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FREQUENCY OF PATTERNS IN CERTAIN GRAPHS AND IN PENROSE TILINGS
J. Peyriere
To cite this version:
J. Peyriere. FREQUENCY OF PATTERNS IN CERTAIN GRAPHS AND IN PENROSE TILINGS.
Journal de Physique Colloques, 1986, 47 (C3), pp.C3-41-C3-62. �10.1051/jphyscol:1986305�. �jpa- 00225715�
JOURNAL DE PHYSIQUE
C o l l o q u e C 3 , suppl&ment au n07, Tome 47, j u i l l e t 1 9 8 6
FREQUENCY OF PATTERNS I N CERTAIN GRAPHS AND I N PENROSE TILINGS
J . PEYRIERE
E q u i p e d l A n a l y s e H a r m o n i q u e , U n i v e r s i t e P a r i s - S u d , ( U . A . 7 5 7 ) M a t h e m a t i q u e s , B a t i m e n t 425, F-91405 O r s a y C e d e x
I - INTRODUCTION
Among t h e a p e r i o d i c t i l i n g s r e c e n t l y used i n c r y s t a l l o g r a p h y [ l - 8 1 some, a s 2D-Fenrose t i l i n g s , a r e i n v a r i a n t by an o p e r a t i o n g e n e r a l i z i n g s e l f s i m i l a r i t y , namely t h e s o c a l l e d i n f l a t i o n ( o r d e f l a t i o n , depending on t h e mood !) procedure.
It is t h e n n a t u r a l t o s t u d y t i l i n g s and more g e n e r a l l y g r a p h s which a r e i n v a r i a n t by s u c h a n o p e r a t i o n . Some of t h e i d e a s and t e c h n i q u e s i n t r o d u c e d by t h e a u t h o r
[9-111 t o b u i l d a n a b s t r a c t s e t t i n g f o r t h e s t u d y of Mandelbrot's s q u i g s [12-151 a r e used a g a i n i n t h e p r e s e n t work.
I n i t we p r o v e theorems on t h e f r e q u e n c y o f a p p e a r a n c e of f i n i t e p a t t e r n s i n c e r t a i n c o l o r e d g r a p h s , examples of which a r e t h e g r a p h s d u a l t o P e n r o s e t i l i n g s .
Here i s t h e main r e s u l t , s t a t e d , f o r t h e s a k e of s i m p l i c i t y , f o r 2D-Penrose t i l i n g s . L e t {%In2 b e a sequence of r e g u l a r p l a n e domains t h e a r e a s of which tend t o i n f i n i t y w h i l e t h e r a t i o of t h e p e r i m e t e r of
Rn t o i t s a r e a t e n d s t o z e r o a s n goes t o i n f i n i t y ( t h i s l a s t c o n d i t i o n means t h a t 63 does n o t f l a t t e n t o o
n
q u i c k l y ) . I f a i s a bounded p a t t e r n a p p e a r i n g i n a c e r t a i n P e n r o s e t i l i n g , l e t u s d e n o t e N ( a ) t h e number of times t h e p a t t e r n a a p p e a r s w i t h i n t h e domain an.
n
Then t h e r a t i o N ( a ) / a r e a ( A ) h a s a non-zero l i m i t .
n n
T h i s r e s u l t is t o b e compared t o Conway's weak p e r i o d i c i t y as w e l l as t h e pro- p e r t y of a l m o s t p e r i o d i c i t y brought i n t o l i g h t b e s e v e r a l a u t h o r s [16-181. But i t is t o b e n o t i c e d t h a t none of t h e s e p r o p e r t i e s i m p l i e s any o t h e r of them a l t h o u g h each one t e l l s something a b o u t t h e c o r r e l a t i o n s w i t h i n such a t i l i n g .
The u s e of g r a p h s may seem c o m p l i c a t e d , b u t i t a l l o w s a u n i f i e d t r e a t m e n t of d i f f e r e n t s i t u a t i o n s , f o r i n s t a n c e d e c o r a t i o n of P e n r o s e t i l i n g s and a p e r i o d i c c o l o r a t i o n of r e g u l a r l a t t i c e s .
For t h e r e a d e r who i s n o t w i l l i n g t o go t h r o u g h a l l mathematical d e t a i l s , t h e p a r t i c u l a r c a s e of word s u b s t i t u t i o n s is r e c a l l e d i n s e c t i o n 11 and t h e o u t l i n e of t h e p r o o f s i s g i v e n i n s e c t i o n 111 f o r a p a r t i c u l a r r e a l i z a t i o n of a 2D-Penrose t i l i n g . The complete s e t t i n g and p r o o f s a r e g i v e n i n s e c t i o n s I V - V I I . Although t h e language of g r a p h t h e o r y is u s e d , no p e r e q u i s i t e s i n t h i s t h e o r y i s needed.
The a u t h o r acknowledges P. Assouad and F. Axel f o r f i r s t i n t r o d u c i n g him t o Penrose t i l i n g s .
I1 - THE ONE DIMENSIONAL CASE : WORD SUBSTITUTIONS
L e t A b e a f i n i t e set which we c a l l " a l p h a b e t w . For e a c h a i n A , a word o(a), c o n s t r u c t e d o v e r t h e a l p h a b e t A , i s g i v e n . Such a d a t a u i s c a l l e d a s u b s t i t u t i o n .
