Maximal pattern complexity for Toeplitz words

Maximal pattern complexity for Toeplitz words (Ergodic Theory and Dynamical Systems 26 (2006), 1-14) Nertila GJINI , Teturo KAMAEy, TAN Boz and XUE Yu-Mei x

Abstract

The notion of the maximal pattern complexity of words is introduced in KZ1, KZ2]. In this paper, we obtain an almost exact formula for the maximal pattern complexity p (k) of Toeplitz words  on an alphabet A de ned by a sequence of coding words ( (n))1 2 (A  f?g)N (n = 1 2    ) including just one ? in their cycles  (n). Using this formula, we characterize pattern Sturmian words (i.e. p (k) = 2k (8k)) in this class. Moreover, we give a characterization of simple Toeplitz words in the sense of KZ2] in term of pattern complexity. In the case where  (1) =  (2) =    , we obtain the value limk!1 p (k)=k. We construct a Toeplitz word  2 A N with ]A = 2 such that p (k) = 2k (k = 1 2    ), while Toeplitz words in our sense always have discrete spectra.

1 Basic notions. Let A be a nite set of letters such that ]A 2, which is called an alphabet. N Let  2 A (N := f0 1 2    g) be an (innite) word on A . Let k be a positive integer. By a k-window  , we mean a subset of N with cardinality k denoting  = f        k; g with  <  <    < k; . We usually, but not always, assume for a window  that  = 0. The k-window  = f0 1     k ; 1g is called the k-block window. For a k-window  = f        k; g and a word  = (0)(1)(2)    , we denote  n +  ] := (n +  )(n +  )    (n + k; ) F ( ) := f n +  ]  n = 0 1 2    g p ( ) := ]F ( ): 0

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University of New York, Tirana, Albania ([email protected]) Matsuyama University, 790-8578 Japan ([email protected]) Department of Mathematics, Huazhong University of Science and Technology, Wuhan 430074, P.R. China (bo [email protected]) x Department of Mathematics, Tsinghua University, Beijing 100084, P.R.China ([email protected]) y z

1

An element in F ( ) is called a  -factor of . The maximal pattern complexity p for a word  has been introduced by the second author together with Zamboni KZ1] as p (k) := sup p ( ) (k = 1 2 3    ) 

where the supremum is taken over all k-windows  , while the block complexity p is dened as p (k) = p (f0 1     k ; 1g): It is known (Morse and Hedlund MH]) that for a word , the following statements are equivalent: (i)  is eventually periodic, (ii) p (k) is bounded in k, (iii) p (k) < k + 1 for some k = 1 2    , while the following parallel statements with respect to the maximal pattern complexity are equivalent KZ1]: (i)  is eventually periodic, (ii0) p (k) is bounded in k, (iii0) p (k) < 2k for some k = 1 2    . A word  with block complexity p (k) = k + 1 (k = 1 2 3    ) is known as a Sturmian word and is studied extensively (see for example Berthe B] and the references therein). A word  with maximal pattern complexity p (k) = 2k (k = 1 2 3    ) is called a pattern Sturmian word and is studied in KZ1, KZ2]. It is known that Sturmian words are always pattern Sturmian words, while simple Toeplitz words dened below are pattern Sturmian words which are not Sturmian words. The value limk!1 (1=k) log p (k) is known to be an invariant of the topological dynamical system arising from  taking values in flog n n = 1 2     ]A g (see Huang and Ye HY]). If the measure-theoretic dynamical system arising from  has a partially continuous spectrum, then this value is nonzero, but the converse is not true. For a nonempty set A of letters and a letter ? which is not included in A, let P (A ?) be the set of periodic words  = 1 2 (A  f?g)N such that every letter in A occurs at least once in while ? occurs in just once. Hence, the length of is the minimum period of . For  2 P (A ?) and  2 P (B ?), we dene   by substituting every occurrence of ? in  by  (0)  (1)  (2)    in the order. Then   2 P (A  B ?). Let n 2 6 SA P (S ?) (n = 1 2    ). We dene 1

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=

   2 (A  f?g)N as the limit of      n as n tends to 1. If the rst occurrence place of ? in n is not 0 for innitely many n's, then  2 A N. In this case, we call  a Toeplitz =

