case of $N=2$ is fundamental. In
this note, we will give a kind of survey at the end of each sections in the caseof $N=2$ .
On the alphabet{l, 2}, we can define Pisot substitution and its matrix. Using the column and row eigenvectors with respect to the maximum
eignvalue :$W^{*}=\bigcup_{n=1}^{\infty}\{1,2.3\}^{n}$$\sigma$$L_{\sigma}$$\lambda$$L_{\sigma}\left\{\begin{array}{l}1\\\alpha\end{array}\right\}=\lambda\left\{\begin{array}{l}l\\\alpha\end{array}\right\}$,${}^{t}L_{\sigma}\left\{\begin{array}{l}1\\\gamma\end{array}\right\}=\lambda\left\{\begin{array}{l}l\\\gamma\end{array}\right\}$

we can define the contracting invariant line
$l_{\sigma}=\{x|\langle x,$

and the projection
tuion satisfying

$\pi$

:

$R^{2}\rightarrow l_{\sigma}$along$l_{\sigma}$,with respect to$L_{\sigma}$

by$\left\{\begin{array}{l}l\\\gamma\end{array}\right\}\rangle=0\}$${}^{t}[1,$$\alpha]$analogously. And for the substi-$\sigma(1)=1W_{1}$$i=1,2$ be projective sets as in Page 4. Then
$i=1,2$ , and $X,$
$X;,$
$X,$
$i=1,2$
which are intervals usually on the following
we find the set

let

$Y,$

$X_{2},$$Y_{i},$

example.

Example 3 In the case of $N=2$ , for each $a\in N$ let us define the substituby
ions$\sigma_{a}$$\rightarrow\hat{11\cdots 1}2atimes$$\sigma_{a}:1$

2
The matrix of$\sigma_{a}$$\rightarrow$1

are given by$L_{\sigma_{a}}=\left\{\begin{array}{ll}a & 1\\l & 0\end{array}\right\}$$a\in N$ are called continued fraction substituions (TheThe substituion
definition of the continued fraction algorithm can be found in the section 6).
$\sigma.,$6

2

Endomorphism

For $x\in Z^{3}$ and $i\in\{1,2,3\}$ let us consider the pair$(x, i^{*})$

by$(x, i^{*}):=\{x+\lambda e_{j}+\mu e_{k}|0\leq\lambda\leq 1, 0\leq\mu\leq 1\}$

where $(j, k)$ is taken as $(i,j, k)\in\{(1,2,3), (2,3,1), (3,1,2)\}$ .

Figure 3: The figure of

All of the pair$(x, i^{*}),$$x\in Z^{3}$$(0, i^{*}),$$i=1,2,3$ and $(x, 1^{*})$

and $i\in\{1,2,3\}$ is denoted by$\Lambda=\{(x, i^{*})|x\in Z^{3},$ $i=\{1,2,3\}\}$

Let$\mathcal{G}$$\Lambda$, that is,.

be the $Z$ -module generated by A as follows:$\mathcal{G}=\{\sum_{\lambda\in\Lambda}m_{\lambda}\lambda$$\#\{\lambda|m_{\lambda}\neq 0\}<+\infty\}$$m_{\lambda}\in Z,$

Let us consider the followin endomorphism$g$$\Theta$of$\mathcal{G}$associated with :,$\Theta(0, i^{*})$$;=\sum_{j=1,23},$$\Theta(x, i^{*})$$;=$$\sigma$$\sum_{W:,\sigma(j)_{-}^{-}1’i\cdot W}.(L_{\sigma}^{-1}(f(W)),j^{*})$$L_{\sigma}^{-1}x+\Theta(0, i^{*})$

where $y+(x, i^{*})$ $:=(y+x, i^{*})$ , and for$\sum_{\lambda\in\Lambda}m_{\lambda}\lambda\in \mathcal{G}$$\Theta(\sum_{\lambda\in\Lambda}m_{\lambda}\lambda):=\sum_{\lambda\in\Lambda}m_{\lambda}\Theta(\lambda)$.

To discuss the geometrical property of , we introduce the stepped surface
of a plane.
$\Theta$7

For any $0<\gamma,$$\delta<1$

let us consider the plane$\mathcal{P}_{\gamma,\delta}:=\{x|\langle x,$

and for each planelows:$\mathcal{P}_{\gamma,\delta}$. that is,$\left\{\begin{array}{l}l\\\gamma\\\delta\end{array}\right\}\rangle=0\}$

let us conider the stepped surface$:=\{(x, i^{*})|\langle x,$$\left\{\begin{array}{l}l\\\wedge(\\\delta\end{array}\right\}\rangle>0,$$(S_{\gamma,\delta}^{-}:=\{(x, i^{*})|\langle x,$$\left\{\begin{array}{l}l\\\gamma\\\delta\end{array}\right\}\rangle\geq 0,$$S_{\gamma,\delta}^{+}$$\mathcal{P}_{\gamma.\delta}$$\langle(x-e_{i}),$$\langle(x-e_{i}),$$\mathcal{S}_{\gamma,\delta}^{+}(S_{\gamma,\delta}^{\sim})$as fol-$\left\{\begin{array}{l}1\\\gamma\\\delta\end{array}\right\}\rangle\leq 0\}$$\left\{\begin{array}{l}1\\\gamma\\\delta\end{array}\right\}\rangle<0\})$and$(x, i^{*})$$\cup$$\mathcal{S}_{\gamma,\delta}^{+}:=$$(X,i.)\in S_{\gamma,\delta}^{+}$$\left(\begin{array}{lll}\mathcal{S}_{\gamma,\delta}^{-}.= & \cup & (x,i^{*})\\ & (X,i.)\in S_{\gamma,\delta}^{-} & \end{array}\right)$

Let$\Gamma_{\gamma,\delta}^{+}(\Gamma_{\gamma,\delta}^{-})$

be$\Gamma_{\gamma,\delta}^{+}:=\{\bigcup_{\lambda\in S_{\gamma,6}^{+}}n_{\lambda}\lambda|$$n_{\lambda}\in\{0,1\},$$\#\{\lambda|n_{\lambda}=1\}<+\infty\}$$(\Gamma_{\gamma,\delta}^{-}:=\{\bigcup_{\lambda\in S_{\gamma,\delta}^{-}}n_{\lambda}\lambda n\lambda\in\{0,1\}, \#\{\lambda|n_{\lambda}=1\}<+\infty\})$is a family of
The setfind the following theorem.
$\Gamma_{\gamma,\delta}^{+}(\Gamma_{\gamma,\delta}^{-})\subset \mathcal{G}$$S_{\gamma,\delta}^{+}(S_{\gamma,\delta}^{+})$

on$S_{\gamma,\delta}^{+}(S_{\gamma,\delta}^{-})$. Then we

Theorem 2.1 ([A-I]) For each Pisot substitution let us consider the endowith respect to the contracting
morphismof associated with and$\Theta$$\mathcal{G}$$\sigma$$\Gamma_{\gamma,\delta}^{+}$

