Some Properties of the Singular Words of the Fibonacci Word

The combinatorial properties of the Fibonacci innnite word are of great interest in some aspects of mathematics and physics, such as number theory, fractal geometry, formal language, computational complexity , quasicrystals etc. In this note, we introduce the singular words of the Fibonacci innnite word and discuss their properties. We establish two decompositions of the Fibonacci word in singular words and their consequences. By using these results, we discuss the local isomorphism of the Fibonacci word and the overlap properties of the factors. Moreover, we also give new proofs for the results on special words and the power of the factors. The combinatorial properties of the Fibonacci innnite word are of great interest in some aspects of mathematics and physics, such as number theory , fractal geometry, formal language, computational complexity, quasicrys-tals etc. See 1, 3, 8, 9, 11]. Moreover, the properties of the subwords of the Fibonacci innnite word have been studied extensively by many authors 2, 4, 5, 6, 9, 10]. In this note, we shall present some new properties of the subwords of the Fibonacci word: as we shall see, the most striking of these properties is that the adjacent singular words of the same order are positively separated. This note is organized as follows. After recalling some preliminary remarks on the Fibonacci word, we introduce the singular words and discuss their elementary properties. Then we establish two decompositions of the Fibonacci word in singular words (Theorems 1 and 2) and their consequences. By using these results, we discuss the local isomorphism of the Fibonacci word (Theorem 4) and the overlap properties of the factors (Theorem 6). Moreover, we also give new proofs for the results on special words (Theorem 5) and the power of the factors (Theorem 3). In this note, we use the following deenitions and terminology. Let A = fa; bg be an alphabet of two letters, let A be the free monoid on A, and let F be the free group generated by A. The elements of A are called words. The neutral element of A is called the empty word which we denote by ". Let w be a word: we denote by jwj the length of w, we denote by jwj a