令α是界在0和1之間的無理數，且其連分數表示法為[0;a1+1,a2,a3,…]，其中an ≧1 (n≧1)。定義q-1=1, q0=1, qn=anqn-1+qn-2 (n≧1)。對於每一個大於等於-1的整數k，我們考慮雙邊無窮特徵字字尾H的k階分解，形式如下：H=ukuk+1uk+2…，其中因子ui的長度為qi (i≧k)。我們證明在這樣的分解中，不是所有ui皆為奇異字，就是存在非負整數q使得H的前q個因子為奇異字，其餘的皆為α-字。更進一步地，這些α-字的標籤可被一非負整數的F-表現法所唯一決定，此非負整數與H在雙邊無窮特徵字上的開始位置有關。

接著，我們定義由一對種子字(u,v)所生成的k階Markov字型(k≧1)，形式如下：Mk(u,v)=z1z2z3…，其中z1=u, z2=v, zi=(zi-1)ai+k-1-1 zi-2 zi-1 (i≧3)。我們證明每一個雙邊無窮特徵字字尾H皆為Markov字型，且其所包含的種子字皆為鄰邊α-字；我們也找到H的所有身為鄰邊α-字的種子字。另一方面，我們證明每一個鄰邊α-字所生成的Markov字型皆為雙邊無窮特徵字字尾。

最後，我們研究鄰邊α-字所形成的集合V。我們分別以循環移位、字典順序、標籤描述V中的元素。對於每一個α-字w，我們找到所有α-字v使得(w,v)為一對鄰邊α-字，也找到所有α-字u使得(u,w)為一對鄰邊α-字。另外，我們建造有向圖Gα，此圖的點集為所有α-字所形成的集合，邊集為所有鄰邊α-字所形成的集合；我們以此圖作為例證，說明鄰邊α-字的相關結果，以及它們所生成的雙邊無窮特徵字字尾。

Let α be an irrational number between 0 and 1 with continued fraction expansion [0;a1+1,a2,a3,…], where an ≧1 (n≧1). Define a sequence of numbers q-1=1, q0=1, qn=anqn-1+qn-2 (n≧1). For each integer k≧-1, we consider the k-order factorization of each suffix H of a two-way infinite characteristic word ofαof the form: H=ukuk+1uk+2…, where the length of the factor ui is qi (i≧k). We show that in such a factorization, either all ui are singular words, or there exists a nonnegative integer q such that H begins with q singular words, and uk+quk+q+1uk+q+2… areα-words. Moreover, the labels of theseα-words are uniquely determined by the F-representation of a nonnegative integer obtained from the position of H in the two-way infinite characteristic word.

Next, define Markov word pattern of order k (k≧-1) generated by a pair of seed words (u,v) as follows: Mk(u,v)=z1z2z3…, where z1=u, z2=v, and zi=(zi-1)ai+k-1-1 zi-2 zi-1 (i≧3). We show that each suffix H of each two-way infinite characteristic word is a Markov word pattern, and the pairs of its seed words obtained are adjacentα-words; we also find all possible pairs of seed words of H which are pairs of adjacent α-words. On the other hand, we show that each Markov word pattern generated by any pair of adjacentα-words is a suffix of a two-way infinite characteristic word.

Finally, we study the set V of all pairs of adjacentα-words. We describe the elements of V in terms of cyclic shifts, lexicographic order, and labels. For eachα-word w, we find all possible words u such that (u,w) (resp., (w,u)) are pairs of adjacentα-words. Also, we construct a directed graph Gα consisting of the vertex set of allα-words and the edge set V, and use such a graph to illustrate the results about adjacentα-words, and the suffixes of the two-way infinite characteristic words that they generate.

Contents
摘要 I
Abstract II
誌謝辭 III
Contents IV
List of Figures VI
List of Tables VII
List of Symbols VIII
Chapter 1 Introduction 1
Chapter 2 Preliminaries 4
2.1 Words 4
2.2 α-words and singular words 5
2.3 Two-way infinite characteristic words 9
Chapter 3 Factorizations of Suffixes of Two-Way Infinite Characteristic Words 12
3.1 The F-representation of nonnegative integers 13
3.2 Factorizations of suffixes ofα-words 16
3.3 (-1)st-order factorization of suffixes of f and f' 24
3.4 kth-order factorizations (k≧-1) of suffixes of f and f' 33
3.5 The case α=[0;1+1,1,1,…] 37
Chapter 4 Markov Word Patterns in Two-Way Infinite Characteristic Words 40
4.1 Markov word patterns in f and f' 40
4.2 More seed words of suffixes of f and f' 45
Chapter 5 Adjacentα-Words 50
5.1 The V+-set 51
5.1.1 Cyclic shifts 51
5.1.2 Lexicographic order 53
5.1.3 Labels 59
5.2 The V--set 61
5.3 Suffixes of f and f' determined by a singleα-words 66
5.4 The set V 67
5.5 The directed graph Gα 69
References 74
Appendices 78
A.1 Using kth-order factorizations to prove Theorem 4.1.2 78
A.2 Using labels in Zn+1 (resp., Zn-1) to rewrite Theorem 5.1.10 (resp., 5.2.5) 79
A.3 Using known V+-sets to prove Theorem 5.2.1 81

List of Figures
Figure 2.1 The defining tree Tαof α-words with α=[0; a1+1, a2, a3,…]. 7
Figure 5.1 The subgraph of Gαwith the elements of W in cyclic shifts andα=[0;3+1,3,1,2,…]. 70
Figure 5.2 The subgraph of Gα with the elements of W in lexicographic order andα=[0;3+1,3,1,2,…]. 71
Figure 5.3 The directed graph Gα with the elements of W in cyclic shifts andα=[0;3+1,3,1,2,…]. 72
Figure 5.4 The directed graph Gα with the elements of W in lexicographic order andα=[0;3+1,3,1,2,…]. 73

List of Tables
Table 3.1 I(r) for each r in S1∪S2∪S3∪S4 with α=[0;3+1,2,1,2,4,…]. 14
Table 3.2 σ( r) for each r in S1∪S2∪S3∪S4 withα=[0;3+1,2,1,2,4,…]. 21
Table 3.3 (-1)st-order factorizations of fm and f’m for some m in [-122,28] withα=[0;2+1,2,3,1,1,1,…], where ai=1 for all i≧4. 31
Table 3.4 1st-order factorizations of fm and f’m for some m in [-122,28] withα=[0;2+1,2,3,1,1,1,…], where ai=1 for all i≧4. 36
Table 4.1 Seed words (in cyclic shifts) of nth-order MWPs in fm, which are pairs of adjacent α-words, where -1≦n≦3 and -9≦m≦100 with α=[0;2+1,2,4,3,2,…]. 47
Table 4.2 Seed words (in cyclic shifts) of nth-order MWPs in f’m, which are pairs of adjacent α-words, where -1≦n≦3 and -9≦m≦100 with α=[0;2+1,2,4,3,2,…]. 48
Table 5.1 The elements of V+( Ti(xn)) for n≧1 and 0≦i≦q_n-1. 52
Table 5.2 The integers i3,k, j3,k, and d3,k for each k in [0, q3-1] withα=[0;3+1,3,1,2,…]. 69

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