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word with single hole provided that it is not eventually periodic, and  1   2     a sequence of its coding words. Here, we only consider Toeplitz words with single hole (that is, one ? in the minimum cycles of all the coding words), so that we call them simply Toeplitz words. Since any Toeplitz word  is not eventually periodic, p (k) 2k (k = 1 2    ) holds by the above equivalence. We remark that any of  = 1 2 3    2 A N with n 2 6=SA P (S ?) (n = 1 2    ) is recurrent. Moreover, it is periodic if and only if there exist a 2 A and h 1 such that i 2 P (fag ?) for any i h. To prove \if" part, let i 2 P (fag ?) for any i h. Then, h h+1    = aaa    holds, so that  is obtained from the periodic word 1 2    h;1 by lling all the ? by the same letter a. Hence,  is periodic. Conversely, assume that ]S 2, where S is the set of a 2 A which appears in i for innitely many i's. Then, for any suciently large h, h h+1    consists only of letters in S , any of which appears innitely often. On the other hand, the minimum period L of the periodic word := 1 2    h;1 containing just one ? in its minimum cycle tends to 1 as h tends to 1. In , all the ? in are lled by the letters in S , while any letter in S is used innitely often. This shows rst that  is recurrent. Also, this shows that we can nd 2 blocks in  with an arbitrary length K  L located at the same place modulo L having dierent numbers of a's (and b's) if a b 2 S with a 6= b, which implies that K is not a period of . Since we can take K = 1 2    ,  is not periodic. A Toeplitz word  2 A N is called a simple Toeplitz word if ]A = 2, say A = fa bg, and it has a sequence of coding words 1 2    2 P (fag ?)  P (fbg ?). It is known that if  is a simple Toeplitz word, then it is a pattern Sturmian word, that is p (k) = 2k (k = 1 2    ). We remark that the proof for it in KZ2] is wrong. Here, we give not only a correct proof for it but also a characterization of the pattern Sturmian words among all Toeplitz words. We also give a characterization for a word to be a simple Toeplitz word in term of pattern complexity.

2 Main results

Let  2 A N be a recurrent word and  2 P (A  ?) with the minimum period r. Let  = f0 =  <  <    < k; g be a k-window. We decompose  into a union of subwindows 0

1

1

=

i2L

 i

where for i = 0 1     r ; 1

 i := fj  j = 0 1     k ; 1 such that j i (mod r)g and L := fi 2 f0 1     r ; 1g   i 6= g: 3

(2.1)

Here, we also denote  i := ( i ; i)=r = f(j ; i)=r  j 2  ig (i 2 L): (2.2) For a 2 A  f?g, S  A ,    n 2 (A  f?g)n and U  (A  f?g)n with n 2 N, we denote  fag if a 6=? S (a) = S if a =? S (    n ) = f    n  i 2 S ( i) (i = 1     n)g S U = S (u): 1

1

1

u2U

We dene

D( L) := ] ;A F (L) ; `]A (2.3)  ` E ( L) := ] A F (L)  fa  a 2 A g ; `]A  where we denote ` = ]L and F (L) is the set of L-factors of . Let  =   2 A N. Let i be such that 0  i  r ; 1 and i0 =?. Then, i =? if and only if i i (mod r). By a constant word, we mean a word in fan a 2 A g for some n 2 N. For U  A n with n 2 N, we denote by C (U ) the set of letters appearing in the constant words in U . It is easy to prove the following Lemma. Lemma 1. Let  =   , where  2 P (A  ?) with the minimum period r and  2 A N is a recurrent word. Let  be a k-window with the decomposition (2:1) and (2:2). Then for any i 2 L, we have F ( i)  F ( i) ;] F (C (i)Fn(Fi))( i)C=(F]A (;i))]C=(FA( i)): We denote F := fu 2 F ( )  uj is a constant word for any i 2 Lg F i := fu 2 F ( )  uj is not a constant wordg (i 2 L) where for u 2 F ( ) with u =  n +  ], we denote uj =  n +  i]. If u 2 F i with i 2 L, then u =  n +  ] holds with n such that n + i i (mod r) since otherwise uj is a constant word. Moreover, if u 2 F i and i0 2 L with i 6= i0, then uj is a xed constant word with letter (j ) such that j i + i0 ; i (mod r) which is independent of u 2 F i. Hence, F and F i's are disjoint each other and any of u 2 F i is determined by uj . Note that if u 2 F i with u =  n +  ] and n + i ; i 0, then uj =  (n + i ; i )=r +  i]: 0