8

invariant plane, that is,$\mathcal{P}_{\gamma.\delta_{i}}$then$\Gamma_{\gamma,\delta}^{+}$

is invariant with respect to the endomorphism$\Theta$

implies$\Theta(\bigcup_{\lambda\in S_{\gamma.\delta}^{+}}n_{\backslash }\lambda)\in\Gamma_{\gamma_{\backslash }\delta}^{+}$$\bigcup_{\lambda\in S_{\gamma,\delta}^{+}}n_{\lambda}\lambda\in\Gamma_{\gamma,\delta}^{+}$$(\bigcup_{\lambda\in S_{\gamma,\text{{\it \’{o}}}}^{\sim}}n_{\lambda}\lambda\in\Gamma_{\gamma,\delta}^{\sim}$

impliesOn Example 1, for the substitution$\Theta\left(\begin{array}{ll}\cup & n_{\lambda}\lambda\\\lambda\in S_{\gamma,\delta}^{-} & \end{array}\right)\in\Gamma_{\gamma.\delta}^{\sim})$$\sigma$1: 2$\rightarrow$$\rightarrow$3$\rightarrow$

12
13 the endomorphism
1$\Theta$

is given by

where$(0,1^{*})$$\rightarrow$$(0.3^{*})+(f_{2},1^{*})+(f_{3},2^{*})$$\Theta:(0,2^{*})$$\rightarrow$$(0,1^{*})$$(0,3^{*})$$\rightarrow$$(0,2^{*})$$L_{\sigma}^{-1}=[100$$-101$$-|0]=[f_{1}, f_{2}, f_{3}]$. The figure offollows:

Figure 4: The figure of

9$\Theta$$\Theta$

is given as

Figure 5: The figure of$\bigcup_{i=1,2,3}\Theta^{n}(0, i^{*}),$

Figure 6: The figure of

10$n=0,1,2,3,4$$\bigcup_{i=1,2,3}\Theta^{8}(0, i^{*})$

$\rightarrow\hat{11\cdots 1}2at\iota mes$

On Example 2, for the substitution$\sigma\left(\begin{array}{l}a\\0\end{array}\right)$

1
:
2
3

$\rightarrow$$3$$\rightarrow$1

and$\rightarrow\hat{11\cdots 1}3atimes$:$\sigma\left(\begin{array}{l}a\\1\end{array}\right)$

1
2

3$\rightarrow$the endomorphisms

1$\Theta\left(\begin{array}{l}a\\0\end{array}\right)$

and$\rightarrow 2$

given by$(0,1^{*})$$\rightarrow(0,3^{*})+\sum_{1-<k\leq a}((e_{1}-ke_{3}), 1^{*})$$\Theta\left(\begin{array}{l}a\\0\end{array}\right):(0,2^{*})$$\rightarrow$$(0,1^{*})$$(0,3^{*})$$\rightarrow$$(0,2^{*})$

and$(0,1^{*})$$\rightarrow(0,2^{*})+\sum_{1\leq k\leq a}((e_{1}-ke_{2}), 1^{*})$$\Theta\left(\begin{array}{l}a\\1\end{array}\right):(0,2^{*})$$\rightarrow$$(0,3^{*})$$(0,3^{*})$$\rightarrow$$(0,1^{*})$

The figures of$\Theta\left(\begin{array}{l}a\\\epsilon\end{array}\right)’\epsilon=0,1$

are as follows:

Figure 7: The figure of

11$\Theta$$\Theta\left(\begin{array}{l}a\\l\end{array}\right)$areFigure 8: The figure of

Figure 9: The figure of$\bigcup_{i=1,2,3}\Theta^{8}(0, i^{*})$$\bigcup_{i=1,2,3}\Theta^{n}(0, i^{*}),$

and$n=0,1,$ $\cdots,5$$\bigcup_{i=1,2,3}\Theta^{10}(0, i^{*})$

in the case of $a=1,$$\epsilon=0$$i=1,2,3$ are
On Example 1 and Example 2, we can see that
simply connected. See [I-O] and [F-I] if you are interested in the geometrical$i=1,2,3$ are not$i=1,2,3$ . In general,properties ofsimply connected.
$\Theta^{n}(0, i^{*}),$$\Theta^{n}(0, i^{*}),$$\Theta^{n}(0, i^{*}),$

12

Example 4 This is an example that $\Theta^{n}(0, i^{*})_{1}i=1,2,3$ are not simply

connected. For the substitution$\sigma$1: 2
3$\rightarrow$$\rightarrow$$\rightarrow$

21
13, the endomorphism
1$\Theta$isgiven bywhere

$L_{\sigma}^{-1}$$(0,1^{*})$$\rightarrow$$(0,1^{*})+(0,3^{*})+(f_{3},2^{*})$$\Theta:(0,2^{*})$$\rightarrow$$(f_{1},1^{*})$$(0,3^{*})$$\rightarrow$$(0,2^{*})$$=\left\{\begin{array}{ll}01 & 0\\00 & 1\\l-l-l & \end{array}\right\}=[f_{1}, f_{2}, f_{3}]$. The figure offollows:

Figure 10: The figure of

13$\Theta$$\Theta$

is given as

Figure 11: The figure of$\bigcup_{t=1,2,3}\Theta^{n}(0, i^{*}),$

Figure 12: The figure of

14$n=0,1,2,3,4$$\bigcup_{i=1,2,3}\Theta^{10}(0, i‘‘)$Notation. For$n_{\lambda}\neq 0$

if$\gamma=\sum_{\lambda\in S_{\gamma,\text{{\it \’{o}}}}^{+}}n_{\lambda}\lambda,$$m_{\lambda}\neq 0$$\delta=\sum_{\lambda\in S_{\gamma,\delta}^{+}}m_{\lambda}\lambda\in\Gamma_{\gamma.\delta}^{+},$, that is, the patch$\delta$

is the subpatch of$\gamma$mea.ns tha.
(See Figure 13).

$\gamma\succ\delta$$t$$\delta$$\succ$$\gamma$Figure 13: The figure of$\gamma\succ\delta$

On the above notation we have the following lemma.
Lemma 2.2 Let$\mathcal{U}:=\bigcup_{i=1,2,3}(e_{i}, i^{*})$,$\mathcal{U}’:=\bigcup_{i=1,2,3}(0, i^{*})$

,

then
(i)
(ii)

$\Theta(\mathcal{U})\succ \mathcal{U}$$\Theta(\mathcal{U}’)\succ \mathcal{U}^{t}$Note. In the case of alphabet {1, 2}, the element
the set is given by$(x, i^{*})\in Z^{2}\times\{1^{*}, 2^{*}\}$$\Lambda$$(x, i^{*})=\{x+\lambda e_{j}|0\leq\lambda\leq 1\}$

where $(i,j)\in\{(1,2), (2,1)\}$ , and$\mathcal{G}=\{\sum_{\lambda\in\Lambda}m_{\lambda}\lambda$

The endomorphism$\Theta$$\mathcal{G}$is given by$m_{\lambda}\in Z,$$\#\{\lambda|m_{\lambda}\neq 0\}<+\infty\}$associated with$\sigma$ofis given analogously by$(L_{\sigma}^{-1}(f(W)),j^{*})$$\Theta(0, i^{*})$$;=\sum_{j=1,2}$$\sum_{W:}$$\sigma(j)=Y\cdot i\cdot W$$\Theta(x, i^{*})$$\mathcal{G}$$;=$$L_{\sigma}^{-1}x+\Theta(0, i^{*})$

15

,

,

of

and for$\sum_{\lambda\in\Lambda}m_{\lambda}\lambda\in \mathcal{G}$$\Theta(\sum_{\lambda\in\Lambda}m_{\lambda}\lambda):=\sum_{\backslash \prime\in\Lambda}m_{\lambda}\Theta(\lambda)$

For any $0<\gamma<1$ , let us consider the line
$l_{\gamma}=\{x|\langle x,$

and for each line

$l_{\gamma}$$l_{\gamma}$.

