Episturmian words: extremal properties & quasiperiodicity

Transcription

1 Episturmian words: extremal properties & quasiperiodicity Laboratoire de Combinatoire et d Informatique Mathématique (LaCIM) Université du Québec à Montréal amy.glen@gmail.com glen Algebra and Geometry Research Group Seminar (Uppsala University, Sweden September 25, 2007)

2 Outline 1 Background Terminology & Notation Sturmian & episturmian words 2 Lexicographic order & extremal words Previous work More on episturmian words A refinement Episturmian Lyndon words 3 Quasiperiodic words Return words & quasiperiodicity Quasiperiodic episturmian words Characterization Non-quasiperiodic episturmian words 4

3 Background Terminology & Notation Sturmian & episturmian words Combinatorics on words: Plays a fundamental role in various fields of mathematics, computer science, physics, and biology. Most renowned among its branches is the theory of a certain family of infinite binary sequences called Sturmian words. Sturmian words: Many fascinating and beautiful properties. Points of view: combinatorial; algebraic; geometric. Applications: symbolic dynamics, the study of continued fraction expansion, and also in some domains of physics (quasicrystal modelling) and theoretical computer science (pattern recognition, digital straightness). Admit several equivalent definitions and have many characterizations natural generalizations on finite alphabets, such as episturmian words.

4 Past & present work Background Terminology & Notation Sturmian & episturmian words Previous work: Inequalities characterizing Sturmian and episturmian words Justin-Pirillo 2002; Pirillo 2003, 2005; Glen-Justin-Pirillo 2006 Present work: Refined characterization of episturmian words via lexicographic orderings. Characterization of strict episturmian words that are infinite Lyndon words. Connection: A Sturmian word is an infinite Lyndon word it is non-quasiperiodic (Levé-Richomme 2007). Now extend to episturmian words: Characterization of quasiperiodic episturmian words in terms of their directive words. If an episturmian word is directed by a regular wavy word, then it is non-quasiperiodic = wider class of non-quasiperiodic episturmian words, besides those that are infinite Lyndon words.

5 Words Background Terminology & Notation Sturmian & episturmian words Let A denote a finite alphabet, i.e., a set of symbols called letters. Finite word over A: w = x 1 x 2 x m, with each x i A Length: w = m w a: number of occurrences of the letter a in u Reversal: ew = x mx m 1 x 1 w is a palindrome if w = ew A : set of all finite words over A ε: the empty word ( ε = 0) A + = A \ {ε}: set of all non-empty finite words over A Factor of w: a finite string of consecutive letters in w Prefix of w: factor occurring at the beginning of w Suffix of w: factor occurring at the end of w F(w): set of all factors of w Terminology extends to infinite words w = x 1 x 2 x 3 A ω : set of all infinite words over A uv ω denotes the ultimately periodic infinite word uvvvvv

6 Sturmian words Definition Background Terminology & Notation Sturmian & episturmian words An infinite word s over {a, b} is Sturmian if there exist real numbers α, ρ [0, 1] such that s is equal to one of the following two infinite words: s α,ρ, s α,ρ : N {a, b} defined by s α,ρ(n) = ( a if (n + 1)α + ρ nα + ρ = 0, b otherwise; ( s α,ρ (n) = a if (n + 1)α + ρ nα + ρ = 0, b otherwise. (n 0) Remark A Sturmian word is: aperiodic if α is irrational; periodic if α is rational; standard (or characteristic) if ρ = α.

7 Sturmian words as cutting sequences Background Terminology & Notation Sturmian & episturmian words Consider a line y = βx + ρ with slope β > 0, ρ R. View 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 line intersects a grid-point: label intersection with ab (or ba) labels x 1, x 2, x 3,... K β,ρ = x 1 x 2 x 3 K β,ρ is a Sturmian word. That is: K β,ρ = s β/(1+β),ρ/(1+β).