I f w = x x 2.. .xv is a word o v e r A , we d e n o t e a(w) t h e word
o(xl)(J(x2). . .'J(xv) o b t a i n e d by p u t t i n g end t o end t h e words G(xl) ,U(x2), . . . , o ( x v ) . These s u b s t i t u t i o n s have been s t u d i e d from d i f f e r e n t p o i n t s of view by many a u t h o r s
[ l l , 19-231.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1986305
C3-42 JOURNAL DE PHYSIQUE
L e t u s t a k e a n example : A = { 0 , 1 ) , u(0) = 011, U(1) = 01. W e t h e n h a v e 2 ( 0 ) = 0110101, 03(0) = OllOlOlOllOlOllOl and s o on. One can remark t h a t t h e word
1 (0) b e g i n s by 0 "(0) ; T h e r e f o r e t h e r e i s a n i n f i n i t e s e q u e n c e E l , E2,. . . , E n , . . .
of 0 ' s and 1 ' s which i s t h e l i m i t i n a s u i t a b l e s e n s e of $ ( o ) . T h i s i n f i n i t e s e q u e n c e is i n v a r i a n t under t h e a p p l i c a t i o n of a.
L e t xn and yn d e n o t e t h e numbers of 0 ' s and 1 ' s i n t h e word Un(0). A s each 0 g e n e r a t e s o n e 0 and two 1's and e a c h 1 g e n e r a t e s one 0 and one 1, one h a s t h e f o l l o w i n g r e c u r s i o n formula
( n o t e t h a t two d i f f e r e n t s u b s t i t u t i o n s may have t h e same m a t r i x M ) . Thus (xn) = Mn (A)
Yn
So, a s a consequence of Perron-Frobenius t h e o r y ( s e e t h e a p p e n d i x ) , we have
X a a
( n, % An ( & , where A i s t h e l a r g e s t e i g e n v a l u e of M and
( & i s a c o r r e s p o n - Yn
d i n g r i g h t e i g e n v e c t o r . So t h e r e l a t i v e f r e q u e n c y of 0 ' s and 1 ' s i s a/B.
E v i d e n t l y s u c h a r e s u l t h o l d s f o r g e n e r a l s u b s t i t u t i o n s . It c a n b e r e p h r a s e d i n t h e f o l l o w i n g form :
1 1 #(o) 1 Q
l i m --- C E . = -
n + a ( 8 ( 0 ) l j=1 J " + B '
where 1$1(0) 1 i s t h e l e n g t h of t h e word 8 ( 0 ) .
The n e x t problem i s t o d e t e r m i n e whether o r n o t t h e s e q u e n c e {E. 1 h a s a n Iun(o) I-k J
a u t o c o r r e l a t i o n s e q u e n c e , i . e . t o d e t e r m i n e i f --L- C E.E. h a s a l8i0) I i = l 3 Jfk l i m i t f o r e a c h p o s i t i v e i n t e g e r k. A way of couAting ;he ngn-zero t e r m s i n t h i s l a s t sum i s t o c o u n t t h e number of o c c u r e n c e s i n on(0) of words of l e n t h k + l which b e g i n and end by a 1. So o n e i s l e d t o s t u d y t h e f r e q u e n c y of a p p e a r a n c e of any word w i n t h e sequence E1,E2, ..., E n'"'
L e t u s e x p l a i n how t h e e x i s t e n c e of a f r e q u e n c y f o r words of l e n g t h 2 c a n b e proved. The r e a d e r would t h e n s u p p l y t h e proof f o r t h e g e n e r a l c a s e ( s e e a l s o
[ l l ] . The t o o l i s t h e change of a l p h a b e t d e s c r i b e d below.
I f w = x x7...xv i s a word o v e r A , l e t u s w r i t e 1 - -
$ = (X x ) (X x ) ( X x ) . . . ( X ~ - ~ X , , ) (xu&) , where E i s a new symbol i n d i c a t i n g ends 1 2 2 3 , 3 4
of words. Then w a p p e a r s t o b e a word o v e r a new a l p h a b e t t h e " l e t t e r s " of which a r e words o f l e n g t h 2 .
L e t u s c o n s i d e r t h e above example and draw t h e f o l l o w i n g diagram showing t h e
"genealogy"
d to, 0 1 1 0 1 0 1
and p e r f o r m t h e change of a l p h a b e t :
% %
It i s t h e n c l e a r t h a t ( d ( 0 ) ) = (O)~(OE) where 0 i s t h e f o l l o w i n g s u b s t i - t u t i o n a c t i n g on t h e a l p h a b e t 2 = 1(01), (lo), (ll), ( 1 ~ ) 1 :
%
O(01) = (01) (11) (10)
'L O(l0) = (01)(10)
'L a(ii) = (OI)(IO)
% U(1E) = (01)(1&) .
T h e r e f o r e t h e number of occurences of words of l e n g t h 2 i n a"(0) i s governed by t h e m a t r i x
0 0 0 1
and t h e i r r e l a t i v e f r e q u e n c i e s a r e t h e components of t h e normalized e i g e n v e c t o r
corresponding t o A -- 1 + 6.
Indeed t h i s a n a l y s i s i s v a l i d i n g e n e r a l , n o t only on t h i s example.
I n f a c t we have only proved t h e f o l l o w i n g p r o p o s i t i o n : i f w i s a word, t h e n t h e r a t i o
--- 1 x number of w ' s i n 8(0) la"(0) I
t e n d s t o a l i m i t a s n goes t o i n f i n i t y . But a s t r o n g e r r e s u l t h o l d s : i f an and bn a r e two sequences of p o s i t i v e i n t e g e r s such t h a t bn t e n d s t o i n f i n i t y , t h e n t h e r a t i o .
- b1 x number of w ' s i n
'a 'a +l... E
n n n an+bn
t e n d s t o a l i m i t , a s n goes t o i n f i n i t y , uniformly w i t h r e s p e c t t o t a n t s . The t r e a t m e n t of Penrose t i l i n g s and graphs f o l l o w s t h e same l i n e s axthough l o t s of t e c h n i c a l i t i e s indeed appear, due t o t h e g r e a t e r c o m b i n a t o r i a l complexity of t i l i n g s and graphs.