, that is,$\left\{\begin{array}{l}1\\\gamma\end{array}\right\}\rangle=0\}$,

let us consider the stepped curve$\mathcal{S}^{+}:=\{(x, i^{*})|\langle x,$$\left\{\begin{array}{l}1\\\gamma\end{array}\right\}\rangle>0,$$(\mathcal{S}^{-}:=\{(x, i^{*})|\langle x,$$\left\{\begin{array}{l}l\\\gamma\end{array}\right\}\rangle\geq 0,$$\langle(x-e_{i}),$$\langle(x-e_{i}),$$\mathcal{S}^{+}(\mathcal{S}^{-})$

as follows:$\left\{\begin{array}{l}l\\\gamma\end{array}\right\}\rangle\leq 0\}$$\left\{\begin{array}{l}l\\\gamma\end{array}\right\}\rangle<0\})$and$S^{+}:=$$(x, i^{*})$$\cup$$(X,i.)\in S+$$(S^{\sim}:=\bigcup_{(X,i)\in S-}(x, i^{*}))$

Let$\Gamma^{+}(\Gamma^{\sim})$be$\Gamma^{+}:=\{\bigcup_{\lambda\in S+}n_{\lambda}\lambda$$n_{\lambda}\in\{0,1\},$$\#\{\lambda|n_{\lambda}=1\}<+\infty\}$$(\Gamma^{\sim}:=\{\bigcup_{\lambda\in S^{-}}n_{\lambda}\lambda n_{\lambda}\in\{0,1\}, \#\{\lambda|n_{\lambda}=1\}<+\infty\})$then the same statement of Theorem 2.1 holds on this frame work.
On Example 3, for the substitution$\rightarrow\hat{11\cdots 1}2atimes$$\sigma:1$

2$\rightarrow$

16

1

,

the endomorphism$\Theta_{a}$is given by$\rightarrow$$(0.2^{*})+\sum_{k=1}^{a}((e_{1}-ke_{2}), 1^{*})$$\rightarrow$$(0,1^{*})$$\Theta_{a}:(0,1^{*})$$(0,2^{*})$

Figure 14: The figure of

Figure 15: The figure of$\bigcup_{i=1,2}\Theta^{n}(0, i^{*}),$

Example 5 For the substitution$\sigma$:

1
2$\Theta$$n=0,1,2,3$ , in the case of $a=1$$\rightarrow$$\rightarrow$121
the endomorphism
12

given by$\Theta:(0,1^{*})\rightarrow(0,1^{*})+(f_{1}+f_{2},1^{*})+(f_{2},2^{*})$$(0,2^{*})\rightarrow(f_{1},1^{*})+(0,2^{*})$

17$\Theta$iswhere$L_{\sigma}^{-1}=\left\{\begin{array}{ll}l & -l\\-l & 2\end{array}\right\}=[f_{1}, f_{2}]$. The figure of

Figure 16: The figure of

Figure 17: The figure of

Example 6 This is an example that$\Theta^{n}(0, i^{*}),$

18

is given as follows:$\Theta$$\bigcup_{i=1,2}\Theta^{n}(0, i^{*}),$On Example 3 and Example 5, we see thatconnected. But in general it is not true.

$\Theta$$n=0,1,2$$\Theta^{n}(0, i^{*}),$$i=1,2$ are simply$i=1,2$ are not simply con-

nected. For the substitution$\sigma$the endomorphism

where$\Theta_{a}$:1$\rightarrow$

112

2$\rightarrow$$21$

is given by$\Theta:(0,1^{*})$$\rightarrow$$(0,2^{*})+(f_{2},1^{*})+(f_{1}+f_{2},1^{*})$$(0,2^{*})$$\rightarrow$$(0,1^{*})+(f_{1},2^{*})$$L_{\sigma}^{-1}=\left\{\begin{array}{l}1-1\\-12\end{array}\right\}=[f_{1}, f_{2}]$.

Figure 18; The figure of

Figure 19: The figure of$\Theta$$\bigcup_{i=1,2}\Theta^{n}(0, i^{*}),$

19$n=0,1,2,3$

We have the following interested theorem.

Theorem 2.3 ([W] and [E-I]) $\Theta^{n}(0, i^{*})_{j}i=1,2$ are simply connected for
all iﬀ the substitution is invertible substitution, that is, there exists theautomorphism : $G\{1.2\}\rightarrow G\{1,2\}$ such that$n$$\sigma$$\theta$$\sigma_{0}\theta=\theta_{0}\sigma$

where $G\{1.2\}$ be a free group of rank 2 generated by $\{1, 2\}$ .

20

3Dynamical system

For each Pisot substitution and its endomorphism let us consider theas follows:
domains on the contracting invariant plane$\Theta$$\sigma$$\mathcal{P}_{\gamma,\delta}$

where
Notice thatknow that$D_{n}^{(i)}$$;=$$\pi(\Theta^{n}(e_{i}, i^{*}))$$D_{n}^{(i\rangle’}$$;=$$\pi(\Theta^{n}(0, i^{*}))$$D_{n}$$;=$

7$D_{n}’$$;=$$\pi(\Theta^{n}(\mathcal{U}’))$$\mathcal{U}=\bigcup_{i=1,2,3}(e_{i}, i^{*}),$$\mathcal{U}\in\Gamma_{\alpha,\beta}^{+},$

Therefore, the sets

, $i=1,2,3$ ,,$(\Theta^{n}(\mathcal{U}))$.
and by Theorem 2.1 and Lemma 2.2 we$\mathcal{U}’=\bigcup_{i=1,2,3}(0, i^{*})$$\mathcal{U}’\in\Gamma_{\alpha,\beta}^{-}$$D_{n}^{(i)},$, $i=1,2,3$ .

$\Theta^{n}(\mathcal{U})$$\succ$$\Theta^{n-1}(\mathcal{U})$$\Theta^{n}(\mathcal{U}’)$$\succ$$\Theta^{n-1}(\mathcal{U}’)$$D_{n}^{(i)’}$.

are well-defined and $D_{n}=D_{n}’$ (See [A-I] indetail).