8 Cutting sequences (cont.) Background Terminology & Notation Sturmian & episturmian words Example: y = x Fibonacci word Fibonacci word f = K ( 5 1)/2,0 = abaababaabaababaababaab A special example of a standard Sturmian word of slope α =

9 Sturmian words & balance Background Terminology & Notation Sturmian & episturmian words Definition A finite or infinite word w over {a, b} is balanced if: u, v F(w), u = v u b v b 1. Morse & Hedlund (1940) All balanced infinite words over a 2-letter alphabet are called Sturmian trajectories. They belong to two classes: 1 aperiodic Sturmian; 2 suffixes of infinite words of the form ϕ(xyx ω ), where ϕ is a pure standard (Sturmian) morphism, x, y {a, b}, x y. Example: aaabaaaaaaaaa Remark Class 2 consists of the periodic Sturmian words and the skew words: ultimately periodic non-recurrent infinite words, all of whose factors are balanced.

10 Episturmian words Background Terminology & Notation Sturmian & episturmian words Introduced by X. Droubay, J. Justin, and G. Pirillo (2001). Interesting natural generalization Sturmian words. Share many properties with Sturmian words. Definition (Justin-Pirillo 2002) An infinite word t over A is episturmian if: F(t) (its set of factors) is closed under reversal, and t has at most one right special factor of each length. t is standard (or epistandard) if all of its left special factors are prefixes of it. Remarks A = 2 Sturmian words (both aperiodic and periodic). Episturmian words are uniformly recurrent.

11 Lexicographic Order Lexicographic order & extremal words Previous work More on episturmian words A refinement Episturmian Lyndon words Suppose A is totally ordered by the relation <. Then we can totally order A by the lexicographic order <. That is: Definition Given two words u, v A +, we have u < v either u is a proper prefix of v or u = xau and v = xbv, for some x, u, v A and letters a, b with a < b. This is the usual alphabetic ordering in a dictionary. We say that u is lexicographically less than v. This notion naturally extends to infinite words.

12 Extremal words Lexicographic order & extremal words Previous work More on episturmian words A refinement Episturmian Lyndon words Definition Let w = w 1 w 2 w 3 be a (right-)infinite word with each w i A. Define min(w) to be the infinite word such that any prefix of min(w) is the lexicographically smallest amongst the factors of w of the same length. Similarly define max(w). Remarks Let min(w k) denote the lexicographically smallest factor of w of length k. Then: min(w) = lim k min(w k). min(w) is an infinite Lyndon word for any aperiodic infinite word w that is uniformly recurrent.

13 Lexicographic order & extremal words Previous work More on episturmian words A refinement Episturmian Lyndon words Inequalities characterizing Sturmian and episturmian words Theorem (Pirillo 2003) An infinite word s on {a, b} (a < b) is standard Sturmian (aperiodic or periodic) as min(s) max(s) bs. That is, standard Sturmian words s on {a, b} are characterized by the inequality: as T k (s) bs, for all k 0, where T is the shift map: T((s n) n 0 ) = (s n+1 ) n 0. In particular, an infinite word s on {a, b} (a < b) is an aperiodic standard Sturmian word (min(s), max(s)) = (as, bs). These Sturmian inequalities date back to Veerman (1986, 1987): symbolic dynamical perspective. Rediscovered numerous times under different guises (survey in preparation with J.-P. Allouche).

14 Lexicographic order & extremal words Previous work More on episturmian words A refinement Episturmian Lyndon words Theorem (Pirillo 2005) Let s be an infinite word over a finite alphabet A. Then the following properties are equivalent: i) s is standard episturmian (or epistandard), ii) for any lexicographic order, we have as min(s) where a = min(a). Moreover, s is a strict epistandard word ii) holds with strict equality for any order. [Justin-Pirillo 2002] Theorem (Glen-Justin-Pirillo 2006) A recurrent infinite word t over A is episturmian there exists an infinite word s such that, for any lexicographic order, we have as min(t) where a = min(a). Question What form is taken by the infinite words s?

