Quasiperiodicities in Fibonacci strings

We consider the problem of finding quasiperiodicities in a Fibonacci string. A factor u of a string y is a cover of y if every letter of y falls within some occurrence of u in y. A string v is a seed of y, if it is a cover of a superstring of y. A left seed of a string y is a prefix of y that it is a cover of a superstring of y. Similarly a right seed of a string y is a suffix of y that it is a cover of a superstring of y. In this paper, we present some interesting results regarding quasiperiodicities in Fibonacci strings, we identify all covers, left/right seeds and seeds of a Fibonacci string and all covers of a circular Fibonacci string.Comment: In Local Proceedings of "The 38th International Conference on Current Trends in Theory and Practice of Computer Science" (SOFSEM 2012

A Note On ℓ\ell-Rauzy Graphs for the Infinite Fibonacci Word

The $\ell$-Rauzy graph of order $k$ for any infinite word is a directed graph in which an arc $(v_1,v_2)$ is formed if the concatenation of the word $v_1$ and the suffix of $v_2$ of length $k-\ell$ is a subword of the infinite word. In this paper, we consider one of the important aperiodic recurrent words, the infinite Fibonacci word for discussion. We prove a few basic properties of the $\ell$-Rauzy graph of the infinite Fibonacci word. We also prove that the $\ell$-Rauzy graphs for the infinite Fibonacci word are strongly connected.Comment: 10 pages, 4 figure

Privileged Words and Sturmian Words

This dissertation has two almost unrelated themes: privileged words and Sturmian words. Privileged words are a new class of words introduced recently. A word is privileged if it is a complete first return to a shorter privileged word, the shortest privileged words being letters and the empty word. Here we give and prove almost all results on privileged words known to date. On the other hand, the study of Sturmian words is a well-established topic in combinatorics on words. In this dissertation, we focus on questions concerning repetitions in Sturmian words, reproving old results and giving new ones, and on establishing completely new research directions. The study of privileged words presented in this dissertation aims to derive their basic properties and to answer basic questions regarding them. We explore a connection between privileged words and palindromes and seek out answers to questions on context-freeness, computability, and enumeration. It turns out that the language of privileged words is not context-free, but privileged words are recognizable by a linear-time algorithm. A lower bound on the number of binary privileged words of given length is proven. The main interest, however, lies in the privileged complexity functions of the Thue-Morse word and Sturmian words. We derive recurrences for computing the privileged complexity function of the Thue-Morse word, and we prove that Sturmian words are characterized by their privileged complexity function. As a slightly separate topic, we give an overview of a certain method of automated theorem-proving and show how it can be applied to study privileged factors of automatic words. The second part of this dissertation is devoted to Sturmian words. We extensively exploit the interpretation of Sturmian words as irrational rotation words. The essential tools are continued fractions and elementary, but powerful, results of Diophantine approximation theory. With these tools at our disposal, we reprove old results on powers occurring in Sturmian words with emphasis on the fractional index of a Sturmian word. Further, we consider abelian powers and abelian repetitions and characterize the maximum exponents of abelian powers with given period occurring in a Sturmian word in terms of the continued fraction expansion of its slope. We define the notion of abelian critical exponent for Sturmian words and explore its connection to the Lagrange spectrum of irrational numbers. The results obtained are often specialized for the Fibonacci word; for instance, we show that the minimum abelian period of a factor of the Fibonacci word is a Fibonacci number. In addition, we propose a completely new research topic: the square root map. We prove that the square root map preserves the language of any Sturmian word. Moreover, we construct a family of non-Sturmian optimal squareful words whose language the square root map also preserves.This construction yields examples of aperiodic infinite words whose square roots are periodic.Siirretty Doriast