We have the following theorem.

Theorem 3.1 Let$L_{\sigma}^{-n}$be denoted by$L_{\sigma}^{-n}=[f_{1}^{(n)},$and the following dynamical system$f_{2}^{(n)},$$W_{n}$

on$f_{3}^{(n)}]$$D_{n}$,

is well-defined:$\rightarrow^{W_{n}}$$D_{n}$$x$$D_{n}$$\mapsto$$x-\pi f_{i}^{(n)}$if$x\in D_{n}^{(t)},$$i=1,2,3$

and the dynamical systems $(D_{n}, W_{n})$ are isomorphic each other.

We will show the figures of$W_{n}$

on the examples.

21

On Example 1, for the substitution$\sigma$1: 2

3$\rightarrow$$\rightarrow$$\rightarrow$as follows:

Figure 20: The figures of22$W_{n}$

12
13
1

the figure of$W_{n}$isOn Example 2. for the substituion

12:$\sigma(_{0}1)$$W_{n}$

3is as follows:

Figure 21: The figure of23$W_{n}$$\rightarrow$$\rightarrow$$\rightarrow$

12
3 , the figure of1

intoTheorem 3.2 The induced transforrreationof$W_{n-1}$cides with
. More precisely, let us denote the substitution by$\nu V_{n}|_{D_{n-1}}$$D_{n1}-$$W_{n}$$\sigma$1$\rightarrow$$s_{1}^{(1)}s_{2}^{(1)}\cdots s_{l(1)}^{(1)}$$\sigma:2$$\rightarrow s_{1}^{(2)}s_{2}^{(2)}\cdots s_{l(2)}^{(2)}$3$\rightarrow s_{1}^{(3)}s_{2}^{(3)}\cdots s_{t(3)}^{(3)}$,then the following relation holds:(1)$VV_{n}^{k-1}(D_{n-1}^{(i)})\subset D_{n}^{(s_{k}^{(i)})}$(2)$W_{n}^{l(i)}(D_{n-1}^{(i)})=D_{n-1}^{(i)’}$, $k=1,2,$.

On Example 1, in the case of $n=3$ :Figure 22:24$\cdots,$$l(i)$

coin-

Corollary 3.3 Let us denote$\sigma^{n}$by$\sigma^{n}(i)=s_{1}^{(n,i)}s_{2}^{(n,i)}\cdots s_{1(n,i)}^{(n,i)}$,$i=1,2,3$ ,

then
(1)
$D_{n}(s_{k}^{(n,i)}),$$\nu V_{n^{-}}^{k1}(D_{0}^{(i\rangle})$$\subset$$W_{n}^{l(n,i)}(D_{0}^{(i)})$$=$$k=1,2,$$D_{0}^{(i)’}$$\cdots,$$l(n, i)$,(See Figure 23), and in particular, we have(2)$\{W_{n}^{k}(0)|k=1,2,$ $\cdots,$ $l(n, 1)\}=\{-\pi\sum_{j_{-}^{-}1}^{k}f_{s_{j}^{(n1)}}^{(n)}$
,

where

$L_{\sigma}^{-}n=[f_{1}^{(n)},$$f_{2}^{(n)},$$f_{3}^{(n)}]$(See Figure 24).

25

$k=1,2,$$\cdots,$$l(n, 1)\}$On Example 1, in the case of $n=3$ :Figure 23:26

Figure 24:

where $\sigma^{3}(1)=1213121$ . and$[f_{1}^{(3)},$$f_{2}^{(3\rangle},$$f_{3}^{(3\rangle}]=\left\{\begin{array}{ll}l-1-l & \\2-1 & 0\\0-l & 2\end{array}\right\}$

Remark 1 The domain exchange transformaiton
with the quasi-periodic motion, that is,$D_{0}$$W_{0}$$\rightarrow$$D_{0}$:$x$$\mapsto$$W_{0}:D_{0}\rightarrow D_{0}$$x-\pi e_{1}$(mod$L_{0}$

where $L_{0}=\{n\pi(e_{2}-e_{1})+m\pi(e_{3^{-}}e_{1})|m, n\in Z\}$ .

Figure 25: The figure of27$W_{0}$).
coincides

$\mapsto$$x$$x-\pi e_{1}$Figure 26: The figure of the quasi-periodic motion

Note. In the case of two alphabets, let us introduce the union of intervals
on the line analogously:$l_{\gamma}$$D_{n}^{(i)}$$D_{n}^{(i)’}$$;=$$\pi(\Theta^{n}(e_{i:}i^{*}))$$;=\pi(\Theta^{n}(0, i^{*})),$. $i=1.2$ .
$i=1,2$ ,$D_{n}$$;=\pi(\Theta^{n}(\bigcup_{i_{--}12},(e_{i}, i^{*})))$$D_{n}’$$;=$$\pi(\Theta^{n}(\bigcup_{-},(0, i^{*})))$

28

,
,

then the followin $g’$ ‘union of intervals” exchange on$D_{n}$$x$$D_{n}$$\rightarrow^{W_{n}}D_{n}$$\mapsto$$x-\pi f_{i}^{(n)}$if$x\in D_{n}^{(i)}$and it is isomorphic to the interval exchange$\rightarrow^{W_{0}}$$D_{0}$

$x$

where

$L_{\sigma}^{-n}=[f_{1}^{(n)},$$D_{0}$$\mapsto$$x-\pi e_{i}$$f_{2}^{(n)}]$29

if$x\in D_{0}^{(i\rangle}$,

is well defined:On Example 5, for the substitution$\sigma$:1$\rightarrow$

121

2$\rightarrow$$12$

as follows:

Figure 27: The figures of30$W_{n}$’

the figure of$W_{n}$isOn Example 6, for the substitution$\sigma$:1
2$\rightarrow$112$\rightarrow$$21$, the figure of$W_{n}$

isas follows:

Figure 28: The figures of$W_{n}$The same statements given by Theorem 3.2, Corollary 3.3 hold in the caseof alphabets {1, 2}.

31

4Renormalization

Let us consider the following limit set in the sense of Hausdorﬀ metric of thefamily of compact subsets of:$\mathcal{P}_{\gamma.\delta}$

$X$

$:=$

$\lim_{n\rightarrow\infty}L_{\sigma}^{n}(\pi(\Theta^{n}(\mathcal{U})))(=\lim_{n\rightarrow\infty}L_{\sigma}^{n}(\pi(\Theta^{n}(\mathcal{U}’))))$$X_{i}$$:=$$\lim_{n\rightarrow\infty}L_{\sigma}^{n}(\pi(\Theta^{n}(e_{i}, i^{*})))$$X_{i}’$$;=\varliminf_{n\infty}L_{\sigma}^{n}(\pi(\Theta^{n}(0, i^{*})))$

,

,

.

Then the following theorem holds.

Theorem 4.1 Let be Pisot substitution and let us assume that there exists$N>0$ , $N\in N$ and such that$\sigma$$i$$\Theta^{N}(e_{i}, i^{*})\succ \mathcal{U}$.