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Conversely, if v 2 F ( i) is not a constant word with v =  n +  i], then v = uj holds for u 2 F i with u =  rn + i ; i +  i]. Thus, we have ]F i = p ( i) ; ]C (F ( i)) = p ( i) ; ]A  where the second equality holds by Lemma 1. Consider the set F . To any word u 2 F , we associate a new word u~ 2 F (L) with the property u~(i) 2 C (uj ) for all i 2 L. Denote F~ = fu~ 2 F (L) u 2 F g then, by the denition of F , it is easy to see that there is a bijection between the sets F and F~ . Also it holds that F~ = (F( ) \ A l )  C ( i ; i + L]) i

0

i

i2L

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0

with Ci := C (F ( i)). Moreover, it holds that C (F ( ))  C ( A (F ( )) and ~ fal a 2 C ( A (F( )))g n fal a 2 C (F ( ))g  A (F(L)) n F: Therefore, we have ;  F~  A (F (L)) n fal a 2 C ( A (F( )))g n fal a 2 C (F ( ))g  while F~ = A (F(L)) holds if Ci = A for any i 2 L. Hence, we have X X p ( ) = ]F + ]F i = ]F~ + ]F i

X(p ( i) ; ] ) i L X = ] F (L) ; ]C ( F(L)) + ]C (F ( )) + (p ( i) ; ] ) i L X i = D( L) + p ( ) ; ]C ( F (L)) + ]C (F ( )) i L P while p ( ) = D( L) + i L p ( i) holds if Ci = for any i 2 L. i2L

i 2L

 ] A F ( ) ; ]C ( A F ( )) + ]C (F ( )) + A

A

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A

A

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2

Thus, we get the following Theorem 1. Theorem 1. For any  2 P (A  ?) with the minimal period r and a recurrent word  2 A N, let  =   and  be a k-window with the decomposition in (2:1) and (2:2). Then, we have X p ( ) ; ]C (F ( ))  D( L) ; ]C ( A (F (L))) + p ( i): Particularly, we have with the equality if C (F

p ( )  D( L) + ( i)) = A

X p ( i) i2L

for any i 2 L.

5

i2L

Corollary 1. With the same setting as in Theorem 1, we have the following state-

ments. (1) It holds that C (F ( i)) = A and p ( i) = p ( i)+ ]A ; ]C (F ( i)) for any i 2 L. In particular, we have p (r ) = p ( ) + ]A ; ]C (F ( )). (2) p (k) p (k) holds for any k = 1 2    . (3) If  =  holds with 2 P (A  ?) having the minimum period s and a recurrent word  2 A N satisfying the following Condition( ) for  , then we have the equality

p ( ) = D( L) +

X p ( i) = D( L) + X p ( i): i2L

i2L

Condition( ) For any i 2 L and j j 0 2  i, rs divides j ; j 0. Proof. The rst part of (1) follows from Lemma 1. The second part follows from the rst part as (r )0 = r and (r )0 =  . For an arbitrary k-window , put  = r . Then, since  0 =  and  0 = , we have p ( ) p ( ) by (1), which proves (2). Let us prove (3). Since  i for any i 2 L is contained in single modulo class of s by the Condition( ), F ( i) contains all constant words with letters appearing in , that is A as 2 P (A  ?). Hence, C (F ( i)) = A for any i 2 L, and by Theorem 1 and (1), we have the equality

X p ( i ) i L X D( L) + p ( i):

p ( ) = D( L) + =

2

i2L

Lemma 2. Let  2 A be any simple Toeplitz word. Then, p ( ) ; ]C (F ( )) + #A  2k N

(2.4) holds for any k-window  . In particular, a simple Toeplitz word is a pattern Sturmian word.

