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 u<v u is lexicographically less than v ∼ reversal operation on A∗ PAL(A) set of all palindromes over A PAL set of all palindromes over {a, b} wp www ···w, p times uω purely periodic infinite word uuuuu ··· (u ∈A+) x ∼ y x is equivalent to y, i.e., Ω(x)=Ω(y) w−1 ‘inverse’ of w P (x,n) number of factors of length n in x (complexity function) h(x,n) number of palindromic factors of length n in x (palindrome 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