Then the limit set satisfies the following properties:(1)(disjoint),$X=\bigcup_{i=1,2,3}X_{i}$that is, $|X_{i}\cap X_{j}|=0(i\neq j)$ in the sense of Lebesgue measure, and(disjoint),$\mathcal{P}_{\gamma,\delta}=$$\bigcup_{z\in L_{0}}(X+z)$

that is, $|(X+z)\cap(X+z’)|=0(z\neq z’)$ in the sense of Lebesgue
measure, where $L_{0}=\{n\pi(e_{2}-e_{1})+m\pi(e_{3}-e_{1})|m, n\in Z\}$ .
(2) The transformation $W:X\rightarrow X$

if$Wx=x-\pi e_{i}$is well-difined and isomorphic to$x\in X_{i}$$W_{0}:D_{0}\rightarrow D_{0}$(3) The induced transformation$W|_{L_{\sigma}X}$.

is isomorphic to $W$ and sat-

isfies$W^{k-1}(X_{i}^{(1)})$$\subset$$X_{s_{k}^{(:)}}$$W^{l(i)}(X_{i}^{(1)})$

where$\sigma(i)=s_{1}^{(i)}\cdots s_{l(i)}^{(i)}$and,$k=1,2,$$\cdots,$$l(i)$$=X_{i}^{(1)’}$$X_{i}^{(1)}:=L_{\sigma}X_{i}(X_{i}^{(1)’} :=L_{\sigma}X_{i}’)$32.

(4) The transformation $T:X\rightarrow X$

if$Tx=L_{\sigma}^{-1}x-L_{\sigma}^{-1}(f(W))+e_{j}$where the word

$L_{\sigma}^{-1}x\in L_{\sigma}^{-1}(f(W))+X_{j}$is given by the following formula:$\Theta(0, i^{*}):=\bigcup_{j=1,2,3}$$\bigcup_{W,\sigma(j\rangle=Y\cdot i\cdot W}(L_{\sigma}^{-1}(f(W)),j^{*})$.

is well-defined and the transformation $T$ is Markov endomorphism whose
stru cture matrix is.
$L_{\sigma}$

Remark. In Theorem 4.1, we find the assumption that there exists $N$ andsuch that.

$i$$\Theta^{N}(e_{i}, i^{*})\succ \mathcal{U}$We don’t have the example which does not hold above. believe that theassumption holds for any Pisot substitution. But it is still open problem evenin the case of two alphabets. If it is . $K$ ., then we can say the dynamicalsystem associated with the substitution has the discrete spectrum, moreover
it is isomorphic to the quasi-periodic motions.
$I$

$0$

On Example 1, the transformation $W$ is given as follows:

Figure 29: The figure of $W$

33

On Example 2 (in the case of$ a=1, \epsilon=0$ ), the transformation
as follows:

Figure 30: The figure of $W$

On Example 4, the transformation $W$ is given as follows:

Figure 31: The figure of $W$34

is given5

Fractal boundaries

The boundary of the domain $X$ seems to be fractal. To observe the property
of boundaries, let us introduce some notations.
Let $(x, i)\in Z^{3}\times\{1,2,3\}$ be$(x, i)=\{x+\lambda e_{t}|0\leq\lambda<1\}$

Figure 32: The figure ofand let us define the boundary map$\partial(x, i^{*})$$:=$$\partial$$(0, i^{*}),$, $i=1,2,3$ ,$i=1,2,3$

by$(x,j)+(x+e_{j}, k)-(x, k)-(x+e_{k},j)$where $\{i,j, k\}$ is taken in $\{\{1,2,3\}, \{2,3,1\}, \{3,1,2\}\}$ . For each
let us define the boundary of by$\gamma=\Sigma_{\lambda\in\Lambda}m_{\lambda}\lambda\in$$\mathcal{G}$$\gamma$$\partial\gamma=\sum_{\lambda\in\Lambda}m_{\lambda}\partial\lambda$,

and the all of the set
is denoted by . Then
is a $Z$ -module.
Now, let us define the boundary endomorphism associated withif
there exists the endomorphism which satisfies the following commutative
relation:$\partial\gamma,$$\gamma\in \mathcal{G}$$\mathcal{G}_{1}$$\mathcal{G}_{1}$$\theta$$\theta$

35$\Theta$

Theorem 5.1 (Ito-Sano) Let us denote Pisot substitution$\sigma(i)=s_{1}^{(i)}\cdots s_{l(i)}^{(i)}$and let us define$\theta$$\sigma$by, $i=1,2,3$ ,by$t\wedge u)1\leq t\leq 3_{s_{l}}m-k(t)$$\sum_{\simeq j,1\leq u\leq 3(u)_{-}}(L_{\sigma}^{-1}(f(S_{l}^{(t)}))+L_{\sigma}^{-1}(f(S_{m}^{(u)})),$$\theta(0, i):=\sum_{\mathcal{S}}$where $\{i, j, k\}$ is taken in $\{\{1,2,3\}, \{2,3,1\}, \{3,1,2\}\}$ and $(x,t\wedge u),$$\{1,2,3\}$ means$t,$$u\in$$(x, t\wedge u)=\left\{\begin{array}{ll}(x, s) & if (t,u)\in\{(2,3), (3,1), (1,2)\}\\-(x, s) & if (t,u)\in\{(3,2), (1,3), (2,1)\}\end{array}\right.$

where $\{s, t, u\}=\{1,2,3\}$ and$\sigma(t)$$S_{l}^{(t)}$means the suﬃx of$=$$s_{1}^{(t)}\cdots s_{l}^{(t)}\cdots s_{l(t)^{(t)}}$$=$$P_{l}^{(t)_{S_{l}}(t)}S_{l}^{(t)}$$s_{l}^{(t)}$in$\sigma(t)$, that is,,.

For $(x, i)$ let us define$\theta(x,i)$$L_{\sigma}^{-1}x+\theta(0,i)$

$;=$

,

$;=$
$\sum_{\lambda\in\Lambda}n_{\lambda}\theta(\lambda)$$\theta(\sum_{\lambda\in\Lambda}n_{\lambda}\lambda)$

where $y+(x, i):=(y+x, i)$ . Then the mapof .

$\theta$

is the bounday endomorphism$\Theta$

On Example 1,$\theta$

is given by the followin manner:$g$$\theta(0,1)$$=$$(0,1\wedge 2)=(0,3)$ ,$\theta(0,2)$$=$$(L_{\sigma}^{-1}(e_{2}),$$=$$-(\left\{\begin{array}{l}1\\0\\-l\end{array}\right\}, 3)+(0,1)$$=$$(L_{\sigma}^{-1}(e_{3}),$$=$$-(\left\{\begin{array}{l}0\\l\\-1\end{array}\right\}, 3)+(0,2)$$\theta(0,3)$36$2\wedge 1)+(0,2\wedge 3)$,$2\wedge 1)+(0,3\wedge 1)$.