Proof. To prove (2.4) for any simple Toeplitz word  2 A N with A = fa bg and any k-window, we use the induction on k. If k = 1 or 2, (2.4) holds clearly. For k 3, assume that (2.4) holds for any `-window with ` < k and any simple Toeplitz word. Take a simple Toeplitz word  2 A N and a k-window  . Since  is a simple Toeplitz word, we may assume without loss of generality that  =   holds for a simple Toeplitz word  2 A N and  2 P (A  ?) such that  = with 2 P (fag ?) and 2 P (fbg ?). Let the minimum periods of and be s and t, respectively. Thus the minimum period of  is r = st.

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Case 1 If  is not a multiple of r, then we have the decomposition of  as in (2.1) and (2.2) with l = ]L induction, we have

2. By Theorem 1, Lemma 1 and the assumption of the

X p ( i) i L X ;p ( i) + 2 ; ]Ci D( L) ; C + 2 + i L X D( L) ; C + 2 + 2] i

p ( ) ; C + 2  D( L) ; C 0 + 2 + =



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0

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i2L

= D( L) ; C + 2 + 2k 0

where we put C := ]C (F ( )) and C 0 := ]C ( A F (L)). Therefore, to prove (2.4), it is sucient to prove

D( L) ; C 0 + 2  0:

(2.5)

Decompose L = fL  L      L` g into the modulo classes of s as follows. 1

2

L=

i2K

Li

where for i = 0 1     s ; 1.

Li := fLj  j = 0 1     k ; 1 such that Lj i (mod s)g and K := fi = 0 1     s ; 1  Li 6= g: For i 2 K , dene

Gi = fu 2 F(L)  u =  n + L] with n such that n i ; i (mod s)g 0

where i is such that 0  i  s ; 1 and (i ) =?. We also dene 0

0

0

G = fu 2 F (L)  u =  n + L] with n such that n 2= i ; K (mod s)g: 0

Note that G = fa`g if G 6= . If u 2 Gi with i 2 K , then ujLnL = a`;]L holds, while ujL consists only of b's except possibly for at one place. Therefore, for any i 2 K , we have ] A Gi = ]Li + 1. Moreover, A Gi contains the constant word a` if and only if ]Li = 1. Hence, we have i

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] A Gi n fa`g = ]Li + 1]L : i

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Dene 1a = 1 or = 0 according to whether A F(L) contains the constant word a` or not. In the same way, we dene 1b = 1 or = 0 according to whether A F (L) 7

contains the constant word b` or not. Then, we have C 0 = 1a + 1b. On the other hand, we have



] AF(L) = ] G 

 =

A Gi

!

i K X 1a + (]Li + 1]L i K X 1 +`+ 1 2

i

2

)

2

a

]L 2 i

i2K

 1a + ` + b`=2c: Thus, we have

D( L) ; C 0 + 2  2 ; 1b ; ` +

X1 i2K

]L 2 i

 2 ; 1b ; d`=2e:

(2.6)

from (2.6). If ` = 2 with ]K = 2, then since PIfi K` 1]L 3, =then0, we(2.5)havefollows (2.5) by (2.6). Finally, if ` = 2 with ]K = 1, then 2

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we have 2 cases, either A F (L) = faa ab bag or faa ab ba bbg. In the rst case, we have C 0 = 1, while in the second case, we have C 0 = 2. Any case, we have ] A F(L) ; C 0 = 2. Thus, D( L) ; C 0 + 2 = 0, which completes the proof of Case 1.

