Thue-Morse Word And Repetitions

Basis of predecessors'works, we study the properties and applicationsof Thue-Morse word and the overlap properties of factors.In the introductory chapter, we ?rst introduces the background of thestudy and then we review the known results and make some remarks.In Chapter 2, we begin with the equivalent de?nitions and basic prop-erties of Thue-Morse word. After that, we discuss its applications on thenumber theory and combinatorics on words. Namely, we solve magic squareproblem and implement it on the computer using Thue-Morse word.In the next chapter, we study the combinatorial properties of Thue-Morse word. Relation between Thue-Morse word and deterministic ?niteautomaton is explained at the same time. Based Axel Thue's work, weintroduce the overlap-free property of Thue-Morse word. Then we ?nd outthe closed-form expression for the complexity function of Thue-Morse wordusing the structure of right special factors. We also discuss its left specialfactors.The last chapter is devoted to dealing with occurrences of squares ingiven ?nite words. We provide bounds for the number of occurrences ofsquares primitive of and for the number of distinct squares in a ?xed ?niteword,by using Three Squares Lemma.