Figure 33: The figure of$\theta$Note. Using the boundary endomorphism , if satisfies some conditions
with respect to the cancellation, then we can caluculate the Hausdorﬀ dimention explicitely. (See [I-K], [I-O 93] and [I-O 91]).
$\theta$37$\theta$6

Diophantine algorithm and substitutions

Let $X$ be the domain given by $X=[0,1)\times[0,1)$ and let us define thetransformation on by$T$$T(\alpha, \beta):=\left\{\begin{array}{ll}\}_{\frac{\frac{\beta}{\alpha 1}}{\beta’}-[\frac{1}{\beta}],\frac{\alpha]}{\beta}I}^{\frac{1}{\alpha}-[\frac{1}{\alpha})} & i_{f}^{f} (\alpha,\beta)\in X_{1}(\alpha,\beta)\in X_{0}-\{(0,0)\}\\(0,0) & if (\alpha, \beta)=(0,0)\end{array}\right.$

where$X_{0}$$=$$\{(\alpha, \beta)|\alpha\geq\beta\}$,$X_{1}$$=$$\{(\alpha, \beta)|\alpha<\beta\}$.

By using the integer value functions$ifif$$a(\alpha, \beta)$$;=$$\epsilon(\alpha, \beta)$$;=$$\{\left\{\begin{array}{l}\frac{1}{\alpha}\\1\\\beta\end{array}\right\}$$(\alpha,\beta)\in X_{1}(\alpha,\beta)\in X_{0}$$\left\{\begin{array}{ll}0 & if (\alpha,\beta)\in X_{0}\\1 & if (\alpha,\beta)\in X_{1}\end{array}\right.$

on $X-\{(0,0)\}$ , we define for each $(\alpha, \beta)\in X-\{(0,0)\}$ a sequence of digits${}^{t}(a_{n},\epsilon_{n})$by${}^{t}(a_{n},\epsilon_{n}):=t(a(T^{n-1}(\alpha,\beta)),$ $\epsilon(T^{n\sim 1}(\alpha, \beta)))$The triple
And we denote$(X, T, (a(\alpha, \beta),\epsilon(\alpha, \beta)))$if $T^{n-1}(\alpha, \beta)\neq(0,0)$ .

is called Modified Jacobi-Perron algorithm.

$(\alpha_{n}, \beta_{n}):=T^{n}(\alpha, \beta)$.

For the modified Jacobi-Perron algorithm, we introduce a transformation
called a natural extension of the modified Jacobi-Perron algorithm.
Letand let us define the transformation on by$(X, \overline{T})$$\overline{X}=X\times X$$\overline{T}$$\overline{X}$$\overline{T}(\alpha, \beta,\gamma, \delta)=\left\{\begin{array}{ll}\}_{-a_{1^{-\frac{}{},\frac{}{})}}}^{\frac{\beta}{\frac{}{}\beta\alpha 1},\frac{1}{\alpha}-a_{1},\frac{\delta}{a_{1}+\delta a_{1}+\gamma 1},\frac{1}{a_{1}+\delta a_{1}+\gamma\gamma})}\alpha\beta’ & ifif (\alpha,\beta)\in X_{1}(\alpha,\beta)\in X_{0}-\{(0,0)\}\\(0,0, \gamma, \delta) & if (\alpha, \beta)=(0,0)\end{array}\right.$

38

Then we know that the transformation$\overline{T}$is bijective from $(X-\{(0,0)\})\times X$

to $X\times(X-\{(0,0)\})$ .

We denote$(\alpha_{n}, \beta_{n}, \gamma_{n}, \delta_{n}):=\overline{T}^{n}(\alpha, \beta, \gamma, \delta)$.

Let us introduce the family of matrices as follows:$A\left(\begin{array}{l}a\\0\end{array}\right)=\left\{\begin{array}{lll}a & 0 & 1\\1 & 0 & 0\\0 & l & 0\end{array}\right\}$

for each integral
formulas:${}^{t}(a,\epsilon),$$a\in N,$,

$A\left(\begin{array}{l}a\\l\end{array}\right)=\left\{\begin{array}{lll}a & 1 & 0\\0 & 0 & l\\1 & 0 & 0\end{array}\right\}$$\epsilon=\{0,1\}$. Then we have the following$\left(\begin{array}{l}l\\\alpha_{n}\\\beta_{n}\end{array}\right)$$A^{-1}\left(\begin{array}{l}a_{1}\\\epsilon_{1}\end{array}\right)\left(\begin{array}{l}1\\\alpha\\\beta\end{array}\right)$$=\frac{1}{\theta\theta_{1}\cdots\theta_{n\sim 1}}A^{-}\left(\begin{array}{l}a_{n}\\\epsilon_{n}\end{array}\right)\left(\begin{array}{l}a_{n1}\\\text{{\it \’{e}}}_{n}1\end{array}\right)\cdots$$\left(\begin{array}{l}l\\\gamma_{n}\\\delta_{n}\end{array}\right)$,$=\frac{1}{\eta\eta_{1}\cdots\eta_{n\sim 1}}{}^{t}A{}^{t}A{}^{t}A\left(\begin{array}{l}a_{n}\\\epsilon_{n}\end{array}\right)\left(\begin{array}{l}a_{n- 1}\\\epsilon_{n- 1}\end{array}\right)\cdots\left(\begin{array}{l}a_{1}\\\epsilon_{1}\end{array}\right)\left(\begin{array}{l}1\\\gamma\\\delta\end{array}\right)$

where$\theta_{k}$$\eta_{k}$$=\max(\alpha_{k}, \beta_{k})$$=$,$\left\{\begin{array}{ll}a_{k}+\gamma_{k\sim 1} & if (\alpha_{k-1}, \beta_{k-1})\in X_{0}\\a_{k}+\delta_{k_{-}1} & if (\alpha_{k-1},\beta_{k-1})\in X_{1}\end{array}\right.$

Let us introduce a transformation:by$R^{3}\rightarrow R^{3}$$\varphi\left(\begin{array}{l}a_{n}\\\epsilon_{n}\end{array}\right)$$\left(\begin{array}{l}x_{n}\\y_{n}\\z_{n}\end{array}\right)=\varphi^{-1}\left(\begin{array}{l}a_{n}\\\epsilon_{n}\end{array}\right)\left(\begin{array}{l}x_{n-1}\\y_{n-1}\\z_{n-1}\end{array}\right):=A^{-1}\left(\begin{array}{l}a_{n}\\\epsilon_{n}\end{array}\right)\left(\begin{array}{l}x_{n-1}\\y_{n-l}\\z_{n-1}\end{array}\right)$

Then we see$\varphi^{-1}\left(\begin{array}{l}a_{n}\\\epsilon_{n}\end{array}\right)\mathcal{P}_{\gamma_{n-1},\delta_{n-1}}=\mathcal{P}_{\gamma_{n},\delta_{n}}$Moreover the substitution$\Theta\left(\begin{array}{l}a\\\epsilon\end{array}\right)$.

satisfies the following property:$\Theta\left(\begin{array}{l}a_{n}\\\epsilon_{n}\end{array}\right)\cdots\Theta\left(\begin{array}{l}a_{2}\\\epsilon_{2}\end{array}\right)\Theta\left(\begin{array}{l}a_{1}\\\epsilon_{1}\end{array}\right)(\mathcal{U})\in\Gamma_{\gamma_{n},\delta_{n}}^{+}$

39.