Case 2 If  is a multiple of r, then by (1) of Corollary 1, we have p ( ) ; ]C (F ( )) + 2 = p (=r) ; ]C (F(=r)) + 2: If  is a multiple of re but not a multiple of re , then we repeat this argument e times until we get a simple Toeplitz word  such that p ( ) ; ]C (F ( )) + 2 = p (=re ) ; ]C (F (=re )) + 2: Since =re is not a multiple of r, we get p ( ) ; ]C (F ( )) + 2 = p (=re ) ; ]C (F (=re )) + 2  2k by Case 1, which completes the rst part of Theorem 2. Since ;]C (F ( )) + #A 0, (2.4) for any simple Toeplitz word  and any kwindow  implies that p (k)  2k (k = 1 2    ) for any simple Toeplitz word . Here, the equality follows since any Toeplitz word is not eventually periodic. Thus, any simple Toeplitz word is a pattern Sturmian word. Theorem 2. Let  2 A N be a Toeplitz word. Then, it is a pattern Sturmian word if and only if either it is a simple Toeplitz word or there exists a simple Toeplitz word  2 A N and  2 P (A  ?) with the minimum period r such that D( L)  0 for any L  f0 1     r ; 1g, and  =   . +1

8

Proof. We prove the \if" part rst. By Lemma 2, it is sucient to prove that for a simple Toeplitz word  2 A N, and  2 P (A  ?) with the minimum period r such that D( L)  0 for any L  f0 1     r ; 1g,  :=   is a pattern Sturmian word. Take any k-window  and use the notations in (2.1) and (2.2). We have already proved in Lemma 2 that p ( i) ; ]C (F ( i)) + 2  2# i : Then, by Theorem 1 and (1) of Corollary 1, we have

X p ( i) i L X ;p ( i) ; ]C (F( i)) + 2 D( L) + X 2ki = 2ikL

p ( )  D( L) + =



2

2

i2L

where we put ki = ] i. Thus,  is a pattern Sturmian word. Let us prove the \only if" part. Assume that a Toeplitz word  is a pattern Sturmian word. Then, just 2 letters appear in , so that we may assume that A = fa bg. By the aperiodicity, we may assume that there exists a sequence        of coding words of  consisting only of elements in P (A  ?). Let ri be the minimum period of i (i = 1 2    ). Denote i = i i    (i = 1 2    ): Then  =  and i = i i , and by (2) of Corollary 1, all i's are pattern Sturmian words. Suppose that D(i  L) > 0 holds for some i = 1 2    and L  f0 1     ri ; 1g. Then, by (3) of Corollary 1 with  = L as Condition ( ) is clearly satised, we have X p (L) = D( L) + p (fj g) = D( L) + 2` > 2` 1

2

+1

1

+1

i

i

j 2L

where ` := ]L. This implies that p (r    ri; L) p (L) > 2` by (1) of Corollary 1, which contradicts the assumption that  is a pattern Sturmian word. Thus, we have the following Lemma 3. Lemma 3. Let  2 P (A  ?) be any word appearing in some sequence of coding words consisting only of elements in P (A  ?) of a Toeplitz word in A N which is a pattern Sturmian word as well. Then, we have D( L)  0 for any L  f0 1     r ; 1g (2.7) where r is the minimum period of . 1

1

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Let us return to the proof of the \only if" part of Theorem 2. We call  2 P (A  ?) a simple coding word, if there exist i 2 P (fag ?)  P (fbg ?) (i = 1 2     h) such that  =    h. Any element in P (fag ?)  P (fbg ?) is also called a simple coding word. We call  2 P (A  ?) irreducible if there does not exist a simple coding word such that either  = or  = holds with some 2 P (A  ?)  P (fag ?)  P (fbg ?). Suppose that one of        , say h , is not simple. Then, either h is irreducible or there exists a simple coding word and an irreducible such that h = . Any case, we have the decomposition that 1

2

2

3

=  with  2 P (A  ?), an irreducible 2 P (A  ?) and a Toeplitz word  . Hence, by Lemma 3, we have D(  rL)  0 for any L  f0 1     s ; 1g, where r s are the minimum periods of  and , respectively. On the other hand, we will prove the following Lemma 4, which contradicts this fact. This implies that any of        2 P (A  ?) in  =       is simple. Therefore,  :=      is a simple Toeplitz word. Moreover, we have  =   with  satisfying (2.7) by Lemma 3, which completes the proof of Theorem 2. Lemma 4. For any  2 P (A  ?) such that ]A = 2 and is irreducible, we have D(  rL) 1 for L = f0 1 2     s ; 1g, where r s are the minimum periods of  and , respectively. 2

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Proof. Let A = fa bg. It is easy to see that