111 - PENROSE TILINGS
There a r e s e v e r a l ways of c o n s t r u c t i n g Penrose t i l i n g s 124-311. We u s e Robinson's approach [ 2 6 ] . A s i m i l a r d e s c r i p t i o n h a s been used by Dekking [29].
We c o n s i d e r t h e i s o c e l e s t r i a n g l e s of f i g u r e 1.
C3-44 30URNAL DE PHYSIQUE
The t i l i n g e l e m e n t s we c o n s i d e r a r e t r i a n g l e s e q u a l t o one of t h o s e , t h e v e r - t i c e s of which a r e c o l o r e d w i t h two c o l o u r s s o t h a t t h e v e r t i c e s of a t r i a n g l e cor- responding t o e q u a l a n g l e s h a v e d i f f e r e n t c o l o u r s . I n t h e s u b s e q u e n t f i g u r e s one of t h e c o l o u r s , s a y t h e w h i t e , w i l l n o t b e r e p r e s e n t e d , t h e o t h e r one, t h e b l a c k , w i l l be i n d i c a t e d by s m a l l c i r c l e s around t h e c o r r e s p o n d i n g v e r t i c e s . We o b s e r v e t h a t , due t o d i f f e r e n t c o l o r a t i o n s and symmetries, t h e r e a r e e i g h t d i f f e r e n t k i n d s of t i l e s a s shown on f i g u r e 2 .
F i g u r e 2
The e i g h t d i f f e r e n t t i l e s w i t h an enumeration of t h e i r s i d e s . The t i l i n g s we a r e c o n s i d e r i n g obey t h e f o l l o w i n g r u l e s :
l o They a r e composed e i t h e r of t i l e s a l , a 2 , b and b2 o r of t i l e s a;, a;, b; and .
b;
1
2' The t i l e s a r e assembled s o t h a t t h e c o l o u r s of t h e i r v e r t i c e s match.
3 O I f two t i l e s a r e i n c o n t a c t along one s i d e t h e v e r t i c e s of which have t h e same c o l o u r , t h e l a r g e r a n g l e s of each t i l e s a d j a c e n t t o t h i s s i d e a l s o match.
L e t u s c o n s i d e r such a t i l i n g of a p l a n e domain. One o b t a i n s a f i n e r t i l i n g by r e p l a c i n g each t i l e according t o t h e r u l e d e s c r i b e d by f i g u r e 3.
gives
gives
F i g u r e 3
F i r s t s t e p of t h e i n f l a t i o n procedure.
T h i s f i g u r e only g i v e s t h e s u b s t i t u t i o n r u l e s f o r two k i n d s of t i l e s , t h e o t h e r ones being deduced by permutation of c o l o r s o r symmetry. I f one expands t h e new t i l i n g by t h e f a c t o r T one g e t s a t i l i n g of a p l a n e domain by t r i a n g l e s of t y p e s I and I1 s a t i s f y i n g t h e above requirements on c o l o r s . The o p e r a t i o n we have j u s t d e s c r i b e d i s commonly c a l l e d i n f l a t i o n . I t e n a b l e s u s , s t a r t i n g from a t e s s e l a t i o n of a p l a n e domain, t o g e t a t e s s e l a t i o n of a l a r g e r one. By i t e r a t i n g i n f l a t i o n and t a k i n g weak limits one g e t s t i l i n g s of t h e p l a n e which a r e i n f l a t i o n i n v a r i a n t . These a r e t h e Penrose t i l i n g s .
Let u s t a k e a n example. We c o n s i d e r OAB a t r i a n g l e of t y p e I w i t h w h i t e v e r t i c e s 0 and B. L e t u s apply t h e i n f l a t i o n procedure t w i c e , u s i n g d i l a t i o n s c e n t e r e d a t 0.
The r e s u l t i s shown on f i g u r e 4. The t i l e OAB a p p e a r s i n t h e new t i l i n g , t o w i t h i n a symmetry, and would have been found e x a c t l y had we a p p l i e d f o u r times t h e i n f l a - t i o n procedure. So t h e sequence of t i l i n g s o b t a i n e d by applying 4n times t h e i n f l a t i o n procedure converges a s n goes t o i n f i n i t y t o a t i l i n g of a s e c t o r of t h e p l a n e of a p e r t u r e T/5. The t e s s e l a t i o n o b t a i n e d by completing t h i s one u s i n g symmetries and r o t a t i o n s of a n g l e 2n/5 is t h e Penrose t i l i n g of t h e p l a n e w i t h pentagonal symmetry.
T h i s o p e r a t i o n of i n f l a t i o n i s a kind of s u b s t i t u t i o n : a t i l e of t y p e a g i v e s one t i l e of t y p e a; , one of t y p e a; and one of t y p e 1
b; and s i m i l a r l y f o r t h e o t h e r types. T h e r e f o r e t h e numbers of each type of t i l e s a f t e r n applica- t i o n s of t h e i n f l a t i o n procedure a r e g i v e n by t h e n-th power of a c e r t a i n m a t r i x .
I n o r d e r t o b e a b l e t o count n o t o n l y each type of t i l e s but a l s o t h e occurences of each f i n i t e p a t t e r n , we a r e going t o d e s c r i b e t i l i n g s by t h e mean of t h e i r d u a l graphs.