Now let us consider the renormalization of$\pi^{n}\Theta\Theta\Theta\left(\begin{array}{l}a_{n}\\\epsilon_{n}\end{array}\right)\cdots\left(\begin{array}{l}a2\\\epsilon_{2}\end{array}\right)\left(\begin{array}{l}a1\\\epsilon_{1}\end{array}\right)(\mathcal{U})$that is,. ..
$\varliminf_{n\infty}L^{-1}\left(\begin{array}{l}a1\\\epsilon_{1}\end{array}\right)$

where$\pi_{n}:R^{3}\rightarrow \mathcal{P}_{\gamma_{n},\delta_{n}}$$L^{-1}\pi_{n}\Theta\Theta(\mathcal{U})\left(\begin{array}{l}a_{n}\\\hat{e}n\end{array}\right)\left(\begin{array}{l}a_{n}\\\epsilon_{n}\end{array}\right)\cdots\left(\begin{array}{l}a_{1}\\\epsilon_{1}\end{array}\right)$

be the projection along$\left\{\begin{array}{l}l\\\alpha_{n}\\\beta_{n}\end{array}\right\}$

Theorem 6.1 ([I-O 93]) For almost everywhere
exist the limit sets
$X_{\alpha,\beta,\gamma,\delta}^{(i)}$,$L^{\sim 1}\pi_{n}\Theta\Theta\left(\begin{array}{l}a_{n}\\\epsilon_{n}\end{array}\right)\left(\begin{array}{l}a_{n}\\\epsilon_{n}\end{array}\right)\cdots\left(\begin{array}{l}a_{1}\\\epsilon_{1}\end{array}\right)(e_{i}, i^{*})$$n\rightarrow\infty\left(\begin{array}{l}a1\\e_{1}\end{array}\right)hmL^{-1}$

. ..

$:=$

there

$\delta\in[0,1)\times[0,1)$

. ..

$:=$

$X_{\alpha,\beta,\gamma,\delta}^{(i)’}$$\gamma,$,$L^{-}1\pi_{n}\Theta\Theta\left(\begin{array}{l}a_{n}\\\text{\’{e}} n\end{array}\right)\left(\begin{array}{l}a_{n}\\\epsilon_{n}\end{array}\right)\cdots\left(\begin{array}{l}a1\\\epsilon_{1}\end{array}\right)(0, i^{*})$$\lim_{n\rightarrow\infty}L^{-1}\left(\begin{array}{l}a1\\\text{\’{e}}_{1}\end{array}\right)$

and satisfy the following property:(1)$X_{\alpha,\beta,\gamma,\delta}=\bigcup_{i=1,2,3}X_{\alpha,\beta,\gamma,\delta}^{(i)}$is the periodic tiling onthat is,$\mathcal{P}_{\gamma,\delta_{j}}$$\cup(X_{\alpha,\beta,\gamma,\delta}+z)=\mathcal{P}_{\gamma,\delta}$$z\in L_{0}$

and

int.

$(X_{\alpha,\beta,\gamma,\delta}+z)\cap int.(X_{\alpha,\beta,\gamma,\delta}+z’)=\phi(z\neq z’)$(2) the domain exchange transformation$W_{\alpha,\beta,\gamma,\delta}$:$X_{\alpha,\beta,\gamma,6}\rightarrow X_{\alpha.\beta,\gamma,\delta}$

such that$W_{\alpha,\beta,\gamma,\delta}(x)=x-\pi_{0}e_{i}$if$x\in X_{\alpha,\beta,\gamma,\delta}^{(:)}$

is well-defined. Moreover, put$X_{n}:=\bigcup_{i=1,2,3}X_{\alpha_{n},\beta_{n},\gamma_{n},\delta_{n}}^{(i)}\subset \mathcal{P}_{\gamma_{n},\delta_{n}}$and$\hat{X}_{n}$

. ..

$;=$

$L^{-1}(X_{n})\subset X_{\alpha,\beta,\gamma,\delta}\left(\begin{array}{l}a_{n}\\\epsilon_{n}\end{array}\right)$$L^{-1}\left(\begin{array}{l}a_{1}\\\epsilon_{1}\end{array}\right)$$\hat{X}_{n,i}$

’

. ..

$;=$

$L^{-1}(X_{\alpha_{n},\beta_{n},\gamma_{n},\delta_{n}}^{(i)})\subset X_{\alpha,\beta,\gamma,\delta}\left(\begin{array}{l}a_{n}\\e_{n}\end{array}\right)$$L^{-1}\left(\begin{array}{l}a_{1}\\e_{1}\end{array}\right)$

40

’

,

then
$\nu V_{\alpha,\beta^{1}\gamma.\delta}^{k-}(X_{\alpha_{n},\beta_{n},\gamma_{n},\delta_{n}}^{(i)})\subset X_{\alpha,\beta,\gamma,\delta_{\dot{}}}^{s_{k}^{(n,i)}}$$1\leq k\leq q_{n}+p_{n}+r_{n}$,$W_{\alpha,\beta,\gamma,\delta}^{q_{n}+p_{n}+r_{n}}(X_{\alpha_{n},\beta_{n},\gamma_{n}.\delta_{n}}^{(i)})=X_{\alpha\beta,\gamma,\delta}^{(i.)’}$

where $\sigma\cdots\sigma(i)=s_{1}^{(n,i)}\cdots s_{l(n.i)}^{(n,i)}$ and$\left\{\begin{array}{lll}q_{n} & * & *\\p_{n} & * & *\\r_{n} & * & *\end{array}\right\}:=A\left(\begin{array}{l}a_{1}\\\epsilon_{1}\end{array}\right)\left(\begin{array}{l}a_{2}\\\epsilon_{2}\end{array}\right)$...
$A\left(\begin{array}{l}a_{n}\\\epsilon_{n}\end{array}\right)$

Note. In the 1-dimensional case, we consider the continued fraction algorithm
on $[0,1]$$T\alpha=\frac{1}{\alpha}-[\frac{1}{\alpha}]$and the contminued fraction expansion:$\alpha=\frac{1}{a_{1}+\frac{1}{a_{2}+\frac{1}{1}}}$$.+\frac{1}{a_{n}+T^{n}\alpha_{n}}$whereInstead of the plane$a_{k}:=[\frac{1}{T^{k-1}\alpha}]$$\mathcal{P}_{\gamma,\delta}$, we introduce the line

