Properties Of The Christoffel Words

Symbolic dynamics was firstly produced in J. Hadamard’s work in1898, and it origi-nated in the topological theory of abstract dynamical systems research. Symbolic dynamicsgradually formed as a method to study general dynamical systems, now its techniques andideas have applied in the study of data linear algebra, storage and transmission, the motionsof the planets and many other areas. The distinct feature in symbolic dynamics is that timeis measured in discrete intervals, so at each time interval the system is in a particular stateand each state is associated with a symbol and the evolution of the system is described byan infinite sequence of symbols.The symbol sequence can also be seen as the formal language on finite alphabet. Thestudy of symbol sequence includes the following two aspects, one is the aspect of its applica-tion in physics, differential equations and chaos; the other is its combinatorial structure andproperties. Christoffel words, the main objects studied in the thesis, are finite words overbinary alphabet. They have a wide range of applications in word combinatorics, differentialgeometry, number theory, computer science and automata and some other areas.The thesis is organized as follows: in the introductory chapter we introduce thebackground of the Fraenkel conjecture, review and make remarks on the pioneer’s work andtheir achievements. A detailed introduction to the Christoffel words and Christoffel paths ispresented in chapter2. Especially, in order to solve the Fraenkel conjecture, we introducethe content of the superimposition of two Christoffel words, which is closely related withthe Fraenkel conjecture. In the next chapter we give a necessary and sufficient condition forthe superimposition of two Christoffel words of same length and the number of differentpossible superimpositions, and then extend the results to the case of the two words havingdifferent lengths. We prove the Frobenius problem, which is also known as money prob-lem, with the help of the geometric interpretation of the Christoffel words in the last chapter.