JOURNAL DE PHYSIQUE
F i g u r e 4
R e s u l t of two a p p l i c a t i o n s of t h e i n f l a t i o n p r o c e d u r e
Given a t i l i n g ( f i n i t e o r n o t ) one can c o n s i d e r t h e g r a p h t h e v e r t i c e s of which a r e t h e elements of t h e t i l i n g , two t i l e s b e i n g l i n k e d by a n edge of t h e graph i f and o n l y i f t h e y s h a r e one of t h e i r s i d e s . One keep i n memory t h e t y p e of each t i l e by i n d i c a t i n g i t a t t h e c o r r e s p o n d i n g node of t h e g r a p h and a l s o t h e way t i l e s a r e connected by t a g g i n g each edge of t h e r e s u l t i n g g r a p h s . The t y p e of a t i l e w i l l b e c a l l e d t h e c o l o u r of t h e c o r r e s p o n d i n g v e r t e x of t h e d u a l graph. F i g u r e 5 shows a t e s s e l a t i o n w i t h i n d i c a t i o n of t y p e s and of t h e t a g g i n g of each s i d e w i t h i n each t i l e . F i g u r e 6 shows t h e coding of t h i s t e s s e l a t i o n as a graph. Such a graph w i l l b e i n t h e s e q u e l c a l l e d a c o l o u r e d tagged g r a p h s .
F i g u r e 6 ( o n t h e r i g h t ) Dual t o f i g u r e 5
F i g u r e 5 (on t h e l e f t ) Same a s f i g u r e 4 , b u t w i t h t h e i n d i c a t i o n of t h e " c o l o u r s " of t r i a n g l e s and of t h e n u m e r a t i o n of t h e i r s i d e s .
C3-48 JOURNAL DE PHYSIQUE
Now, we have t o d e s c r i b e i n t h i s s e t t i n g t h e o p e r a t i o n of i n f l a t i o n . Consider f o r i n s t a n c e t h e s p l i t t i n g of a t i l e of t y p e
a; a s shown on f i g u r e 7 .
gives
F i g u r e 7 ( i n f l a t i o n r u l e f o r t i l e a;)
The coding of t h i s s i t u a t i o n is shown on f i g u r e 8. A s t h i s graph is t o b e l i n k e d t o o t h e r s , i t has dangling bonds. These dangling bonds have been s e p a r a t e d i n t o t h r e e c l a s s e s w1 , w2 and ~ 3 , t a k i n g i n t o account which edge of t h e o r i g i n a l t i l e t h e y come from.
The arrows i n d i c a t e an o r d e r of enumeration of t h e s e d a n g l i n g bonds i n each c l a s s . The r e s u l t i n g f i g u r e i s a n example of what is c a l l e d an i o n i n t h e sequel.
F i g u r e 8
F i g u r e 9 shows a t w o - t i l e s t e s s e l a t i o n and t h e c o r r e s p o n d i n g graph. F i g u r e 10 shows t h e i n f l a t i o n p r o c e s s on t h i s t e s s e l a t i o n and t h e c o r r e s p o n d i n g o p e r a t i o n on g r a p h s
F i g u r e 9
F i g u r e 1 0 Binding of i o n s
T h i s o p e r a t i o n c o n s i s t s i n r e p l a c i n g each node of t h e o r i g i n a l graph by a n i o n and b i n d f n g t h e s e i o n s a s shown on f i g u r e 10 : each edge of t h e o r i g i n a l graph d i r e c t s t h e b i n d i n g of c o r r e s p o n d i n g d a n g l i n g bonds. T h i s o p e r a t i o n i s analogous t o t h e w o r d - s u b s t i t u t i o n s p r e v i o u s l y c o n s i d e r e d . F i g u r e 11 shows t h e r e s u l t of
C3-5 0
t h i s o p e r a t i o n .
JOURNAL DE PHYSIQUE
F i g u r e 11
Up t o now, we h a v e j u s t g i v e n a n a l t e r n a t i v e d e s c r i p t i o n of t i l i n g s and i n f l a - t i o n p r o c e d u r e , b u t t h i s e n a b l e s u s t o p r o v e t h a t p a t t e r n s o c c u r w i t h a w e l l d e t e r m i n e d f r e q u e n c y . As p r e v i o u s l y f o r w o r d s - s u b s t i t u t i o n s , t h e proof r e l i e s upon changes of c o l o u r s . Here i s t h e s k e t c h of t h e p r o o f .
By s u c c e s s i v e a p p l i c a t i o n s of t h e i n f l a t i o n p r o c e d u r e we g e t a s e q u e n c e
-(Gnjn> of c o l o r e d tagged g r a p h s . I f we s u i t a b l y d e f i n e new c o l o u r s , t h e s e q u e n c e Gn o b t a i n e d by changing c o l o u r s i n Gn c a n b e g e n e r a t e d by a new i n f l a t i o n pro- c e d u r e , which p r o v i d e s i n f o r m a t i o n on f r e q u e n c i e s of new c o l o u r s .
W e u s e two changes of c o l o u r s . T h i s f i r s t o n e aims a t s t u d y i n g t h e growth of t h e boundarz, aGn of Gn. It is shown t h a t t h e number of e l e m e n t s of a~~ grows a s Tn
where i s t&e Perron-Frobenius e i g e n v a l u e of a c e r t a i n m a t r i x . I n t h e P e n r o s e c a s e one h a s X < A ( t h i s c a n a l s o b e shown by u s i n g a r e a - p e r i m e t e r argument). I n t h e g e n e r a l c a s e we have t o assume X < A. Then we perform a second change of co- l o u r s : t h e new c o l o u r of a v e r t e x d e s c r i b e s i t s s u r r o u n d i n g s up t o d i s t a n c e k.