A F  (rL) = A s  A F (L): The set A F (L) contains ( (n) (n + 1)    (n + s ; 1))a and ( (n) (n + 1)    (n + s ; 1))b (n = 0 1     s ; 1) where the word (   )a or (   )b denotes the word obtained from the original word    by replacing all occurrences of ? in    by a or b, respectively. We prove that all 2s words in this list are dierent from each other. Since ( (n) (n +1)    (n + s ; 1))a and ( (m) (m +1)    (m + s ; 1))b contain dierent numbers of letters a for any n m, they are dierent from each other. Suppose that ( (n) (n + 1)    (n + s ; 1))a = ( (m) (m + 1)    (m + s ; 1))a holds for some n < m. Then with the minimum s = m ; n as this, we have the factorization s = s s and 1

1 2

( (0) (1)    (s ; 1))a = ( (0) (1)    (s ; 1))sa2 : 1

10

There exists just one i with 0  i  s ; 1 such that (i ) =?. Let i = i + i s with 0  i < s and 0  i < s . Then we can write = 0 00 with  (i) i 6 i (mod s ) 0 (i) = ? i i (mod s )  i (mod s ) 00(i) = a? ii 6 i (mod s )  which contradicts the assumption that is irreducible. Thus, all elements in the list ( (n) (n + 1)    (n + s ; 1))a (n = 0 1     s ; 1) are dierent from each other. In the same way, all elements in the list ( (n) (n + 1)    (n + s ; 1))b (n = 0 1     s ; 1) are dierent from each other. Therefore, we have 2s elements in A F(L), which contains at most one constant word. Thus, we have ] A F  (rL) 2s + 1, which implies that D(  rL) 1. Theorem 3. Let  2 A N be a Toeplitz word. It is a simple Toeplitz word if and only if (2:4) holds for any k-window  . Proof. The \only if" part has been already proved in Lemma 2. Here, we prove the \if" part. Assume that (2.4) holds for a Toeplitz word  2 A N. Then,  is a pattern Sturmian word. Hence, we may assume that ]A = 2 Suppose that  is not a simple Toeplitz word. Then by Theorem 2, there exists a simple Toeplitz word  2 A N and  2 P (A  ?) with the minimum period r such that D( L)  0 for any L  f0 1     r ; 1g, and  =   . If  is a simple coding word, then  is a simple Toeplitz word contradicting our supposition. Hence,  is not a simple coding word. Then, there exists an irreducible word such that either  = or there exists a simple coding word 0 such that  = 0. Any case, we have  =  with an irreducible coding word and a simple Toeplitz word  . Let s be the minimum period of . Then, by the same argument as in the proof of Lemma 4, ] A F ( ) = 2s holds for the s-window  := f0 1     s ; 1g and F ( ) has at most one constant word. Since D(  ) = 0 and the window  satises the condition ( ) with some and  , we have p ( ) = 2s and ]C (F ( ))  1. This implies that p ( ) ; ]C (F ( )) + 2 > 2s and (2.4) does not holds, which contradicts our assumption. Thus,  is a simple Toeplitz word. Example 1. Let  := (a?)1 (b?)1 (a?)1 (b?)1    = abaaabababaaabaa     which is a simple Toeplitz word. Let  = (ab?)1. Then,  satisfy (2.7). Therefore, by Theorem 3,  :=   = abaabbabaabaabaabbabaabbabaabbabaaba    0

1

1

0

2

0

2

11

1

1

1

1

2

2

2

2

0

1

2 1

is a pattern Sturmian word. It is not a simple Toeplitz word since F (f0 1 2g) = faab aba baa abb bab bbag and hence,

p (f0 1 2g) ; #C (F (f0 1 2g)) + 2 = 8 > 2  3 and (2.4) is not satised. Theorem 4. Let  =      2 A N with  2 P (A  ?). Then, we have p (k) = ]A + max E ( L)  lim 01 1 ]L ; 1 k!1 k 2 Lf   r; g ]L

where r is the minimum period of , and E ( L) is dened in (2.3). Proof. Note that E ( L) = D(  rL) for any L  f0 1     r ; 1g. Let E ( L) : E := 0max 1 1 ]L ; 1 2 Take L0 with ]L0 = `0 2 attaining the maximum of E ( L)=(]L ; 1). Without loss of generality, we may assume that 0 2 L0. Let L00 = L0 nf0g. Dene a sequence of windows by  (1) := rL0 and Lf   r; g ]L

