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 Contents

List of Symbols vi

Abstract x

Signed Statement xi

Acknowledgements xii

1 Introduction 1

1.1OutlineofThesis...... 2

1.2FutureDirections...... 4

2 Preliminaries 5

2.1Words...... 5

2.1.1 FiniteWords...... 5

2.1.2 Infinite Words ...... 6

2.1.3 Factors...... 6

2.1.4 LexicographicOrder...... 7

2.1.5 Powers...... 8

2.1.6 Conjugation...... 8

2.1.7 Inverses...... 9

i 2.2Morphisms...... 9

2.3SturmianWords...... 11

2.3.1 StandardSequences...... 13

2.3.2 PropertiesofCharacteristicSturmianWords...... 15

2.4EpisturmianWords...... 17

3 Some Decompositions of the Fibonacci Word 18

3.1TheFibonacciWord...... 18

3.1.1 PalindromesintheFibonacciWord...... 19

3.2PreliminaryResults...... 21

3.3SingularWords...... 24

3.4CircularWords...... 28

4 Conjugates of Characteristic Sturmian Words Generated by Morphisms 37

4.1Preliminaries...... 38

4.1.1 SturmianMorphisms...... 38

4.1.2 Conjugation of Infinite Words and Morphisms ...... 38

4.1.3 Characteristic Sturmian Words cα andSingularWords...... 39

4.2CharacteristicSturmianWordsGeneratedbyMorphisms...... 41

4.3 Conjugates of cα with α =[0;2, r]...... 44

4.3.1 Conjugates of c1−α ...... 49

4.4SomeRemarks...... 50

5 Occurrences of Palindromes in cα 51

5.1Preliminaries...... 52

5.1.1 TerminologyandNotation...... 52

ii 5.1.2 Some Singular Decompositions of cα ...... 52

5.2Palindromes,ReturnWords,andOverlaps...... 55

5.2.1 Structure of Palindromes in cα ...... 55

5.2.2 ReturnWordsandOverlappingOccurrences...... 56

5.3 Decompositions of cα intoPalindromes...... 58

5.3.1 UsefulResults...... 58

5.3.2 SomeLemmas...... 58

5.4MainResult...... 62

5.5 Occurrences of Factors of Length qn in cα ...... 67

6 Some Properties of the Tribonacci Sequence 70

6.1 The Tribonacci Sequence ζ ...... 70

6.2ConjugatesoftheTribonacciSequence...... 71

6.2.1 ConjugacyandtheTribonacciMorphism...... 71

6.2.2 RelationtoEpisturmianMorphisms...... 72

6.2.3 Tribonacci Numbers (Ti)i≥0 ...... 73

6.2.4 Decompositions of Conjugates of ζ ...... 74

6.3PowersintheTribonacciSequence...... 76

6.3.1 UsefulResults...... 76

6.3.2 A Decomposition of ζ ...... 78

6.3.3 Factors of Length Tn ...... 79

6.3.4 Factors of Length Tn + Tn−1 ...... 80

6.3.5 Squares in ζ ...... 86

6.3.6 The Number of Distinct Squares in An ...... 90

6.3.7 Cubes and Higher Powers in ζ ...... 95

iii 7PowersinaClassofA-Strict Episturmian Words 96

7.1EpisturmianWords...... 97

7.1.1 StrictEpisturmianWords...... 99

7.2AClassofStrictStandardEpisturmianWords...... 99

7.2.1 TwoSpecialIntegerSequences...... 102

7.3GeneralizedSingularWords...... 103

7.3.1 UsefulResults...... 103

7.3.2 Singular n-words of the r-thKind...... 107

7.4Index...... 111

7.5Powers...... 113

7.5.1 Squares...... 122

7.5.2 CubesandHigherPowers...... 127

7.5.3 Examples...... 128

7.5.4 SomeRemarks...... 130

7.5.5 The Number of Distinct Squares in sn ...... 130

7.6ADivisionProperty...... 137

7.6.1 PreviousResults...... 138

7.6.2 The k-bonacciWord...... 139

7.6.3 Bi-IdealSequences...... 141

7.6.4 Bi-Idealness and the Palindromic Words Dn ...... 142

8 Transcendence of Episturmian Continued Fractions 145

8.1AKeyResult...... 146

8.2SemigroupsofMatrices...... 147

8.3 Transcendence of Certain k-ary Continued Fractions...... 155

iv 8.4Examples...... 158

8.4.1 EpisturmianContinuedFractions...... 158

8.4.2 Thue-MorseContinuedFractions...... 159

9 Invertible Substitutions on a Finite Alphabet 163

9.1Preliminaries...... 163

9.2Motivations...... 164

9.3AConjectureandExamples...... 166

9.3.1 TerminologyandNotation...... 166

