On Sturmian and Episturmian Words, and Related Topics

Transcription

1 On Sturmian and Episturmian Words, and Related Topics by Amy Glen Supervisors: Dr. Alison Wolff and Dr. Robert Clarke A thesis submitted in fulfilment of the requirements for the degree of Doctor of Philosophy April 2006 SCHOOL OF MATHEMATICAL SCIENCES DISCIPLINE OF PURE MATHEMATICS

2 Contents List of Symbols vi Abstract x Signed Statement xi Acknowledgements xii 1 Introduction Outline of Thesis Future Directions Preliminaries Words Finite Words Infinite Words Factors Lexicographic Order Powers Conjugation Inverses i

3 2.2 Morphisms Sturmian Words Standard Sequences Properties of Characteristic Sturmian Words Episturmian Words Some Decompositions of the Fibonacci Word The Fibonacci Word Palindromes in the Fibonacci Word Preliminary Results Singular Words Circular Words Conjugates of Characteristic Sturmian Words Generated by Morphisms Preliminaries Sturmian Morphisms Conjugation of Infinite Words and Morphisms Characteristic Sturmian Words c α and Singular Words Characteristic Sturmian Words Generated by Morphisms Conjugates of c α with α =[0;2, r] Conjugates of c 1 α Some Remarks Occurrences of Palindromes in c α Preliminaries Terminology and Notation ii

4 5.1.2 Some Singular Decompositions of c α Palindromes, Return Words, and Overlaps Structure of Palindromes in c α Return Words and Overlapping Occurrences Decompositions of c α into Palindromes Useful Results Some Lemmas Main Result Occurrences of Factors of Length q n in c α Some Properties of the Tribonacci Sequence The Tribonacci Sequence ζ Conjugates of the Tribonacci Sequence Conjugacy and the Tribonacci Morphism Relation to Episturmian Morphisms Tribonacci Numbers (T i ) i Decompositions of Conjugates of ζ Powers in the Tribonacci Sequence Useful Results A Decomposition of ζ Factors of Length T n Factors of Length T n + T n Squares in ζ The Number of Distinct Squares in A n Cubes and Higher Powers in ζ iii

5 7 Powers in a Class of A-Strict Episturmian Words Episturmian Words Strict Episturmian Words A Class of Strict Standard Episturmian Words Two Special Integer Sequences Generalized Singular Words Useful Results Singular n-words of the r-th Kind Index Powers Squares Cubes and Higher Powers Examples Some Remarks The Number of Distinct Squares in s n A Division Property Previous Results The k-bonacciword Bi-Ideal Sequences Bi-Idealness and the Palindromic Words D n Transcendence of Episturmian Continued Fractions A Key Result Semigroups of Matrices Transcendence of Certain k-ary Continued Fractions iv

6 8.4 Examples Episturmian Continued Fractions Thue-Morse Continued Fractions Invertible Substitutions on a Finite Alphabet Preliminaries Motivations A Conjecture and Examples Terminology and Notation Indecomposable and Simple Substitutions A Characterization of Indecomposable Substitutions A Characterization of Incidence Matrices Towards a proof of Conjecture Nielsen Cancellation Theory Mixed Substitutions Some Lemmas Bibliography 179 v

7 List of Symbols In the table below, letters that are not further qualified have the following significance: i, j, k non-negative integers m, n, p positive integers r, t integers γ real number α positive irrational (0 <α<1) w finite word u, v finite or infinite words x, y infinite words ψ morphism M square matrix Formal Symbolism Meaning Z integers N natural numbers (i.e., non-negative integers) Z +, N + positive integers Z m ring of integers modulo m Z + m Z m \{0} = {1, 2,...,m 1} R real numbers γ greatest integer γ γ least integer γ γ if γ 0, γ absolute value of γ; γ = γ if γ<0 r t (mod m) r is congruent to t modulo m r t (mod m) r is incongruent to t modulo m end of proof, end of example gcd(a, b), (a, b) greatest common divisor of a and b summation product S cardinality of a finite set S A finite alphabet A free monoid generated by A; set of all finite words over A vi

8 Formal Symbolism Meaning ε identity of A (empty word) A + A ω A free semigroup A \{ε} set of all infinite words over A the set A A ω A k the alphabet {a 1,a 2,...,a k } (k 2) Σ k the alphabet {0, 1,...,k 1} w number of letters in w; lengthofw w a number of occurrences of the letter a in w w z number of occurrences of a finite word z in w F (w) first letter of w L(w) last letter of w w p v w is a prefix of v u s v u is a suffix of v Ω(u) set of all factors of u Ω n (u) set of all factors of u of length n (if u n) w u w is a factor of u w u w is not a factor of u Alph(u) alphabet of u (i.e., set of all letters occurring in u) Ult(x) set of all letters occurring infinitely many times in x u < v u is lexicographically less than v PAL(A) reversal operation on A set of all palindromes over A PAL set of all palindromes over {a, b} w p www w, p times u ω purely periodic infinite word uuuuu (u A + ) x y x is equivalent to y, i.e., Ω(x) =Ω(y) w 1 P (x,n) h(x,n) ψ inverse of w number of factors of length n in x (complexity function) number of palindromic factors of length n in x (palindrome complexity function) length of the morphism ψ on A; x A ψ(x) ψ w ψ n (w) ψ ω (x) s α,ρ c α w-length of the morphism ψ on A; x A ψ(x) w ψ 0 (w) =w, ψ 1 (w) =ψ(w), ψ n (w) =ψ(ψ n 1 (w)) infinite word, beginning with x A, generated by ψ Sturmian word of slope α and intercept ρ characteristic Sturmian word of slope α vii

