数理解析研究所講究録 1437 巻 2005 年 145-154

145

Tilings and Fractals from Pisot substitutions
Shunji ITO(Kanazawa University)

This is the note for the lecture at RIM

1(Kyoto University).

Definition of Pisot Unit Subsitutions( $G\{1$ ,$\cdots$,$d\}$$A$$:=$$\{1, 2, \cdots, d\}$$A^{*}$$:=$$n=0\cup A^{n}\infty$

(1)

a(i)

(alphabet)(free monoid, i.e., the set of finite words):afree group of rank d)

Definition 1.1(2)$\mathrm{S}$$=$$\sigma$: $A^{*}-A^{*}$ is$W^{(i)}\in A^{*}$$a$substitution if,$w_{k}^{(i)}\ldots w_{l_{i}}^{(i)},w_{k}^{(i)}W^{(i)}\neq\emptyset.\in A$$\sigma$$(w_{1}$$=$$w_{1}^{(i)}\ldots$$=$$P_{k}^{(i)}w_{k}^{(i)}S_{k}^{(i)}$. . . $w_{k}):=\sigma(w_{1})\cdots\sigma(w_{k})$ for(A substitutionLet$L_{\sigma}$;$\sigma$$w_{1}$. .. $w_{k}\in A^{*}$ .

is invertible if a is an automorphism on$G\{1_{\mathrm{t}}2$,$\cdots$$d\}$

be a matrix of a that is,$L_{\sigma}$

Example 1.1$(\mathrm{i},j)$the number of the letter contained in a (j) .
$\mathrm{i}$$:=$(Fibonacci substitution)$\sigma_{F}$Example 1.2

:

$11$

$arrowarrow$$121$$L_{\sigma_{F}}=\{\begin{array}{ll}1 1\mathrm{l} 0\end{array}\}$’.

(Rauzy substitution)$\sigma_{R}$

:

$111$

$arrowarrowarrow$$12131$$L_{\sigma_{R}}=\ovalbox{\tt\small REJECT}$’$011011$$0\ovalbox{\tt\small REJECT} 10^{\cdot}$. )14GAssumption

For the substitution ,$\mathrm{a}$

:.

(1)

$L_{\sigma}$is primitive, that is,(2)$L_{\sigma}$is unirnodular, that is,(3)$L_{\sigma}$

is Pisot type, that is, for eigenvalues$\exists N$$L_{\sigma}^{N}>0_{\rangle}$$\det L_{\sigma}=\pm 1$;$\lambda=\lambda_{1}$$\lambda=\lambda_{1}>1>|\lambda_{\iota}|$(4) the characteristic polynomialof$\Phi_{\sigma}(x)$$L_{\sigma}$,,$\lambda_{2}$,$\cdots$$\mathrm{i}=2$,,$\cdots$$\lambda_{d}$of$L_{\sigma}$,, ;$d$is irreducible;(5) $\sigma(1)=1W’$ .

We say the substitution a satisfying Assumption the Pisot Unit subsitution.
On Assumption (5),$w_{\sigma}:= \lim_{narrow\infty}\sigma^{n}(1)=s_{1}s_{2}\cdots s_{m}\cdots$.

is a fixed point of , that is,.
: $A’arrow Z^{d}$ is a homomorphism given by$\sigma$$\sigma(w_{\sigma})=$$w_{\sigma}$$f$$f(\mathrm{i}):=e_{i}$and$f(w_{1} \cdots w_{h}):=\sum_{j=1}^{k}f(w_{j})$.

Lemma 1.1 The following relation holds:$A^{*}$$f$$\downarrow$$Z^{d}$

Let
that is,$v$$arrow\sigma$$A^{*}$$f$$\downarrow$$arrow L_{\sigma}$$Z^{d*}$$>0$ be a positive eigenvector of A and $P$ be.

$R^{d}:=\mathcal{L}(v)$ $\oplus P$

The projection$\pi$

is given by$\pi$: $Z^{d}arrow P$ along .
$v$

Lemma 1.1 The following relation holds:$Z^{d}$$\pi$$arrow L_{\sigma}$$\downarrow$$P$$Z^{d}$$\downarrow$$arrow L_{\sigma}$$P$$\mathrm{y}\mathrm{r}$$L_{\sigma}$-invariant contractive plain,

147

Figure 1: The figure of $\{f (s_{0}s_{1}\cdots s_{k})|k=0, 1, 2, \cdots\}$ on

For the fixed point

where$s_{0}=\epsilon$$w_{\sigma}=$

SIS2$\ldots$$s_{n}\cdots$$\sigma_{F}$

.