$l_{\gamma}:=\{\left\{\begin{array}{l}x\\y\end{array}\right\}|\langle\left\{\begin{array}{l}x\\y\end{array}\right\},$and whose stepped surfaces$\overline{T}$$S_{\gamma}$$l_{\gamma}$,$\left\{\begin{array}{l}l\\\gamma\end{array}\right\}\rangle=0\}$. We also introduce the natural extension ofby$\overline{T}(\alpha,\gamma):=(\frac{1}{\alpha}-a_{1}, \frac{1}{a_{1}+\gamma})$

Let us introduce a map$\varphi_{a}:R^{2}\rightarrow R^{2}$by$\left(\begin{array}{l}x_{n}\\y_{n}\end{array}\right)=\varphi_{a}^{-1}\left(\begin{array}{l}x_{n-1}\\y_{n}1\end{array}\right):=A_{a}^{-1}\left(\begin{array}{l}x_{n\sim 1}\\y_{n-1}\end{array}\right)$

41where$A_{a}^{-1}=\left\{\begin{array}{ll}a_{1} & 1\\l & 0\end{array}\right\}=\left\{\begin{array}{ll}0 & 1\\l & -a_{1}\end{array}\right\}$

Then the map$\varphi_{a}$satisfies$\varphi_{a_{n}}^{-}1l_{\gamma_{n-}1}=l_{\gamma_{n}}$where$(\alpha_{n},\gamma_{n}):=T^{n}(\alpha,\gamma)$

Then we know$\Theta_{a_{n}}(\mathcal{S}_{\gamma_{n-1}})=\mathcal{S}_{\gamma_{n}}$where$\Theta_{a}$is given in Note of the section 3.

42

.

7

Applications

7.1

Quasi-periodic tiling related to the stepped surfaceLet$\mathcal{P}_{\gamma.\delta},$$0<\gamma,$ $\delta<1$

be the plane given by$\mathcal{P}_{\gamma,6}=\{x|\langle x,$

and letLet$S_{\gamma,\delta}$$\left\{\begin{array}{l}1\\\gamma\\\delta\end{array}\right\}\rangle=0\}$be the stepped surface with respects to$\pi:R^{3}\rightarrow \mathcal{P}_{\gamma,\delta}$

along$\left\{\begin{array}{l}l\\\alpha\\\beta\end{array}\right\}$,$\mathcal{P}_{\gamma,\delta}$.

, then we have a tiling$\pi S_{\alpha,\beta}$

generated$i=1,2,3$ and whose translations.
by three parallelograms. Let
Let us denote the above tiling by
be the family ofpatches which is generated by parallelograms and simply conneted, that is,$\pi(0, i^{*}),$$\Gamma_{n}$$\mathcal{T}_{\alpha.\beta}(=\pi \mathcal{S}_{\alpha,\beta})$$n$$\Gamma_{n}=${$\gamma|\gamma\prec \mathcal{T}_{\alpha,\beta},$$\#\gamma=n,$$\gamma$

is simply connected}.

tilingDefinition 7.1
of a plane is said to be quasi-periodic if for any$n>0$ there exists $R>0$ such that any configuration
occurs somewhere$R$
in a neighbourhood of any point of the radius .
$A$$\mathcal{T}$$\gamma\in\Gamma_{n}$Theorem 7.2 Letbe the linearly independent with respect to $Q$ , then
the tilingis a quasi-periodic tiling.
$(1, \gamma, \delta)$$\mathcal{T}_{\gamma,\delta}$

The essential idea is coming from the following fact: for eachexists a sequence$(1, \gamma, \delta)$

there$\left(\begin{array}{lll}a_{1} & a_{2} & \cdots\\\epsilon_{1} & \epsilon_{2} & \cdots\end{array}\right)$

such that the stepped surface of$\mathcal{P}_{\gamma,5}$is given by$\lim_{n\rightarrow\infty}\Theta\left(\begin{array}{l}a_{1}\\\epsilon_{1}\end{array}\right)\Theta\left(\begin{array}{l}a_{2}\\\epsilon_{2}\end{array}\right)\cdots\Theta\left(\begin{array}{l}a_{n}\\\epsilon_{n}\end{array}\right)(\mathcal{U})$.

where the sequence is obtained by the modified Jacobi-Perron algorithm, andis appeared in Example 2 (See [I-O 94] in detail).
$\Theta\left(\begin{array}{l}a\\\epsilon\end{array}\right)$

43

7.2

Markov partition of group automorphisms on$T^{3}$Let us consider the following special matrix $A=L_{\sigma}$ which is given by Pisot
substitution, that is, which satisfies the assumption of Theorem 4.1. On the$i=1,2,3$ ofbyassumption, let us define the sets$R^{3}$$\overline{X}_{i},$$\overline{X}_{i}=\{(x, y)|x\in X_{i}, y=\lambda(e_{i}-\pi e_{i}), 0\leq\lambda<1\}$then the domain$\triangle$,:$\triangle:=\bigcup_{i=1,2,3}\overline{X}_{i}$is the 3-dimensional toms, that is,(1) $\bigcup_{z\in Z^{3}}(\triangle+z)=R^{3}$(2) int. $(\triangle+z)\cap int.(\triangle+z’)=\phi$ if $z\neq z’$ .

Theorem 7.3 Letthat is,$\xi$

be the partition of the 3-dimensional torus$\xi=\{\overline{X}_{i},$then the partition$\xi$$\triangle(\simeq T^{3})$,$i=1,2,3\}$ ,be the AMarkov partition with structure matrix${}^{t}L_{\sigma}$.

Note. The existence of Markov partitions of group automorphisms on
are discussed in [A-W] and [S]. Bowen claims that the boundary of Markov
partition of 3-dimensional group automorphisms must not be smooth in [Bo].
Theorem 7.3 says that how we can constmct the (not smooth) Markov partition (analogous discussion can be found in [Be]).
The following question is reasonable. For any element of $A\in SL(3, Z)$ ,satisfying the assumption in Thedoes there exist the substituion and$A$ ?
We only know with the private
isomorphic toorem 4.1? and isdiscussion between I and FURUKADO that for any there exists $N>0$
such thatwhich satisfies the assumption in Theorem 4.1.
$T^{n}$$L_{\sigma}$$\sigma$$L_{\sigma}$$A$$A^{N}$

44

7.3

Let

Diophantine approximation$\langle 1,$$\beta\rangle$$\alpha,$be the integer basis of the cubic field$Q(\lambda)$

given by

$A\left\{\begin{array}{l}1\\\alpha\\\beta\end{array}\right\}=\lambda\left\{\begin{array}{l}l\\\alpha\\\beta\end{array}\right\}$

for some $A\in SL(3, Z)$ , and let us assume that be a complex Pisot number.
On the above settin , let us consider the limit set of the points$\lambda$$g$$\{\sqrt{q}(q\alpha-pq\beta-r)|(q,p, r)\in Z^{3},$ $q>0\}$

Theorem 7.4 The limit set of above points consists of the family of ellipses.
Therem 7.4 is found by the method of algebraic geometry in [A]. But by
using the substitution, we can give another proof (See [F] in detail).

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[S] YA.G.SINAI, Construction of Markov partitions, Funk. Anal. Pril. 2, no.3
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