9 Formal Symbolism Meaning [0; a 1,a 2,a 3,...] simple continued fraction expansion of α α n,r [0; a n+1 + r, a n+2,a n+3,...] [0; a 1,a 2,...,a m 1 ] purely periodic continued fraction expansion [0; a 1,...,a n 1, a n,...,a n+m 1 ] eventually periodic continued fraction expansion p i q i i-th convergent numerator i-th convergent denominator ϕ Fibonacci morphism on the alphabet {a, b} f infinite Fibonacci word f n F n C j (w) C(w) w (i) n-th finite Fibonacci word n-th Fibonacci number j-th conjugate of w (where 0 j w 1) set of all conjugates of w i-th appearance of w in some infinite word x E exchange morphism on A = {a, b} E xy morphism on A that exchanges the letters x and y E i,j exhange morphism E ai a j on A k i-th right conjugate of ψ ψ i Id A identity morphism on A Id k F(A) identity morphism on A k free group over A F k free group over A k θ Tribonacci morphism on the alphabet {a, b, c} ζ Tribonacci sequence T n n-th Tribonacci number morphism: a a and z az, z A, z a Ψ a Ψ a w (+) (t) s morphism: a a and z za, z A, z a shortest palindrome of which w is a prefix (palindromic right-closure of w) directive word of a standard episturmian word t standard episturmian word over A k with (s) =a d 1 1 ad 2 2 ad k k ad k+1 1 a d 2k k ad 2k+1 1, d i > 0 η k infinite k-bonacci word (k 2) ind(w) index of a factor w of some infinite word x ind(x) index (or critical exponent) of x P(m; l) set of all factors w of s of length m such that w l s p(m; l) P(m; l) Pref m (u) prefix of u of length m Suff m (w) suffix of w of length m viii

10 Formal Symbolism Meaning min{a 1,a 2,...,a n } minimum of a 1, a 2,..., a n Z max{a 1,a 2,...,a n } maximum of a 1, a 2,..., a n Z M T transpose of M det M determinant of M tr(m) trace of M ρ(m) spectral radius of M M spectral norm of M x[i] (i + 1)-st letter x i of x = x 0 x 1 x 2 x 3 x[i...i+ j] the factor x i x i+1 x i+j of x = x 0 x 1 x 2 x 3 IS(A k ) monoid of invertible substitutions on A k SL(k, N) set of all k k matrices over N with determinant equal to 1 Aut(F k ) group of automorphisms of the free group F k S m (n) sum of the digits in the base-m representation of n N t m,k infinite sequence given by (S m (n) modk) n 0 ix

11 Abstract In recent years, combinatorial properties of finite and infinite words have become increasingly important in fields of physics, biology, mathematics, and computer science. In particular, the fascinating family of Sturmian words has become an extremely active subject of research. These infinite binary sequences have numerous applications in various fields of mathematics, such as symbolic dynamics, the study of continued fraction expansion, and also in some domains of physics (quasicrystal modelling) and computer science (pattern recognition, digital straightness). There has also been a recent surge of interest in a natural generalization of Sturmian words to more than two letters - the so-called episturmian words, which include the well-known Arnoux-Rauzy sequences. This thesis represents a significant contribution to the study of Sturmian and episturmian words, and related objects such as generalized Thue-Morse words and substitutions on a finite alphabet. Specifically, we prove some new properties of certain palindromic factors of the infinite Fibonacci word ; establish generalized singular decompositions of suffixes of certain morphic Sturmian words; completely describe where palindromes occur in characteristic Sturmian words; explicitly determine all integer powers occurring in a certain class of k-strict episturmian words (including the k-bonacci word); and prove that certain episturmian and generalized Thue-Morse continued fractions are transcendental. Lastly, we begin working towards a proof of a characterization of invertible substitutions on a finite alphabet, which generalizes the fact that invertible substitutions on two letters are exactly the Sturmian morphisms. x

12 Signed Statement This work contains no material which has been accepted for the award of any other degree or diploma in any university or other tertiary institution and, to the best of my knowledge and belief, contains no material previously published or written by another person, except where due reference has been made in the text. I give consent to this copy of my thesis, when deposited in the University Library, being available for loan and photocopying. Signed:... Date:... xi

13 Acknowledgements First and foremost, I wish to thank my supervisors, Alison Wolff and Bob Clarke, for their encouragement and support; in particular, I am grateful for their reading of my work and generosity of their time. Thanks also to Alison for introducing me to Sturmian words. A most heartfelt thanks goes to my family for their unwavering support and love. I extend a warm thanks to my friends: Jono Tuke, Jason Ellul, David Butler, and Tony Scoleri. Without you, the past year or two would have certainly been a much darker period in my life. Thanks also to Dr. Liz Cousins for her help and Margaret Vaughton for being concerned about my general well-being. I am very grateful for the financial support provided by the George Fraser Scholarship. I am also thankful to the School of Mathematical Sciences and Bradford College for the numerous tutoring and lecturing opportunities. Lastly, I would like to thank past teachers and lecturers who sparked my interest in mathematics and inspired me to pursue it further. xii