,

$Y$

$:=$

$\pi\{f(s_{0}s_{1}\cdots s_{k})|k=01, \cdots\}\}$ ’$Y_{i}$$:=$$\pi\{f(s_{0}s_{1}\cdots s_{k-1})|\exists k :$Y_{i}’$$:=$$\pi\{f(s_{0}s_{1}\cdots s_{k})|\exists k :s_{k}=\mathrm{i}, k=1,2, \cdots\}$,s_{k}=\mathrm{i}, k=0, 1, \cdots\}$(the empty word).

Definition 1.2 $X_{i}:=$ the closure of) andthe closure of(substitution .
$\pi Y_{i}’$$X_{i}’.--$$\pi Y_{i}$is called atomic surfaces
is called atomic surface of the$X:= \bigcup_{i=1}^{d}X_{i}(=\bigcup_{i=1}^{d}X_{i}^{J})$$\sigma$

On Example 1.1 and Example 1.2, we have the figures (see Figure 2 and Figure 3
respectively).

Remark There is the theorem. Theorem ([E-I]): On d $=2$ ,iff is invertible.

$X_{i}$

is interval is the interval$\sigma$1$\bigcup_{\mathrm{n}=1,2}X_{i}’$$\bigcup_{i=1,2}X_{i}$Figure 2: The figure of the atomic surface on$\sigma_{F}$: $1\mapsto 12,2\mapsto 1$ .

148$\bigcup_{i=1,2,3}X_{i}’$$\bigcup_{i=1,2,3}X_{\mathrm{i}}$Figure 3: The figure of the atomic surface on$\sigma_{R}$; $1\mapsto 12,2\mapsto 13,3\mapsto 1$.

Theorem 1 ([A-I], [I-R]) Atomic surfaces satisfy(1) $X$ ,$X_{i}$,$X_{i}’$are compact sets;(2) $int.X=X$ ;(3)$L_{\sigma}^{-1}X_{i}= \bigcup_{j=1}^{d}\bigcup_{kW_{k}^{(j)}=\iota}(X_{j}+L_{\sigma}^{-1}(f(P_{k}^{(j)})))$(4) ’ $(|X_{1}|, \ldots, |X_{d}|)$ is the eigenvector of$|B|$ is the volumof thle set .

$L_{\sigma}$(non-overlapping);with respect to A $=\lambda_{1}>1$ where$B$$\mathrm{e}$Sketch of proof. On the notation,$(x, \mathrm{i})$

$:=$

{

$\sigma(0, \mathrm{i})$$:=$$\Sigma_{k=1}^{l_{t}}(f(P_{k}^{(i)})’$$\overline{w_{\sigma}}$A$x+\lambda e_{i}|0\leq$:$\leq 1$},,( the broken line starting from 0),$W_{k}^{(i)})$$:= \lim_{narrow\infty}\sigma^{n}(0,1)$,$1\leq \mathrm{i}\leq d$

we define$(Y_{i}, \mathrm{i})$

,

$:=$

$\{(y, \mathrm{i})|(y, \mathrm{i})\in\overline{w}\}$$=$$\{(y, i)|(y)\mathrm{i})\in\sigma(j=\mathrm{d}|(Y_{j}1d, j))\}$$=$$\bigcup_{J^{=1}}^{d}\{(y, \mathrm{i})|(y, \mathrm{i})\in\sigma(Y_{j}, j)\}$$=$$j=1_{k:W_{k}^{(J)}=i}\cup^{d}\cup\{$($\overline{w}=\cup i=1d(Y_{i}, \mathrm{i})$$L_{\sigma}y+f(P_{k}^{(j)})$,$\mathrm{i}$)$|y\in Y_{j}\}$.

149

Taking only the starting points of line segments, we get Theorem 1 (3).
Question Are$X_{i}^{r}$,$\mathrm{i}=1$, 2,$\ldots$

Definition 1.3 Substitution,$d$$\sigma$non-overlapping?

satisfies the coincidence condition if$\exists n$, k:(i) $f(P_{k}^{(n,1)})=f(P_{k}^{(n,2)})=\cdots=f(P_{k}^{(n,d)}.)$(2) $w_{k}^{(n,1)}=w_{k}^{(n,2)}=$ . . .
$\sigma^{n}(1)$.

where.
$\cdot$$\sigma^{n}(d)$$w_{1}^{(n,1)}\cdot$

$=$

..