 (k + 1) := r(r (k)  L0 ) (k = 1 2    ): Let us apply (3) of Corollary 1 with   for ,  for  ,  for ,  for and rL for L. Then,  (k) (k = 1 2    ) satisfy Condition( ). Moreover,  (k + 1) =  (k) (k = 1 2    ). Since D(  rL ) = (` ; 1)E , we have by (3) of Corollary 1 that 0

0

0

0

p ( (1)) = D(  rL ) + 0

0

X p (fig)

i2rL0

= (` ; 1)E + `]A p ( (k + 1)) = D(  rL ) + p ( (k)) + 0

0

X p (fig)

i2rL0 0

= (` ; 1)E + p ( (k)) + (` ; 1)]A  0

0

and hence,

p ( (k)) = (]A + E )(K ; 1) + ]A (k = 1 2    ) where K := k(` ; 1) + 1 is the size of the window  (k). Thus, we have lim inf p (k)=k klim p ( (k))=(K ; 1) = ]A + E : k!1 !1 0

Now, let us prove that lim supk!1 p (k)=k  ]A + E . For any k 2, let p (k) = p ( 0) for some k-window  0. By Lemma 1, we may assume that  0 is divisible by r but not by r . Hence, we put  0 = r with  which is not divisible by 2

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r. We use the decomposition of r given in (2.1) and (2.2) for   instead of . Then by Theorem 1, we have p (k) = p (r ) X  D(  rL) + p (r i) i L X  E ( L) + p (ki) i L P 1 for any i 2 L. Since k = i L ki with ]L 2

2

(2.8)

where ki = ] i 2, any of ki's for 2 i 2 L is less than k. Hence, we can apply the induction on k to prove that p (k) + E  (]A + E )k: (2.9) For k = 1, (2.9) holds since p (k) = ]A . Let k 2. Assume that (2.9) holds for 1 2     k ; 1. Then by (2.8) and the assumption of the induction, we have

p (k) + E  E ( L) +

 (` ; 1)E =



X p (ki) + E i L X + p (ki) + E 2

X(p (ki) +iEL ) i L X (] + E )ki = (] 2

2

i2L

A

A

+ E )k:

Thus, we have (2.9) for k = 1 2    , and hence, lim supk!1 p (k)=k  ]A + E , which completes the proof. Example 2. Let  = (anbban ?)1 and  =       for n 1. Then, E ( L) = 0 if ]L = 2, E ( L)  2 if ]L 3 with the equality for L = f0 1 n + 2g. Hence, we have maxL ]L E ( L)=(]L ; 1) = 1. Thus, limn!1 p (k)=k = 3. +1

2

3 Further examples and open problems By (3) of Corollary 1, we can calculate almost exact values of the maximal pattern complexity p (k) of the Toeplitz words  just by calculating D( L). It is well known that the measure-theoretic dynamical systems arising from our Toeplitz words have discrete spectrum JK]. On the other hand, it is known KZ1] that if the maximal pattern complexity of a word increases in less than exponential order, then the measure-theoretic dynamical system arising from it has a discrete spectrum but the converse is not true. Here, we give such examples of Toeplitz words  2 A N with ]A = 2 and p (k) = 2k (k = 1 2    ). We also give  2 A N with p (k) increasing in the polynomial order of given degree 1. 13

It is known K] that for any word  2 A N with ]A = 2, either n   X k (k = 1 2    ) p (k)  ;1

i

i=0

for some n = 1 2    , or

p (k) = 2k (k = 1 2    )

holds.