It p r o v i d e s u s w i t h a m a t r i x which g o v e r n s t h e numbers of p a t t e r n s of s i z e k i n Gn. A f t e r t h a t t h e r e is s t i l l some work t o o b t a i n t h e r e s u l t f o r any l i m i t p o i n t o f t h e s e q u e n c e {G,) and f o r any sequence {Rnl of domains.
I V - TAGGED GRAPHS, IONS, BINDINGS 1. Tagged graphs.
A f i n i t e non-empty s e t F i s g i v e n t h r o u g h o u t t h e p a p e r . The g r a p h s we a r e g o i n g t o c o n s i d e r a r e n o n - d i r e c t e d and t h e i r v e r t i c e s h a v e o r d e r s less t h a n o r e q u a l t o # F , t h e c a r d i n a l i t y of F.
D e f i n i t i o n s u b s e t of
. A t a g g e d g r a p h i s a c o u p l e Z = (V,E) where V V X F X V X F SO t h a t
a ) (a,m,b,n) e E i m p l i e s a # b ,
b) ( a , m , b , n ) E E i m p l i e s ( b , n , a , m ) E E ,
c ) if a & b a r e e l e m e n t s of V , t h e n t h e s e t F ; (a,m,b,n) e E ) h a s o n e element a t most,
d ) i f (a,m) i s a n element of Vx F , t h e n t h e s e t F ; ( a , m , b , n ) E ~ ] h a s o n e element a t most.
s e t and
- .
The element of V a r e t h e v e r t i c e s of Z , t h o s e of E i t s e d g e s .
As we s h a l l h a v e t o c o n s i d e r s e v e r a l tagged g r a p h s s i m u l t a n e o u s l y , we s h a l l b e more s p e c i f i c i f needed : i f :: is a tagged g r a p h , VZ w i l l d e n o t e t h e s e t of i t s
v e r t i c e s and EZ t h a t of i t s edges.
-
To e a c h tagged graph Z we a s s o c i a t e a n o r d i n a t y g r a p h ZQ i n t h e f o l l o w i n g way. The g r a p h Z~ h a s t h e same v e r t i c e s a s Z . The s e t E$ of i t s edges i s s o
z
d e f i n e d : ( a , b ) E VZ xVZ b e l o n g s t o EZ i f and o n l y i f t h e r e e x i s t s (m,n) i n
F X F s u c h t h a t (a,m,b,n) b e l o n g t o
Ez.
A tagged g r a p h Z i s s a i d t o be connected i f Z' is. Then t h e d i s t a n c e w i t h i n Z , denoted d7 , of two of i t s v e r t i c e s i s , by d e f i n i t i o n , t h e i r g e o d e s i c d i s t a n c e a l o n g Z4 (i.;. t h e minimum number of e d g e s t o go t h r o u g h t o c o n n e x t them).
I f Z i s a t a g g e d g r a p h we s e t
W = {(a,m) EVZ X F ; t h e r e e x i s t s no ( b , n ) i n VZXF
a s u c h t h a t (a,m,b,n) E E
z
and
a Z = { a g ~ , ; t h e r e e x i s t s m i n F s u c h t h a t (a,m)EwZ}.
F i g u r e 6 , i f we f o r g e t t h e names of t h e nodes, shows a p i c t u r e of a t a g g e d graph.
L e t u s c o n s i d e r a tagged g r a p h Z = (V z z , E ) and U a s u b s e t o f VZ. We a r e g o i n g t o d e f i n e a tagged g r a p h which we c a l l sub t a g g e d g r a p h of Z a s s o c i a t e d
t o U : i t s s e t o f v e r t i c e s i s U and i t s s e t o f edges i s {(a,m,b,n) ; ( ~ , ~ ) E u x u } . I f x i s a v e r t e x of Z and r a p o s i t i v e i n t e g e r , B Z ( x , r ) s t a n d s f o r t h e s u b t a g g e d g r a p h of Z a s s o c i a t e d t o t h e b a l l VZ ; d Z ( x , y ) r1. I t w i l l b e c a l l e d t h e b a l l of c e n t e r x and r a d i u s r of Z.
Two t a g g e d g r a p h s Z1 e t Z 2 a r e isomorphic i f t h e r e e x i s t s a one-to-one mapping rp from V o n t o V s u c h t h a t ( ~ ( a ) ,m,lp(b) ,n) b e i n VZ i f and
z2 2
o n l y i f (a,m,b,n) i s i n "zl'
2. C o l o r a t i o n s , i o n s .
L e t A b e a f i n i t e s e t , which we c a l l s e t of c o l o u r s . An A - c o l o r a t i o n of a t a g g e d g r a p h i s a mapping g from VZ t o A. From now on, a n A-colored t a g g e d g r a p h w i l l b e simply c a l l e d a n A-graph.
I f Z i s a n A-graph and U a s u b s e t of V , Z U d e n o t e s t h e A-graph o b t a i n e d by r e s t r i c t i n g t h e c o l o r a t i o n of Z t o t h e sub t a g g e d g r a p h of Z a s s o c i a t e d t o U. B Z ( x , r ) w i l l a l s o d e n o t e t h e b a l l B Z ( x , r ) d e f i n e d above endowed w i t h t h e c o l o r a t i o n of Z .
An isomorphism Ip o f t h e A-graph Z = (VZ,EZ,gZ) on t h e A-graph Z ' i s a n isomorphism of t h e u n d e r l y i n g tagged g r a p h s which i s c o m p a t i b l e w i t h t h e c o l o r a - t i o n s : i . e . g Z , . Ip = gZ.
D e f i n i t i o n . We s h a l l c a l l a f i n i t e A-graph Z w i t h a p a r t i t i o n { q } m E F - of Wz by non-empty s e t s and, f o r any m & F , a t o t a l o r d e r i n g of WZ a n A-ion.