$w_{k}^{(n,1)}\cdots w_{l(n,1)}^{(n,1)}$

.
.

$=$

$\cdot$$w_{1}^{(n,d)}\cdots w_{k}^{(n,d)}\cdots$ $w_{l(n,d)}^{(n,d)}$$=$.

If a satisfies the coincidence condition, thenProposition 1.1
non-overlapping.

Conjecture$=w_{k}^{(n,d)}$$X_{i}$,$\mathrm{i}=1,2$,$\ldots$,

$d$

are

Any Pisot unimodular substitutions satisfy thle coincidence condition.

Remark On $d=2$ , the conjecture is proved by Barge and Diamond ([B-D]).
If(1)$X_{i)}\mathrm{i}=1,2$$T$,$\ldots$,

are non-overlapping, then we have two dynamical systems on :$\mathrm{X}$$d$: $Xarrow X$ ,if $x\in X_{i}$ , E3,$Tx=L_{\sigma}^{-1}x-L_{\sigma}^{-1}\pi f(P_{k}^{(j)})$Therefore,
(2)

$W$

: $X$

$T$

$arrow$$k$:$L_{\sigma}^{-1}x$$\in X_{I}+L_{\sigma}^{-1}\pi f(P_{k}^{(j)})$is Markov endomorphism with the structure matrix${}^{t}L_{\sigma}$.

.

$X$

is well-defined
if $x\in X_{i}$and $W$ is called the domain exchange transformation (later, we will see).
rotation on$wx$$x\iota v$$\mapsto$$-\pi e_{i}$$T^{d-1}$.

$\mathrm{W}\simeq$

theFrom Markov endomorphism (1), we have the following numerical expression,
Corollary 1.1 Using Markov erulornorp hisrn,$X_{i}$

is presented by$X_{i}=\{\pi_{c}f(P_{k_{0}}^{(j_{0})})+\pi_{c}L_{\sigma}f(P_{k_{1}}^{(j_{1})})+\pi_{c}L_{\sigma}^{2}f(P_{k_{2}}^{(j_{2})})+\cdots+\pi_{c}L_{\sigma}^{n}f(P_{k_{n}}^{(j_{n})})+\cdots$

where $(*)$ is defined by(see Figre 4)$(j_{0}k_{0}j_{1}k_{1}$$\cdot$$\cdot\cdot$$\cdot$..

$k_{n}j_{n}$$\cdot$...

$)$is given by$w_{k_{n}}^{(j_{n})}=\mathrm{j}\mathrm{n}-\mathrm{i}$and$|(*)\}$$w_{k_{0}}^{(j\mathrm{o})}=\mathrm{i}$.

150$arrow T$$\downarrow L_{\sigma}^{-1}$$L_{\sigma}^{-1}X_{1}$$=X_{1}\mathrm{U}X_{2}\cup X_{3}$$L_{\sigma}^{-1}X_{2}$$=$$\pi e_{3}+X_{1}$$L_{\sigma}^{-1}X_{3}$$=$$\pi e_{3}+X_{2}$$arrow W$$X_{i}’$$X_{i}$

Figure 4: Figures of Markov endomorphism$W$ on Example 1.2$T$

and the domain exchange transformation

1512

Stepped Surfaces and Tiling Subsitutions

For$(x, \mathrm{i}^{*})\in Z^{d}\cross$$\{1^{*}, \ldots, d^{*}\}$,

we give a geometrical meaning such that$x+\mu_{j}e_{j}|0\leq\mu_{j}\leq 1\}$$(x, \mathrm{i}^{*}):=\{j=1j’\neq.:\sum_{\prime}.d$,(see Figure 5).

$(0,1^{*})$$-\mathrm{K}$$(0,2^{*})$

Figure 5: The figures of$(0, \mathrm{i}^{*})$.

