On the Frequency and Periodicity of Infinite Words On the Frequency and Periodicity of Infinite Words

This work contributes to two aspects of the understanding of infinite words: the frequency of letters in a morphic sequence and periodicity considerations on infinite words. First, we develop a necessary and sufficient criteria for the existence of the frequency of a letter in a morphic sequence, and give some applications of this result. We show that the frequencies of all letters exist in pure binary morphic sequences. We also show that the frequency of all factors exist in polynomially generated morphic sequences. Second, we investigate the set of positive integers, called a period set, which is the set of the least periods of the factors of an infinite word. We characterize the period set of two famous infinite words: the Fibonacci word and Thue-Morse word. Then we generalize the result for the Fibonacci word to all Sturmian words. This enables us to give new proofs, tightenings, and generalizations for some known properties of Sturmian words. Third, we study the topic of global and local periodicity of infinite words from a new perspective. We define the notion of an everywhere αrepetitive sequence, a sequence in which every position starts an α-repetition of bounded length. We study borderlines between ultimate periodicity and aperiodicity by restricting the number of different minimal α-repetitions in an α-repetitive sequence. In the theory of everywhere α-repetitive sequences, Sturmian words play a fundamental role. We will show that Sturmian words are optimally squareful and overlapful sequences.