Two A-ions Z = (VZ ,EZ , {w;, ImE F,gZj ) , ( f o r j = 1 , 2 ) , a r e isomorphic i f
j j j J
t h e r e e x i s t s a one-to-one mapping 9 from V
Onto V ~ 2
s u c h t h a t rp b e a n z1
isomorphism of t h e u n d e r l y i n g tagged g r a p h s and, f o r any m i n F , a n isomorphism of t h e o r d e r e d sets
"; and $2.
F i g u r e 8 shows a n i o n .
C3-52 JOURNAL DE PHYSIQUE
3. B h d i n g of i o n s .
W e a r e g i v e n a tagged g r a p h Z = (V,E) a n d , f o r any x i n V , a n A-ion
Zx = (vZx,EZx, {$ ImE F , g z 2 i n such a way t h a t whenever (x,m,y,n) is i n E t h e
X
s e t s $ and W; have t h e same c a r d i n a l i t y . Then we d e f i n e a n A-graph
X v
2 ' = ( v ' , E 1 , g ' ) by t h e f o l l o w i n g t h r e e c o n d i t i o n s .
a ) V i s t h e d i s j o i n t u n i o n of t h e s e t s { v ~ ~, ) ~ ~ b) g ' e x t e n d s any of t h e mappings g ,
Zx 'L
c ) E' i s t h e d i s j o i n t union of t h e s e t s { E ~ ~and of t h e s e t E } ~ ~
s o d e f i n e d : ,I'
( ~ , j , 0,k) i s i n E i f and o n l y i f t h e r e e x i s t s ( x , m , ~ , n ) i n E s u c h t h a t ( ~ , j ) b e i n W; , (6,k) b e i n W; and (U,j) and (0,k) match
Y a c c o r d i n g t o t h e o r d e r s on $x and W; .
X Y
The A-graph Z' w i l l b e c a l l e d t h e b i n d i n g of t h e i o n s { z ~ } d i r e c t e d by ~ ~ t h e tagged g r a p h 2.
The b i n d i n g p r o c e s s i s i l l u s t r a t e d by f i g u r e s 9 , 10 and 11.
V - PENROSE TILINGS AS A-GRAPHS
L e t F b e t h e set {1,2,3). The set A h a s e i g h t elements a l , a 2 , b l , b2 ,
a; , a; , b; , b; which c o r r e s p o n d t o t r i a n g l e s a s shown i n f i g u r e 2. I n f i g u r e 2 a l s o a p p e a r s a numbering of t h e edges.
L e t u s c o n s i d e r f o r i n s t a n c e t h e t e s s e l a t i o n o b t a i n e d by a p p l y i n g t w i c e t h e i n f l a t i o n p r o c e d u r e t o a t r i a n g l e of t y p e a l a s shown i n f i g u r e 4 . F i g u r e 5 t h e n shows t h e t y p e s of t r i a n g l e s which a p p e a r and t h e numbering of t h e i r edges. The d u a l t o f i g u r e 5 is f i g u r e 6. I n o t h e r words i t i s a n A-graph Z :
V I - SUBSTITUTIONS
1. D e f i n i t i o n of s u b s t i t u t i o n s .
As p r e v i o u s l y two f i n i t e sets A and F are g i v e n . An A - s u b s t i t u t i o n is a mapping (J , which a s s o c i a t e s a n A-ion t o e a c h element of A, t o g e t h e r w i t h a set
of A-graphs, s u b j e c t e d t o t h e f o l l o w i n g r e q u i r e m e n t s :
a ) $)- c o n t a i n s A , t h e e l e m e n t s of which a r e i d e n t i f i e d t o A-graphs w i t h a s i n g l e v e r t e x ,
b) i f Z = (V,E,g) i s i n 8- , t h e n t h e b i n d i n g o f t h e f a m i l y
{ Q ( ~ ( x ) ) IxE d i r e c t e d by Z can b e done and t h e r e s u l t i n g A-graph, d e n o t e d O(Z), i s i n 8. .
I n o t h e r words, one p a s s e s from Z t o o(Z) by r e p l a c i n g e a c h node of Z by a g r a p h a c c o r d i n g t o i t s c o l o r . The v e r t i c e s of t h e i o n by which a v e r t e x x of Z h a s been r e p l a c e d a r e c a l l e d t h e d e s c e n d e n t s of t h e f i r s t g e n e r a t i o n of x.
T h i s n o t i o n g e n e r a l i z e s t h a t of s u b s t i t u t i o n o p e r a t i n g on w o r d s .
I t h a s been i n t r o d u c e d i n [ l o ] i n o r d e r t o g i v e a n a b s t r a c t s e t t i n g f o r M a n d e l b r o t ' s s q u i g s [12-15 I.
S e v e r a l f a c t s a r e t o b e n o t i c e d :
- I f t h e A-graph Z is c o n n e c t e d , s o i s O(Z).
- O d o e s n o t d e c r e a s e d i s t a n c e s . I t means t h e f o l l o w i n g : l e t x and y b e two e l e m e n t s of V , t h e n i f x' and y ' a r e d e s c e n d e n t s of x and y r e s p e c t i v e l y o n e h a s (x' , Y O d Z ( x , y ) .
- O c a n b e i t e r a t e d : i f Z E 9 , t h e n we g e t a s e q u e n c e *(z) of
A-graphs. We a r e m o s t l y i n t e r e s t e d i n t h e b e h a v i o u r of on(a) f o r a i n A.