Definition 2.1 $S:=\{(x, \mathrm{i}^{*})|(x, u)\geq 0, (x-e_{i}, u)<0\}$ is called the stepped surwhere the row vector $u>=0$ is the eigenvector such thatface of the contract plainis given by $P=\{x|(x, u)=0\}$ .
$uL_{\sigma}=Au$ and the contract plainis called a tiling of P from the stepped
Definition 2.2 $\pi S:=\{\pi$ (x, i’) |(x,surface S (see Figure 6).
$P$

$P$

$\mathrm{i}^{*})\in S\}$(Tiling Substitution)

3

Dual Subsitution$S^{*}:=${ the finite sum of elements of }$\sigma^{*}$$\mathrm{S}$$\simeq$

Let us define the dual (tiling) substitution{ the patches of tiles of the tiling$\sigma^{*}$$\sigma^{*}\pi(x, i^{*})=L_{\sigma}^{-1}\pi x+\sum_{j=1(}^{d}kj$(see Figure 7 and Figure 8).

by,$). \sum_{W_{k}^{(j)}=i}.\pi_{c}(L_{\sigma}^{-1}f(P_{k}^{(j\}})$$j^{*})$$\pi S$}.

152

Figure 6: The figure of the tiling
1.2.

$\pi S$

of $P$ from the stepped surface$\pi(0, 1^{*})$$arrow\sigma^{*}$$\pi(0, 3^{*})$

Figure 7: The figure of$\sigma_{R}^{*}\pi(0, \mathrm{i}^{*})$.

$S$

on

$\sigma_{R}$in Example

153

Figure 8: The figure of
Theorem 2 ([A-I])(1) a’ :(2)$S^{*}arrow S^{*}$$\in S$$\forall(x, \mathrm{i}’)$$\sigma$$\sigma_{R}^{*n}\bigcup_{i=1,2,3}\pi(e_{i_{1}}\mathrm{i}^{*})$

on cry.

: An Pisot Unit subsitution, thenis well-defined;, $\exists(y, j^{*})\in S:(x, i^{*})\in\sigma^{*}(y)j’)$ ;(3) $(x, i^{*})\neq(y_{1}j^{*})\in S\Rightarrow\sigma^{*}(x, i^{*})\cap\sigma^{*}(y, j^{*})=\emptyset$ .

Theorem 3([I-R]) Let&(1)$\sigma^{*}\mathcal{U}\succ\$(2) if$d(\partial(\sigma^{*7}’\ ), 0)$

coincides with$arrow\infty${ ( ,$\pi$$\pi S$$x$( $narrow$ oo ), then$j^{*}$)$|\pi(x,$$j^{*})\in\sigma^{*n}\pi(e_{i}$,$\mathrm{i}^{*})$

for some$n$andand a quasi-periotdic tiling;$-X_{i}= \lim_{narrow\infty}L_{\sigma}^{n}\sigma^{*n}\pi(e_{i}, \mathrm{i}^{*})$(4) 7 $:=$

|i $=1,$2, \cdots , d}, then,;$\tau’:=$(3)$=\{\pi(e_{i}, \mathrm{i}^{*})$;$\{\mathrm{y}\mathrm{r}x-X_{j}|\pi(x, j^{*})\in\tau’\}$is also a quasi-periodic tiling of$P$.

$j^{*}$}

154

Figure 9: The figure of the quasi-periodic tiling .
$\tau$

References[A-I] P.ARNOUX and S.Ito, Pisot substitu tions and Rauzy fractals, Bull. Belg. Math. Soc,no.2, 181-207.
[B-D]$\mathrm{S}(2001)$,M.BARGE and B.DIAMOND, Coincidence for substitutions of Pisot type, Bull. Soc. Math.
France 130 (2002), no.4, 619-626.

[E] H.El, Some properties of invertible substitutions of rank d, and higher dimensional substitutions,
Osaka J. Math. 40 (2003), 543-562.
[E-I] H.El and S.Ito Decomposition theorem on invertible substitutions, Osaka J. Math. 35 (1998)

no.4, 821-834.

[E-I] H.El and S.Ito Tiling fvorn some non-irreducible Pisot substitutions, to appear in Discrete Math.
& Th. Computer Science.
[I-R] S.Ito and H.RAO, Atomic surfaces, tilings and coincidence I. Irrecuciblle case, to appear inIsrael J..
[S-A-I] Y. SANO, P.ARNOUX and S . ITO higher dimensional extensions of substitutions and their dual
maps, J. Anal Math. 83 (2001), 183-20