디오판틴 근사, 연분수, 동역학적 스펙트럼

학위논문(박사) -- 서울대학교대학원 : 자연과학대학 수리과학부, 2023. 2. 임선희.Diophantine approximation is a rational approximation to an irrational number, which has been investigated using continued fractions. In the thesis, we deal with three topics related to Diophantine approximation and continued fractions. The first topic is the Markoff and Lagrange spectrum associated with the Hecke group. The classical Markoff and Lagrange spectrum is associated with the modular group PSL(2,$\mathbb Z$)=H_3, which has been studied using the regular continued fraction. We consider the Markoff and Lagrange spectrum associated with H_4 and H_6. We use the Romik dynamical system to show that some results on the classical Markoff and Lagrange spectra appear in the Markoff and Lagrange spectra associated with the Hecke group. The second topic is the exponents of repetition of Sturmian words. The exponent of repetition of a Sturmian word gives the irrationality exponent of the Sturmian number associated with the Sturmian word. For an irrational number $\theta$, we determine the minimum of the exponents of repetition of Sturmian words of slope $\theta$. We also investigate the spectrum of the exponents of repetition of Sturmian words of the golden ratio. The last topic is quasi-Sturmian colorings on regular trees. We characterize quasi-Sturmian colorings of regular trees by its quotient graph and its recurrence function. We obtain an induction algorithm of quasi-Sturmian colorings which is analogous to the continued fraction algorithm of Sturmian words.디오판틴 근사는 무리수의 유리수 근사를 뜻하는데 연분수를 사용하여 연구되어 왔습니다. 이 논문에서는 디오판틴 근사와 연분수에 관련된 세 가지 주제를 다루고 있습니다. 첫 번째 주제는 헤케군에 관련된 마르코프와 라그랑지 스펙트럼입니다. 고전적인 마르코프와 라그랑지 스펙트럼은 모듈러군 PSL(2,$\mathbb Z$)=H_3와 관련이 있는데, 단순연분수를 사용하여 연구되어 왔습니다. 우리는 H_4 와 H_6에 관련된 마르코프와 라그랑지 스펙트럼을 다룹니다. 우리는 로믹 동역학을 이용하여 고전적인 마르코프와 라그랑지 스펙트럼에서 발견된 결과가 헤케군에 관련된 마르코프와 라그랑지 스펙트럼에서도 나타남을 보입니다. 두 번째 주제는 스터미안 단어의 반복지수입니다. 스터미안 단어의 반복지수는 그 스터미안 단어와 연관된 스터미안 수의 비합리성 지수를 줍니다. 주어진 무리수 $\theta$에 대해, 우리는 기울기가 $\theta$인 스터미안 단어의 반복지수 중 최소값을 밝힙니다. 또한 우리는 황금비를 기울기로 갖는 스터미안 단어의 반복지수들의 스펙트럼을 연구합니다. 마지막 주제는 정규나무 위에서의 준-스터미안 채색입니다. 우리는 정규나무의 준-스터미안 채색을 이것의 몫 그래프와 회귀함수로 구분짓습니다. 우리는 스터미안 단어의 연분수 알고리즘과 유사한 준-스터미안 채색의 귀납적 알고리즘을 밝힙니다.1 Introduction 1 2 Diophantine approximation 7 2.1 Continued fraction 8 2.1.1 Basic properties 8 2.1.2 Gauss map 10 2.2 The Markoff and Lagrange spectra 11 3 The Markoff and Lagrange spectra associated with the Hecke group 16 3.1 The Markoff and Lagrange spectra on H_4 17 3.1.1 The Markoff and Lagrange spectra of the index 2 sublattice 17 3.1.2 The Markoff spectrum and the Romik expansion 23 3.1.3 Closedness of the Markoff spectrum 33 3.1.4 Hausdorff dimension of the Lagrange spectrum 34 3.1.5 Gaps of the Markoff spectrum 37 3.1.6 Halls Ray 40 3.2 The Markoff and Lagrange spectra on H_6 48 3.2.1 The Markoff spectrum and the Romik expansion 48 3.2.2 Closedness of the Markoff spectrum 53 3.2.3 Hausdorff dimension of the Lagrange spectrum 54 3.2.4 Gaps of the Markoff spectrum 57 4 Combinatorics on words 68 4.1 Sturmian words 68 4.2 The exponent of repetition 71 5 The spectrum of the exponents of repetition 74 5.1 The exponents of repetition of Sturmian words 74 5.2 The spectrum of the exponents of repetition of Fibonacci words 87 6 Colorings of regular trees 98 6.1 Sturmian colorings of trees 98 6.2 Quasi-Sturmian colorings 101 6.2.1 Quotient graphs of quasi-Sturmian colorings 102 6.2.2 Evolution of factor graphs 105 6.2.3 Quasi-Sturmian colorings of bounded type 110 6.2.4 Recurrence functions of colorings of trees 112 Bibliography 118 Abstract (in Korean) 122 Acknowledgement (in Korean) 123박