On the Periodicity of Morphic Words
نویسندگان
چکیده
Given a morphism h prolongable on a and an integer p, we present an algorithm that calculates which letters occur infinitely often in congruent positions modulo p in the infinite word h(a). As a corollary, we show that it is decidable whether a morphic word is ultimately p-periodic. Moreover, using our algorithm we can find the smallest similarity relation such that the morphic word is ultimately relationally p-periodic. The problem of deciding whether an automatic sequence is ultimately weakly R-periodic is also shown to be decidable.
منابع مشابه
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 pur...
متن کاملPseudopalindrome closure operators in free monoids
We consider involutory antimorphisms θ of a free monoid A∗ and their fixed points, called θ-palindromes or pseudopalindromes. A θ-palindrome reduces to a usual palindrome when θ is the reversal operator. For any word w ∈ A∗ the right (resp. left) θ-palindrome closure of w is the shortest θ-palindrome having w as a prefix (resp. suffix). We prove some results relating θ-palindrome closure operat...
متن کاملExtremal words in morphic subshifts
Given an infinite word x over an alphabet A, a letter b occurring in x, and a total order σ on A, we call the smallest word with respect to σ starting with b in the shift orbit closure of x an extremal word of x. In this paper we consider the extremal words of morphic words. If x = g(fω(a)) for some morphisms f and g, we give two simple conditions on f and g that guarantees that all extremal wo...
متن کاملOn Almost Periodicity Criteria for Morphic Sequences in Some Particular Cases
In some particular cases we give criteria for morphic sequences to be almost periodic (=uniformly recurrent). Namely, we deal with fixed points of non-erasing morphisms and with automatic sequences. In both cases a polynomial-time algorithm solving the problem is found. A result more or less supporting the conjecture of decidability of the general problem is given.
متن کاملA little more about morphic Sturmian words
Among Sturmian words, some of them are morphic, i.e. fixed point of a non-identical morphism on words. Berstel and Séébold (1993) have shown that if a characteristic Sturmian word is morphic, then it can be extended by the left with one or two letters in such a way that it remains morphic and Sturmian. Yasutomi (1997) has proved that these were the sole possible additions and that, if we cut th...
متن کامل