Combinatorics On Sturmian Words. Amy Glen. Major Review Seminar. February 27, 2004 DISCIPLINE OF PURE MATHEMATICS

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1 Combinatorics On Sturmian Words Amy Glen Major Review Seminar February 27, 2004 DISCIPLINE OF PURE MATHEMATICS

2 OUTLINE Finite and Infinite Words Sturmian Words Mechanical Words and Cutting Sequences Infinite Words and Morphisms The Fibonacci Word & Characteristic Sturmian Words Decompositions of Sturmian Words into Palindrome Words Powers of Factors of Sturmian Words The Tribonacci Sequence Future Endeavours Application to Continued Fractions 1

3 FINITE AND INFINITE WORDS Combinatorial properties have become significantly important Applications in physics, biology, maths & computer science Family of Sturmian words a very active subject of research Named after Charles François Sturm ( ) Morse and Hedlund (1940): symbolic dynamics Largest developments in last twenty years Points of view: geometrically, combinatorially, algebraically Combinatorial approach is most common 2

4 WHAT ARE FINITE AND INFINITE WORDS? Alphabet: A denotes a finite set of symbols called letters. Finite/Infinite word: concatenation of letters from A. Let u, w denote finite words & x, y denote infinite words. If x = uwy w is a factor of x, u is a prefix of x, y is a suffix of x. The length of w is number of letters it contains, denoted by w. The empty word of length 0 is denoted by ε. The set A of all finite words over A is a monoid (Id = ε). 3

5 WHAT IS A STURMIAN WORD? An infinite word s with exactly n + 1 distinct factors of length n, n 0 = s is over a two-letter alphabet, say A = {a, b}. Sturmian words are not eventually periodic they are aperiodic words of minimal complexity many characterizations and numerous properties of Sturmian words Applications: symbolic dynamics, continued fraction expansion, physics (crystallography), computer science (pattern recognition) Equivalent definitions: mechanical words & cutting sequences 4

6 IRRATIONAL MECHANICAL WORDS An infinite word s over A is Sturmian an irrational α (0, 1) & ρ R such that s is either: s α,ρ = s(0)s(1)s(2) or s α,ρ = s (0)s (1)s (2), where s(n) = a if (n + 1)α + ρ nα + ρ = 0, s(n) = b otherwise; s (n) = a if (n + 1)α + ρ nα + ρ = 0, s (n) = b otherwise. Note: denotes floor function & denotes ceiling function α is called the slope & ρ is the intercept ρ = 0 s α,0 = ac α & s α,0 = bc α where c α is the characteristic Sturmian word of slope α 5

7 CUTTING SEQUENCES An interpretation of mechanical words Line: y = βx + ρ with irrational β > 0, ρ R consider this ray in the positive quadrant of R 2 overlay quadrant with an integer grid & construct infinite word K β,ρ vertical grid-line crossed: label intersection with a horizontal grid-line crossed: label intersection with b Labels: x 0, x 1, x 2,... K β,ρ = x 0 x 1 x 2 is a Sturmian word, i.e. K β,ρ = s β/(1+β),ρ/(1+β). Note: c α = K β,0 α = β/(1 + β) 6

8 y = x Fibonacci word b a a b b a a b a a Fibonacci word: f = K ( 5 1)/2,0 = abaababaabaababaababaab special example of a characteristic Sturmian word of slope α = Properties of f may be extended to Sturmian words 7

9 INFINITE WORDS & MORPHISMS Let (u n ) n 0 be a sequence of words from A s.t. u n is a proper prefix of u n+1. This gives obvious meaning to lim n u n as an infinite word. A morphism on A is a map ψ : A A such that ψ(uv) = ψ(u)ψ(v), u, v A. If ψ(c) = cw, c A, w A ψ n (c) is a proper prefix of ψ n+1 (c) and (ψ n (c)) n 0 converges to a unique infinite word: We say that x is generated by ψ. x = lim n ψn (c). 8

10 PALINDROME WORDS Palindrome: a finite word that reads the same backwards as forwards Examples: aa, abaaba, aba, ababa Objects of great interest in computer science Important tools used in the study of factors of Sturmian words Question: Where exactly do palindromes occur in a Sturmian word? Example: f = ab(aa)bab(aa)b(aa)bab(aa)bab(aa), where aa occurs at positions: 2, 7, 10, 15, 20,... Distances: 5, 3, 5, 5,... Fibonacci word over the alphabet {5, 3} I have described precisely where palindromes occur in a Sturmian word 9

11 FINITE FIBONACCI WORDS Define the Fibonacci numbers (F n ) n 1 by F 1 = F 0 = 1, F n = F n 1 + F n 2. Let f n denote the prefix of f of length F n and set f 1 = b. Then f 0 = a, f n = f n 1 f n 2 and f = lim n f n. E.g. f 1 = ab, f 2 = aba, f 3 = abaab Singular words of f : w 2n 1 = af 2n 1 b 1, w 2n = bf 2n a 1 Examples: w 0 = b, w 1 = aa, w 2 = bab, w 3 = aabaa palindromes Wen & Wen (1994): f = aw 0 w 1 w 2 w 3 = a b aa bab aabaa 10

12 CONTINUED FRACTIONS Combinatorial structure of c α has close relationship with the continued fraction (CF) of its slope α Recall: every irrational α (0, 1) has a unique CF expansion 1 α = [0; a 1, a 2, a 3,...] = 1 a 1 + a 2 + where all the a i Z + are called partial quotients. 1 a 3 +, 11