15 Epistandard words Lexicographic order & extremal words Previous work More on episturmian words A refinement Episturmian Lyndon words Theorem (Droubay-Justin-Pirillo 2001) An infinite word s A ω is epistandard there exists an infinite word = x 1 x 2 x 3... (x i A), called the directive word of s, such that s = lim n u n, with u 1 = ε, u i+1 = (u i x i ) (+) for all i 1, where w (+) denotes the shortest palindrome having w as a prefix. Remarks uniquely determines the epistandard word s. An infinite word t isepisturmian if F(t) = F(s) for some epistandard word s. Example = (ab) ω directs the infinite Fibonacci word: f = abaababaabaababaababaabaaba Each u n is followed by a letter x n in purple.

16 Episturmian morphisms Lexicographic order & extremal words Previous work More on episturmian words A refinement Episturmian Lyndon words For each a A, define the morphisms ψ a, ψ a on A by ψ a : j a a x ax, ψa : j a a x xa for all x A \ {a}. Monoid of episturmian morphisms: generated by all the ψ a, ψ a, and permutations on A. Submonoid of pure episturmian morpisms: generated by the ψ a, ψ a only.

17 Spins Spinned words Lexicographic order & extremal words Previous work More on episturmian words A refinement Episturmian Lyndon words Let Ā = { x x A}. x is x with spin 1 x is itself with spin 0. Spinned word : a finite or infinite word w over A Ā. Write w = x 1 x 2 x 3, where x i = j xi if x i has spin 0, x i if x i has spin 1. Spinned morphisms For x A, let µ x = ψ x and µ x = ψ x. Extend to a pure episturmian morphism: for a finite spinned word w = x 1 x 2 x n define µ w := µ x1 µ x2 µ xn with µ ε = Id. Any episturmian word can be infinitely decomposed over the pure episturmian morphisms...

18 Characterization by morphisms Lexicographic order & extremal words Previous work More on episturmian words A refinement Episturmian Lyndon words Theorem (Justin-Pirillo 2002) An infinite word t A ω is episturmian there exists a spinned infinite word = x 1 x 2 x 3 (x i A) and an infinite sequence (t (i) ) i 0 of recurrent infinite words such that t (0) = t and t (i 1) = µ xi (t (i) ) for all i > 0. is called a spinned directive word for t or t is directed by. Each t (i) is an episturmian word directed by T i ( ) = x i+1 x i+2. For each i, F(t (i) ) = F(s (i) ) where s (i) is the (unique) epistandard word directed by T i ( ) = x i+1 x i+2. From now on: directive word of an epistandard word s spinned version of directing a factor-equivalent episturmian word t

19 Directive words Lexicographic order & extremal words Previous work More on episturmian words A refinement Episturmian Lyndon words In general, a spinned word directing an episturmian word is not unique. Example The Tribonacci word or Rauzy word (1982) is the epistandard word: r = abacabaabacababacabaabacabacaba with directive word (abc) ω. It is also directed by (abc) n ā b c(a b c) ω for each n 0, as well as infinitely many other spinned words. But all of the spinned infinite words directing an non-periodic episturmian word t are spinned versions of the same A ω which directs the unique epistandard word s with F(s) = F(t). Definition A spinned infinite word w is said to be wavy if it has infinitely many spins 0 and infinitely many spins 1. Proposition Any episturmian word has a wavy directive word.

20 Directive words (cont.) Lexicographic order & extremal words Previous work More on episturmian words A refinement Episturmian Lyndon words Notation Let O(t) denote the shift orbit closure (or subshift) of an episturmian word t. Then O(t) = {x A ω F(x) = F(t)}, i.e., the set of all (episturmian) infinite words that are factor-equivalent to t. Ult( ) : set of letters occurring infinitely often in. Proposition Suppose s is an epistandard word directed by = x 1 x 2 x 3 and let a be a letter. Then as is an episturmian word a Ult( ). Moreover, as is the (unique) episturmian word in O(s) directed by a which is with all spins 1 except when x i = a. Example With = (abc) ω, ar is directed by a = (a b c) ω where r is the Tribonacci word.