- I f Z = (V,E) i s a n element of and i f U i s a s u b s e t of V ,
t h e n o"(zU) = ( $ ( z ) ) ~ , where U i s t h e n t h - g e n e r a t i o n o f f s p r i n g of t h e n
element of U. n
2 . Example.
The above c o n s t r u c t i o n of P e n r o s e t i l i n g s c a n b e r e p h r a s e d i n t e r m s of s u b s t i - t u t i o n : i n c h a p t e r V t h e s e t i l i n g s h a v e been coded as A-graphs and t h e i n f l a t i o n p r o c e d u r e g i v e s t h e r u l e s of s u b s t i t u t i o n . L e t u s f o r i n s t a n c e d e f i n e t h e i o n o ( a ' ) . F i g u r e 7 shows t h e i n f l a t i o n of a t i l e of t y p e
a; and f i g u r e 8 shows i t s 2
c o d i n g s a s a n i o n :
( t h e o r d e r of enumeration of t h e e l e m e n t s of
':(a:) d e f i n e s t h e t o t a l o r d e r i n g
L
of t h i s s e t ) . T h i s p a r t i t i o n of W and t h e c o r r e s p o n d i n g o r d e r i n g s a r e o(a:)
L
o b t a i n e d by a n a l y s i n g from which edge of t h e i n i t i a l t r i a n g l e t h e pending bonds come, and i n which o r d e r t h e y a p p e a r a c c o r d i n g t o t h e o r i e n t a t i o n induced by t h e numbering of t h e edges of t h e i n i t i a l t r i a n g l e .
The o t h e r i o n s { ~ ( a ) I a E A a r e d e f i n e d i n t h e same way. I f a i s a n element -
of A, t h e A-graph d l ( a ) d e s c r i b e s t h e P e n r o s e t i l i n g o b t a i n e d a f t e r a p p l y i n g n t i m e s t h e i n f l a t i o n p r o c e d u r e t o a t r i a n g l e of t y p e a .
3 . M a t r i x of a s u b s t i t u t i o n
I f Z is a n A-graph L(Z) d e n o t e s t h e v e c t o r i n IRA which d e s c r i b e s t h e c o m p o s i t i o n i n c o l o r s of t h e v e r t i c e s o f Z : t h e component La(Z) of L(Z) c o r r e s p o n d i n g t o t h e element a i n A i s t h e number of v e r t i c e s of Z t h e c o l o r of which i s a .
A s q u a r e m a t r i x M indexed by AXA i s a s s o c i a t e d t o t h e s u b s t i t u t i o n 0 i n t h e f o l l o w i n g way : t h e column of M c o r r e s p o n d i n g t o t h e element b o f A
C3-54 JOURNAL DE PHYSIQUE
is L(U(b)). F o r any Z i n 9 t h e f o l l o w i n g r e l a t i o n h o l d s : L ( Q ( 2 ) ) = ML(Z).
For i n s t a n c e t h e m a t r i x of t h e P e n r o s e s u b s t i t u t i o n , d e s c r i b e d i n t h e p r e v i o u s p a r a g r a p h , i s
i f t h e c o l o r s a r e s o o r d e r e d : al,a2,bl,b2,a;,a;,b;,b; .
4. F i r s t change of c o l o u r s
C o n s i d e r t h e s e t A % = AX 2F , where 2F s t a n d s f o r t h e s e t o i s u 2 s e t s of F.
If Z = (V,E,g) i s a n A-graph, we d e f i n e on i t a n A - c o l o r a t i o n g : g ( x ) = ( g ( x ) ,B) where B i s t h e s e t B = { m ~ F ; t h e r e i s no ( y , n ) i n V X F s u c h t h a t
(x,m,y,n) b e i n E). The A-graph s o obt-ained w i l l be-denoted 5 .
We aim a t defining a s u b s t i t u t i o n 0 a c t i n g on A i n such a w2y it-be e q u i v a l e n t t o 0. Tke set of A-graphs on which it w i l l o p e r a t e i s 4. = {Z ; Z E ?I.
L e t u s now d e f i n e u(a,B) f o s a i n A and B i n 2F.
- f o r Q ( a ) and O(a,B) t h e u n d e r l y i n g t a g g e d g r a p h s a r e t h e same,
- i f x i s a v e r t e x of E ( ~ , B ) ( t h e r e f o r e a l s o of O ( a ) ) , i t s c o l o u r is (g(x) ,B') where
- t h e o r d e r e d s e t s W' and ~i
Q(a) u ( a , B ) a r e i d e n t i c a l .
'L 'L %
For any Z i n % we have U(Z) = (O(Z)) . The p a r t i t i o n
A = (Ax { 0 1 ) _ ~ (X\(AZ {@I>) i n d u c e s t h e f o l l o w i n g d e c o m p o s i t i o n i n t o b l o c k s of t h e m a t r i x M of O :
It i s t o b e n o t i c e d t h a t t h e m a t r i x M of 0 a p p e a r s a s one of t h e s e
b l o c k s . A,
As p r e v i o u s l y , a v e c t o r L(H) .-, i n lRA i n a s s o c i a t e d t o any x-graph H. I f Z i s a n A-graph, t h e n t h e sum of t h e components of L(Z) which c o r r e s p o n d t o
AX If$)) is t h e number o f e l e m e n t s o f 32.
5. Hypotheses
Terminology and a few f a c t s a b o u t non-negative m a t r i c e s can b e found i n t h e appendix.
The h y p o t h e s e s we s h a l l assume t o h o l d from now on, u n l e s s o t h e r w i s e s p e c i f i e d , are t h e f o l l o w i n g :
a ) t h e m a t r i x M i s p r i m i t i v e and i t s Perron-Frobenius e i g e n v a l u e A is s t r i c t l y g r e a t e r t h a n 1 ,
b) t h e l a r g e s t e i g e n v a l u e of M" i s s t r i c t l y l e s s t h a n A.