13 CHARACTERISTIC STURMIAN WORDS Let α (0, 1) be irrational with α = [0; 1 + d 1, d 2, d 3,...]. Define standard sequence (s n ) n 1 of words by: s 1 = b, s 0 = a, s n = s d n n 1 s n 2. n 0, s n is a prefix of s n+1 lim n s n is a well-defined infinite word in fact, each s n is a prefix of c α and c α = lim n s n Fibonacci word: α = (3 5)/2 = [0; 2, 1, 1, 1,...] Singular words of c α : w 2n 1 = as 2n 1 b 1, w 2n = bs 2n a 1 Melançon (1999): defined palindrome words v n such that c α = av 0 v 1 v 2 v 3 v 4 12

14 EXAMPLE Consider c α with α = ( 3 1)/2 = [0; 2, 1, 2, 1, 2, 1, 2,...]. Standard sequence: s 1 = ab, s 2 = s 1 s 0 = aba, s 3 = s 2 2 s 1 = abaabaab,... c α = abaabaababaabaabaababaabaabaab Singular words: w 0 = b, w 1 = aa, w 2 = bab, w 3 = aabaabaa,... Decomposition in terms of w 2 : c α = abaabaa(bab)aabaabaa(bab)aabaabaa(bab)aabaa(bab)aabaabaa = abaabaa(bab)z 1 (bab)z 2 (bab)z 3 (bab)z 4, where z 1 z 2 z 3 is the characteristic Sturmian word of slope β = [0; 1, 2] = 3 1 over the alphabet {aabaa, w 3 } 13

15 CONJUGATES OF f AND c α Let x, x be infinite words such that x = wx, w finite, w = k. Then x is called the k th conjugate of x; i.e. x is obtained from x by deleting the first k letters of x. Levé and Séébold (2003): a factorization of each conjugate of f as an infinite concatenation of generalized singular words I have extended this result to conjugates of c α for α = [0; 2, r, r, r,...] and α = [0; 1, 1, r, r, r,...] Paper accepted for publication (European J. Combinatorics) 14

16 POWERS IN STURMIAN WORDS Interested in finite words w and integers p Z + such that w p = www w (p times) is a factor of c α Application: spectral properties of associated quantum mechanical quasi-crystal models (Physics) Damanik and Lenz (2003): explicitly determined all integer powers of words occurring in c α also considered the # of distinct squares of words contained in s n combinatorial method: based on properties of the building blocks s n 15

17 EXACT NUMBER OF SQUARES IN f n A square is a word of the form uu; u a finite word Example: f 4 = abaababa contains 4 squares: (aba) 2, (ab) 2, (ba) 2, a 2 Fraenkel and Simpson (1999): exact # of distinct squares in f n is 2(F n 2 1), n 5 Recently extended to the case of the factors s n of c α u p (p 2) is a factor of f = u = F n for some n I have extended the above results to the Tribonacci sequence paper in preparation 16

18 TRIBONACCI SEQUENCE ξ Infinite word over A = {a, b, c}: ξ = abacabaabacababacabaabacab a generalization of the Fibonacci word Define the Tribonacci numbers (T n ) n 1 by T 1 = T 0 = 1, T 1 = 2, T n = T n 1 + T n 2 + T n 3. Let A n denote the prefix of ξ of length T n and set A 1 = c. Then A 0 = a, A 1 = ab, A n = A n 1 A n 2 A n 3 and ξ = lim n A n. E.g. A 2 = (ab)ac, A 3 = (abac)(ab)a, A 4 = (abacaba)(abac)(ab) Yet to determine exactly where palindromes occur in ξ 17

19 EXACT NUMBER OF SQUARES IN A n Define P 0 = ε, P n = A n 1 A n 2 A 1 A 0, n 1 E.g. P 1 = a, P 2 = aba, P 3 = abacaba palindrome prefixes of ξ Exact # of distinct squares in A n (n 6): n 3 m=0 ( P m + 1) + P n 5 + P n Example: A 5 = abacabaabacababacabaabac contains 7 squares: aa, abab, baba, abaaba, (abacab) 2, (bacaba) 2, (abacaba) 2 A 6 has (0 + 1) + (1 + 1) + (3 + 1) + (7 + 1) = 17 squares 18

20 SQUARES AND HIGHER POWERS IN ξ Let u be a factor of ξ with T n u < T n+1. u p # u s.t. u p in ξ T n 2 T n T n + T n 1 2 P n T n 3 P n T n + T n I am now extending this to episturmian words Episturmian words: a generalization of Sturmian words to an arbitrary finite alphabet 19

21 FUTURE ENDEAVOURS The study of episturmian words is a new area of research I hope to extend my results to the family of episturmian words Application to continued fractions and transcendence: γ R is transcendental if it is not a zero of a polynomial with integer coefficients Allouche et al. (2001): Let a, b Z + (a b) & let x = (x n ) n 0 be an infinite sequence over {a, b}. Then γ := [0; x 0, x 1, x 2,...] is transcendental if x is a Sturmian word over {a, b}. Question: What if the partial quotients form an infinite episturmian word over a k-letter alphabet? 20

22 SOME INTERESTING READING Allouche et al. (2001): Transcendence of Sturmian or morphic continued fractions Berstel and Séébold (2002): Chapter 2 of Algebraic Combinatorics on Words Damanik and Lenz (2003): Powers in Sturmian sequences Justin and Pirillo (2002): Episturmian words & episturmian morphisms Melançon (1999): Lyndon words & singular factors of Sturmian words Wen and Wen (1994): Some properties of the Fibonacci word 21