21 A refinement Lexicographic order & extremal words Previous work More on episturmian words A refinement Episturmian Lyndon words Theorem For any recurrent infinite word t A ω, the following properties are equivalent. i) t is an episturmian word directed by. ii) There exists an infinite word s such that, for any lexicographic order, we have as min(t) where a = min(a). Moreover, s is the (unique) epistandard word in O(t) directed by, with the property that as = min(t) a = min(a) belongs to Ult( ). Corollary Suppose t A ω is an episturmian word directed by and s is the unique epistandard word in O(t) with directive word. Then, for any lexicographic order, as T k (t) for all k 0, where a = min(a). Moreover, s has the property that as = inf k T k (t) a = min(a) belongs to Ult( ).

22 Episturmian Lyndon words Lexicographic order & extremal words Previous work More on episturmian words A refinement Episturmian Lyndon words We characterize strict episturmian words that are infinite Lyndon words. Definition (Strict episturmian words) An epistandard word s, or any factor-equivalent (episturmian) word, is strict if Ult( ) = Alph( ). That is, every letter in Alph(s) appears infinitely often in. Definition (Lyndon words) A finite or infinite word w over A is a Lyndon word if and only if it is (strictly) lexicographically smaller than all of its proper suffixes for the given order on A. That is, w is a Lyndon word iff, for the given order on A, w < T i (w) for all i > 0 (where T acts circularly on finite words). Notes Now assume that A 2 since on a 1-letter alphabet there are no infinite Lyndon words. Infinite Lyndon words are not (purely) periodic. Periodic episturmian words Ult( ) = 1... so assume Ult( ) > 1.

23 Results Lexicographic order & extremal words Previous work More on episturmian words A refinement Episturmian Lyndon words Proposition Suppose t = as where a = min(a) for the given order on A = Alph(t) and s is an aperiodic epistandard word with a Ult( ). Then t is an episturmian Lyndon word. Examples 1 s = ψ d (r) is the non-strict epistandard word directed by = d(abc) ω. So, for any order with a = min{a, b, c, d}, as = adadbdadcdadbd is an episturmian Lyndon word directed by d(a b c) ω. 2 = d(abc) ω directs the non-strict episturmian word : ψ d (r) = adbdadcdadbdadadbdadcdadbdadbdadcdadbd which is not a Lyndon word for any order on {a, b, c, d}. Note: ψ d (r) O(s) and as < ψ d (r) for any order with a = min{a, b, c, d}.

24 Results (cont.) Lexicographic order & extremal words Previous work More on episturmian words A refinement Episturmian Lyndon words Theorem An A-strict episturmian word t is an infinite Lyndon word t = as where a = min(a) for the given order on A and s is an (aperiodic) A-strict epistandard word. Remarks Let s be any A-strict epistandard word. Theorem shows that there exist exactly A 2 infinite Lyndon words that are factor-equivalent to s. That is, for any order with min(a) = a, O(s) contains a unique infinite Lyndon word beginning with a, namely as. as is the episturmian word directed by a which is with all spins 1, except when x i = a.

25 Corollaries Lexicographic order & extremal words Previous work More on episturmian words A refinement Episturmian Lyndon words Corollary An A-strict episturmian word t is an infinite Lyndon word t is directed by a where a = min(a) for the given order on A. Example For a < b < c and a < c < b, ar is an episturmian Lyndon word, directed by (a b c) ω. Corollary An A-strict episturmian word t is an infinite Lyndon word it can be infinitely decomposed over {ψ a, ψ x x A \ {a}} where a = min(a) for the given order on A. Sturmian case (Levé-Richomme 2007) A Sturmian word over {a, b} is an infinite Lyndon word it can be infinitely decomposed over {ψ a, ψ b } for a < b or { ψ a, ψ b } for b < a. A Sturmian word is an infinite Lyndon word it is non-quasiperiodic.

26 Quasiperiodic words Quasiperiodic words Return words & quasiperiodicity Quasiperiodic episturmian words Characterization Non-quasiperiodic episturmian words Definition A finite or infinite word w is quasiperiodic if there exists a proper factor u of w such that the occurrences of u in w entirely cover w. That is, every position of w falls within some occurrence of u in w. We say that: u covers w; u is a quasiperiod of w; w is u-quasiperiodic. Any quasiperiod of a quasiperiodic word must be a prefix of it. Examples 1 ababa is aba-quasiperiodic. 2 The infinite Fibonacci word f = abaababaabaaba, directed by (ab) ω, has infinitely many quasiperiods, which were characterized by Levé and Richomme in 2004 (generalized to epistandard words here). The smallest quasiperiod of f is aba.