It r e s u l t s from h y p o t h e s i s a ) t h a t , f o r any a i n A , t h e v e c t o r h - n ~ ( X n ( a ) ) , a s n g o e s t o i n f i n i t y , t e n d s t o a v e c t o r any component o f which i s p o s i t i v e . H y p o t h e s i s b) i m p l i e s t h a t %-n c a r d ( a ( * ( a ) ) ) h a s a polynomial growth a s a
f u n c t i o n of n.
Example 1.
T h i s example shows t h a t h y p o t h e s e s a ) and b) a r e independent. L e t F = {1,2,31, A = { a ) and d e f i n e t h e i o n U(a) = ( v , E , w ~ ,w2,w3) s o :
V = { u , ~ ] E = { ( u , l , v , l ) , ( v , l , u , l ) ]
A,
It i s e a s i l y checked t h a t A = ?I = 2.
Example 2 : t h e Penrose s u b s t i t u t i o n
The m a t r i x Mp i s n o t p r i m i t i v e . But, i f we c o n s i d e r 2 , we s e e t h a t when s t a r t i n g from a l , i t i s a s u b s t i t u t i o n i n f a c t a c t i n g on {al , a 2 , b l , b 2 } a s s e t of c o l o r s w i t h m a t r i x (M:)~ , which i s p r i m i t i v e .
r *
I t i s easy t o determine and f o r t h e Penrose s u b s t i t u t i o n . S t a r t i n g from a t r i a n g l e of t y p e I and applying n times t h e i n f l a t i o n procedure we g e t a t e s s e l a t i o n of T , a t r i a n g l e of t y p e I expanded by t h e f a c t o r Tn. So, t h e number of t i l e s and t h e a r e a of T have t h e same o r d e r of magnitude and so have i t s perimeter-and t h e number of t i l e s touching i t s boundary. T h e r e f o r e we have A = T 2 and h = T .
6 . Second change of c o l o u r s .
Let r be a nonnegative i n t e g e r . Let u s c o n s i d e r t h e s e t
= {(Z,x) ; ZE 9- , x E V z 1 and d e f i n e a n equivalence r e l a t i o n dir on 3k : (Z,x) and (Z' , x ' ) a r e e q u i v a l e n t i f t h e r e e x i s t s a mapping P from { y € v Z ; d Z ( x , y ) G r } onto i Y ' E v Z , ; d Z , ( x ' , y ' ) G r }
such t h a t we have P ( x ) = x ' and 9 b e an isomorphism of t h e A-graph B Z ( x , r ) o n t o B Z , ( x t , r ) .
The q u o t i e n t space %/ar i s denoted A . I n o t h e r words, Ar i s t h e s e t of c l a s s e s of b a l l s of r a d i i r of elements of 8. , modulo isomorphisms of A-graphs c a r r y i n g c e n t e r s onto c e n t e r s .
Any A-graph Z i n g i v e s an Ar-graph z ( ~ ) i n t h e f o l l o w i n g way :
- t h e underlying tagged graphs a r e t h e same f o r Z and Z(') ,
- i f x i s a v e r t e x , i t s c o l o u r i n z ( ~ ) is t h e c l a s s modulo
of (BZ ( x , r ) ,XI. 'r
I n t h e same s p i r i t a s i n paragraph 4 we a r e going t o d e f i n e a n A - s u b t i c u t i o n or such t h a t , f o r any Z i n 9 , we have o ~ ( z ( ~ ) ) = (o(Z))('). Thi: s u b s t i t u t i o n w i l l a c t on t h e s e t & = LZ(') ; Z 9). Let a b e an element of Ar. We now proceed t o t h e d e f i n i t i o n of o r ( a ) . L e t u s suppose t h a t a i s r e p r e s e n t e d by (BZ ( x , r ) ,x) where Z E and x c VZ. The s u b s t i t u t i o n procedure g i v e s an isomorphism cp from t h e tagged graph l y i n g under 5(g&x)) onto t h e sub tagged graph of 5(Z) t h e v e r t i c e s of which a r e t h e descendents of x i n u(Z). We now d e f i n e a new c o l o r a t i o n g ( r ) on 5 ( g Z ( x ) ) : i f y i s a v e r t e x of O(gZ ( x ) ) , t h e n g ( r ) (y) is t h e c l a s s modulo 5 of (B5(Z) (P(Y) , r ) , P ( Y ) ) . I t
. .
r e s u l t s from t h e f a c t t h a t 5 does n o t d e c r e a s e d i s t a n c e s t h a t t h e c o l o r a t i o n gr does n o t depend on t h e p a r t i c u l a r c h o i c e of t h e r e p r e s e n t a n t of a . Therefore, we can d e f i n e Qr((y.) a s t h e i o n U(gZ(x)) t h e c o l o r a t i o n of which h a s been r e p l a c e d by g"). It i s easy t o check t h a t , a s claimed above, f o r any Z i n ? , we have % ( z ( ~ ) ) = ( G ( z ) ) ( ~ ) and t h u s * ( z ( ~ ) ) = ( 8 ( ~ ) ) ( ~ ) .
Let u s c o n s i d e r t h e following s u b s e t of &i :
&r = ( ( 2 , ~ ) ; Z , x E VZ , d z (x, 3Z) > r) and denote Ar 1 t h e s e t of c l a s s e s modulo $ of elements of &r. We s e t A* = A? A:. Corresponding t o t h e p a r t i t i o n Ar = A:uA: t h e m a t r i x M of Or deco:poses i n t o blocks :