9.3.2 Indecomposable and Simple Substitutions ...... 167

9.3.3 A Characterization of Indecomposable Substitutions ...... 169

9.3.4 ACharacterizationofIncidenceMatrices...... 171

9.4TowardsaproofofConjecture9.1...... 173

9.4.1 NielsenCancellationTheory...... 174

9.4.2 Mixed Substitutions ...... 175

9.4.3 SomeLemmas...... 177

Bibliography 179

v 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

Zm ring of integers modulo m + Zm Zm\{0} = {1, 2,...,m− 1} R real numbers γ greatest integer ≤ γ γ least integer ≥ γ ⎧ ⎨ γ if γ ≥ 0, |γ| γ |γ| absolute value of ; = ⎩ −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 Formal Symbolism Meaning ε identity of A∗ (empty word) A+ free semigroup A∗ \{ε} Aω set of all infinite words over A A∞ the set A∗ ∪Aω

Ak the alphabet {a1,a2,...,ak} (k ≥ 2)

Σk 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

Ωn(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 upalindrome complexity function) ψ length of the morphism ψ on A; |ψ(x)| x∈A ψ w w-length of the morphism ψ on A; x∈A |ψ(x)|w ψn(w) ψ0(w)=w, ψ1(w)=ψ(w), ψn(w)=ψ(ψn−1(w)) ψω(x) infinite word, beginning with x ∈A, generated by ψ sα,ρ Sturmian word of slope α and intercept ρ cα characteristic Sturmian word of slope α

vii Formal Symbolism Meaning

[0; a1,a2,a3,...] simple continued fraction expansion of α

αn,r [0; an+1 + r, an+2,an+3,...]

[0; a1,a2,...,am−1] purely periodic continued fraction expansion

[0; a1,...,an−1, an,...,an+m−1] eventually periodic continued fraction expansion pi i-th convergent numerator qi i-th convergent denominator ϕ Fibonacci morphism on the alphabet {a, b} f infinite Fibonacci word fn n-th finite Fibonacci word

Fn n-th Fibonacci number

Cj(w) j-th conjugate of w (where 0 ≤ j ≤|w|−1) C(w) set of all conjugates of w w(i) i-th appearance of w in some infinite word x E exchange morphism on A = {a, b}

Exy morphism on A that exchanges the letters x and y E E A i,j exhange morphism aiaj on k

ψi i-th right conjugate of ψ

IdA identity morphism on A

Idk identity morphism on Ak F(A) free group over A

Fk free group over Ak θ Tribonacci morphism on the alphabet {a, b, c} ζ Tribonacci sequence

Tn n-th Tribonacci number

Ψa morphism: a → a and z → az, ∀z ∈A, z = a

Ψa morphism: a → a and z → za, ∀z ∈A, z = a w(+) shortest palindrome of which w is a prefix (palindromic right-closure of w) ∆(t) directive word of a standard episturmian word t s standard episturmian word over Ak with d1 d2 dk dk+1 d2k d2k+1 ∆(s)=a1 a2 ···ak a1 ···ak a1 ···, di > 0

ηk 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 wl ≺ s p(m; l) |P(m; l)|

Prefm(u) prefix of u of length m

Suffm(w) suffix of w of length m

viii Formal Symbolism Meaning min{a1,a2,...,an} minimum of a1, a2, ..., an ∈ Z max{a1,a2,...,an} maximum of a1, a2, ..., an ∈ 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 xi of x = x0x1x2x3 ··· x[i...i+ j] the factor xixi+1 ···xi+j of x = x0x1x2x3 ···

IS(Ak) monoid of invertible substitutions on Ak SL(k, N) set of all k × k matrices over N with determinant equal to 1

Aut(Fk) group of automorphisms of the free group Fk

Sm(n) sum of the digits in the base-m representation of n ∈ N tm,k infinite sequence given by (Sm(n)modk)n≥0

ix Abstract

In recent years, combinatorial properties of finite and infinite words have become in- creasingly 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 var- ious 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 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 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.

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