有關克里斯託費爾字的理論已發展超過兩百年，在這期間，數學家們對此已
發表了許多與之相關的主題以及重要的結果。至今，克里斯託費爾字不論在數學
領域以及電腦科學領域皆有相當廣泛的應用。
我們主要考慮的主題是克里斯託費爾字的共軛的分解。在文獻[1][2]中提及
一個眾所周知有關克里斯託費爾字的分解的結果，亦即每個長度大於等於二的克
里斯託費爾字都會有一個標準分解，換言之，每個長度大於等於二的克里斯託費
爾字都可以唯一地表示為兩個克里斯託費爾字的乘積。在這篇論文中，我們進一
步考慮一個更廣義的問題，亦即克里斯託費爾字的共軛分解。由數值結果觀察分
析，我們發現每一個長度大於等於二的克里斯託費爾字的共軛皆可以表示為兩個
克里斯託費爾字的共軛的乘積。而且這種分解方式並非唯一(至少兩種以上)，此
外，我們發現某些特定的克里斯託費爾字的共軛其實是有規律可循。對此，我們
提出了一個克里斯託費爾字的共軛的分解之猜想公式。雖然這篇論文只提供了有
限數量的表格，但我們也有使用電腦程式驗證了這個猜想公式，所以這個猜想的
正確性以及可靠性是非常高的。不過，我們未來的研究會再進一步使用嚴謹的數
學工具以及文獻上已有的結果來嚴格證明我們的猜想公式是正確的。

It is well known in the literature [1][2] that every nontrivial Christoffel word has a standard
factorization, namely, every nontrivial Christoffel word can be uniquely expressed as
the product of two Christoffel words. In this thesis, we consider a more general case, the
factorizations of conjugates of Christoffel words. From numerical results, we find that a conjugate
of a nontrivial Christoffel word can be expressed as the products of two conjugates
of Christoffel words. Furthermore, such factorizations of conjugates of Christoffel words are
not unique, and we make a conjecture on such factorizations at the end of the thesis.

Abstract i
Contents ii
1 Introduction 1
2 Preliminaries 3
3 The Standard Factorization of Christoffel words 8
4 The Factorizations of Conjugates of Christoffel Words 10
5 Conclusion 14
Appendix A The Simplified Table of Cℓ(13, 8) 15
Appendix B The Simplified Table of Cℓ(16, 7) 17
Bibliography 18

[1] J. Berstel, A. Lauve, C. Reutenauer, and F. Saliola, Combinatorics on Words: Christoffel
Words and Repetitions in Words, Providence, USA: American Mathematical Society
Press, 2008.
[2] J.-P. Borel and F. Laubie, “Quelques mots sur la droite projective r´eell,” Journal de
Th´eorie des Nombres de Bordeaus, vol. 5, pp. 23–51, 1993.
[3] M. Aigner, Markov’s Theorem and 100 Years of the Uniqueness Conjecture, Berlin,
Germany: Springer Press, 2013.
[4] G. Melan¸ccon, “On a Class of Lyndon Words Extending Christoffel Words and Related
to a Multidimensional Continued Fraction Algorithm,” Journal of Integer Sequences,
vol. 16, article 13.9.7, 2013.
[5] A. D’Aniello, A. de Luca and A. De Luca, “On Christoffel and standard words and their
derivatives,” Journal of Theoretical Computer Science, vol. 658, part A, pp. 122–147,
January 2017.
[6] M. Lothaire, Combinatorics on Words, Boston, USA: Addison-Wesley Press, 1983.
[7] J. Berstel and A. de Luca, “Sturmian words, Lyndon words and trees,” Journal of
Theoretical Computer Science, vol. 178, pp. 171–203, May 1997.
[8] M. Lothaire, Algebraic Combinatorics on Words, Cambridge, UK: Cambridge University
Press, 2002.
[9] J. -P. Borel and C. Reutenauer, “On Christoffel classes,” Journal of RAIRO–Theoretical
Informatics and Applications, vol. 40, number 1, pp. 15–27, 2006.
[10] V. Berth´e, A. de Luca, and C. Reutenauer, “On an involution of Christoffel words and
Sturmian morphisms,” European Journal of Combinatorics, vol. 29, issue 2, pp. 535–553,
February 2008.
[11] C. Reutenauer, “Studies on finite Sturmian words,” Journal of Theoretical Computer
Science, vol. 591, pp. 106–133, October 2015.