Example 3. Let A = fa bg. For n = 1 2    , we can take n 2 A n with length

n2n such that n contains every block in A n . For example, = ab = aaabbabb = aaaaabababaaabbbabbbabbb    Let n = ( n ?)1 2 P (A  ?). Then, we have D(n  f0 1     n ; 1g) = 2n ; 2n: Let k = k k   (k) = fi`k  i = 0 1     k ; 1g (k = 1 2    ) 1

2

3

+1

where ` := 1 and `k := 1

Y(i2i + 1):

k;1 i=1

Then, by Corollary 1, we have p ( ) = p ( (k)=`k ) = D(k  f0 1     k ; 1g) + 2k = 2k : Thus, p (k) = 2k (k = 1 2    ). Example 4. (Xue X]) Let aabb  : ab ! ! aaab k

be a substitution on A = fa bg. The xed point aabbaabbaaabaaab    is a Toeplitz word such that  = (aa?b)1 (bb?a)1(aa?b)1 (bb?a)1    : Let  = (aa?b)1 (bb?a)1 = aabbaabba?abaaab: Then, E ( L) = 0 if ]L = 2 or 3, E ( L)  2 if ]L 4 with the equality for  = f0 1 2 3g. Thus, maxL ]L E ( L)=(]L;1) = 2=3 and limn!1 p (k)=k = 8=3. This gives an alternative proof of the well known fact (Goodman G], for example) that the measure-theoretic dynamical system arising from this  has a discrete spectrum since p (k) increases less than exponentially as the function of k. 2

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Open Problems: 1. What is the maximal pattern complexity of the Toeplitz words  =       with  2 P (A  ?) having more than one hole in the minimum cycle. Is

there a Toeplitz word in this extended class having the maximal pattern complexity increasing exponentially? Cassaigne and Karhum!aki CK] obtained the increasing order k of the block complexity in this class. For example,  := (ab?a?)1 has the block complexity increasing in the order k = ; . The maximal pattern complexity should increase at least as this. We don't know even whether the order is sub-exponential or not. log 5 (log 5 log 2)

2. All the pattern Sturmian words known so far except for the simple Toeplitz

words fail to satisfy (2.4). Does the property (2.4) characterize the simple Toeplitz words among all the words?

3. Is there a Toeplitz word with one hole such that the maximal pattern complexity increases in a polynomial order with degree > 1? Acknowledgement: The authors thank Professor Wen Zhi-Ying (Tsinghua Uni-

versity) for inviting 2 of the authors (Kamae, Tan) to Morningside center of Mathematics (CAS) giving them a chance to complete the paper which has been suspended for 3 years. Also thanks go to Professor Rao Hui (Tsinghua University) for helpful discussions.

References

B]

CK]

G]

JK]

HY]

K]

KR]

V. Berthe, Sequences of low complexity: automatic and Sturmian sequences, London Math. Soc. Lecture Note Ser. 279, Cambridge Univ. Press, 2000, 1-34. J. Cassaigne and J. Karhum!aki, Toeplitz words, generalized periodicity and periodically iterated morphisms, European J. Combin. 18:5 (1997), 497{510. T.N.T.Goodman, Topological sequence entropy, Proc. London Math. Soc. (3) 29 (1974), 331-350. K. Jacobs and M. Keane, 0 ; 1-sequences of Toeplitz type, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 13 (1969), 123-131. Wen Huang and Xiangdong Ye, Maximal pattern entropy and null systems, preprint. T. Kamae, Maximal pattern complexity as topological invariants, preprint. T. Kamae and H. Rao, Pattern complexity over ` letters, to appear in European J. Combinatorics 27 (2006), 125-137. 15

KX] T. Kamae and Y.M. Xue, Two dimensional word with 2k maximal pattern complexity, Osaka J. Math. 41 (2004), 257-265.

KZ1] T. Kamae and L. Zamboni, Sequence entropy and the maximal pattern complexity of innite words, Ergodic Theory and Dynamical Systems 22:4 (2002), 1191-1199.

KZ2] T. Kamae and L. Zamboni, Maximal pattern complexity for discrete systems, Ergodic Theory and Dynamical Systems 22:4 (2002), 1201-1214.

MH] M. Morse and G.A. Hedlund, Symbolic dynamics II: Sturmian sequences, Amer. J. Math. 62 (1940), 1-42.

X] Y.M. Xue, Sequence entropy and maximal pattern complexity, Doctoral Thesis at Osaka City University, 2003.

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