27 Remarks on quasiperiodicity Quasiperiodic words Return words & quasiperiodicity Quasiperiodic episturmian words Characterization Non-quasiperiodic episturmian words Interest: study of DNA sequences; musicology. Apostolico-Crochemore (2002): survey of quasiperiodicity in strings. Marcus (2004) posed several questions that were answered by Levé and Richomme (2004). Levé and Richomme (2007) characterized the non-quasperiodic Sturmian words. We do the same for episturmian words via a different method: return words versus (epi)sturmian morphisms. Allows us to easily show that any epistandard word is quasiperiodic and describe all of the quasiperiods.

28 Return words & quasiperiodicity Quasiperiodic words Return words & quasiperiodicity Quasiperiodic episturmian words Characterization Non-quasiperiodic episturmian words Definition (Durand 1998, Holton-Zamboni 1999) Let v be a recurrent factor of w A ω. Then a return word to v in w is a factor of w beginning at an occurrence of v and ending exactly before the next occurrence of v in w. Thus, a return word to v in w is a non-empty factor r of w such that rv contains exactly two occurrences of v as a prefix and as a suffix. As episturmian words are uniformly recurrent, any factor has only a finite number of return words. In particular, any factor of an A-strict episturmian word has exactly A return words (Justin-Vuillon 2000).

29 Equivalent definition of quasiperiodicity Quasiperiodic words Return words & quasiperiodicity Quasiperiodic episturmian words Characterization Non-quasiperiodic episturmian words Theorem A finite word u is a quasiperiod of an infinite word w u is a recurrent prefix of w such that all of the return words to u in w have lengths at most u. Noteworthy Fact A quasiperiodic infinite word is not necessarily recurrent (Marcus 2004), although it must have a prefix that is recurrent in it.

30 Return words in episturmian words Quasiperiodic words Return words & quasiperiodicity Quasiperiodic episturmian words Characterization Non-quasiperiodic episturmian words Justin and Vuillon (2000) completely determined the return words to any factor of an episturmian word. It suffices to consider only the palindromic prefixes (u i ) i 1 of the epistandard word s directed by = x 1 x 2 x 3. Let m be the minimal integer such that Alph(x 1 x m) = Alph( ) = Alph(s). Then, for n m, any return word to u n+1 has length at most u n+1. Hence: Proposition Any epistandard word s is quasiperiodic. In particular, if s is directed by = x 1 x 2 x 3 and m is the minimal integer such that Alph(x 1 x 2 x m) = Alph(s), then u n+1 is a quasiperiod of s for all n m. Moreover, we can describe all of the quasiperiods of s by determining all of its prefixes that have no return words longer than themselves. We give only an example...

31 Example Quasiperiodic words Return words & quasiperiodicity Quasiperiodic episturmian words Characterization Non-quasiperiodic episturmian words Epistandard word directed by a(abc) ω : µ a(r) = aabaacaabaaabaacaabaabaacaabaaabaacaabaaca Palindromic prefixes: u 1 = ε u 2 = a u 3 = aa u 4 = aabaa u 5 = aabaacaabaa u 6 = aabaacaabaaabaacaabaa u 7 = aabaacaabaaabaacaabaabaacaabaaabaacaabaa.

32 Example (cont.) Quasiperiodic words Return words & quasiperiodicity Quasiperiodic episturmian words Characterization Non-quasiperiodic episturmian words Quasiperiods: u 5 a 1 = aabaacaaba u 5 = aabaacaabaa u 6 (aa) 1 = aabaacaabaaabaacaab u 6 a 1 = aabaacaabaaabaacaaba u 6 = aabaacaabaaabaacaabaa u 7 (aabaa) 1 = aabaacaabaaabaacaabaabaacaabaaabaac u 7 (abaa) 1 = aabaacaabaaabaacaabaabaacaabaaabaaca u 7 (baa) 1 = aabaacaabaaabaacaabaabaacaabaaabaacaa u 7 (aa) 1 = aabaacaabaaabaacaabaabaacaabaaabaacaab u 7 a 1 = aabaacaabaaabaacaabaabaacaabaaabaacaaba u 7 = aabaacaabaaabaacaabaabaacaabaaabaacaabaa.

33 Quasiperiodic words Return words & quasiperiodicity Quasiperiodic episturmian words Characterization Non-quasiperiodic episturmian words Words directing quasiperiodic episturmian words Lemma If an episturmian word t is directed by a spinned infinite word = x 1 x 2 x 3 with all spins ultimately 0, then t is quasiperiodic. Example µ w (s) is quasiperiodic for any spinned word w and epistandard word s. Lemma If an episturmian word t is directed by = v y for some spinned infinite word y and word v such that Alph(v) = Alph( ), then t is quasiperiodic. Example Recall the Tribonacci word r, directed by (abc) ω. = aadbc d(a b c) ω directs the quasiperiodic episturmian word µ aadbc d(ar). Smallest quasiperiod: u 6 u 1 = aadaabaadaacaadaabaada (same as the 2 epistandard word directed by ).

34 Quasiperiodic words Return words & quasiperiodicity Quasiperiodic episturmian words Characterization Non-quasiperiodic episturmian words Characterization of quasiperiodic episturmian words Theorem An episturmian word is quasiperiodic it has a directive word of the form wv y, for some spinned infinite word y and words w, v with Alph(v) = Alph(vy). Examples Recall the Tribonacci word r, directed by = (abc) ω. 1 ar is an infinite Lyndon word directed by a = (a b c) ω non-quasiperiodic. 2 µ bc (ar), µ cb (ar), µā d(ar) are quasiperiodic episturmian words, directed by Notes on Example 2 bc(a b c) ω = bca b( ca b) ω, cb(a b c) ω = cba b( ca b) ω, ā d(a b c) ω. First two spinned words take the form v y where y is regular wavy and Alph(v) = Alph(vy). ā d(a b c) ω directs the same episturmian word as adabcā( b ca) ω.

35 Regular spinned words Quasiperiodic words Return words & quasiperiodicity Quasiperiodic episturmian words Characterization Non-quasiperiodic episturmian words Recall The (unique) aperiodic episturmian word xs directed by x (with x Ult( ) and Ult( ) > 1) is an infinite Lyndon word for any order with x = min(a). Any Lyndon word is not quasiperiodic = x directs a non-quasiperiodic episturmian word. x is regular in the following sense. Definition A spinned version w of a finite or infinite word w is regular if, for each letter x Alph(w), all occurrences of x in w have the same spin (0 or 1). Examples Regular: a baa c b and (a bc) ω. Not regular: a baā cb and (a bā) ω., x and their opposites are regular.

36 Regular wavy words Quasiperiodic words Return words & quasiperiodicity Quasiperiodic episturmian words Characterization Non-quasiperiodic episturmian words A regular wavy word necessarily has Ult( ) > 1. Lemma If is a regular wavy word, then is the unique directive word for exactly one (aperiodic) episturmian word. Theorem If an episturmian word t is directed by a regular wavy word, then t is non-quasiperiodic. Example With = (abcd) ω, the regular wavy words a, b, c, d, (ā bcd) ω, (āb cd) ω, (ābc d) ω and their opposites direct non-quasiperiodic episturmian words that are factor-equivalent to the 4-bonacci word. Hence, unlike the Sturmian case, there is a much wider class of episturmian words that are non-quasiperiodic, besides those that are infinite Lyndon words.

37 Further work: Self-episturmian numbers. Characterization of all episturmian Lyndon words. Description of all the quasiperiods of a quasiperiodic episturmian word. Related work: F. Levé and G. Richomme have independently established another characterization of the quasiperiodic episturmian words using morphic decompositions. They have also characterized the epistandard morphisms that are strongly quasiperiodic.