We give necessary and sufficient conditions for a language to be the language of finite words that occur infinitely many times in an infinite word.

We give necessary and sufficient conditions for a language to be the language of finite words that occur infinitely many times in an infinite word.

This paper consists of three parts. In the first part we prove a general theorem on the image of a language $K$ under a substitution, in the second we apply this to the special case when $K$ is the language of balanced words and in the third part we deal with recurrent Z-words of minimal block growth.

We show that any positive integer is the least period of a factor of the Thue-Morse word. We also characterize the set of least periods of factors of a sturmian word. In particular, the corresponding set for the Fibonacci word is the set of Fibonacci numbers. As a by-product of our results, we give several new proofs and tightenings of well-known properties of sturmian words.

We show that any positive integer is the least period of a factor of the Thue-Morse word. We also characterize the set of least periods of factors of a Sturmian word. In particular, the corresponding set for the Fibonacci word is the set of Fibonacci numbers. As a by-product of our results, we give several new proofs and tightenings of well-known properties of Sturmian words.

We prove that for each positive integer $n,$ the finite commutative language $E_n=c(a_1{a}_2\cdots a_n)$ possesses a test set of size at most $5n.$ Moreover, it is shown that each test set for $E_n$ has at least $n-1$ elements. The result is then generalized to commutative languages $L$ containing a word $w$ such that (i) $\text{alph}\left(w\right)=\text{alph}\left(L\right);$ and (ii) each symbol $a\in \text{alph}\left(L\right)$ occurs at least twice in $w$ if it occurs at least twice in some word of $L$: each such $L$ possesses a test set of size $11n$, where $n=\text{Card}\left(\text{alph}\left(L\right)\right)$. The considerations rest on the analysis of some basic types of word equations....

We prove that for each positive integer n, the finite commutative language En = c(a1a2...an) possesses a test set of size at most 5n. Moreover, it is shown that each test set for En has at least n-1 elements. The result is then generalized to commutative languages L containing a word w such that (i) alph(w) = alph}(L); and (ii) each symbol a ∈ alph}(L) occurs at least twice in w if it occurs at least twice in some word of L: each such L possesses a test set of size 11n, where n = Card(alph(L))....

Motivated by striking properties of the well known Fibonacci word we consider pictures which are defined by this word and its variants via so-called turtle graphics. Such a picture can be bounded or unbounded. We characterize when the picture defined by not only the Fibonacci recurrence, but also by a general recurrence formula, is bounded, the characterization being computable.

Motivated by striking properties of the well known Fibonacci word we consider pictures which are defined by this word and its variants via so-called turtle graphics. Such a picture can be bounded or unbounded. We characterize when the picture defined by not only the Fibonacci recurrence, but also by a general recurrence formula, is bounded, the characterization being computable.

We study the logarithmic frequency of letters and words in morphic sequences and show that this frequency must always exist, answering a question of Allouche and Shallit.

The $LS$ (Look and Say) derivative of a word is obtained by writing the number of consecutive equal letters when the word is spelled from left to right. For example, $LS\left(11233\right)=211223$ (two $1$, one $2$, two $3$). We start the study of the behaviour of binary words generated by morphisms under the $LS$ operator, focusing in particular on the Fibonacci word.

The LS (Look and Say) derivative of a word is obtained by writing the number of consecutive equal letters when the word is spelled from left to right. For example, LS( 1 1 2 3 3) = 2 1 1 2 2 3 (two 1, one 2, two 3). We start the study of the behaviour of binary words generated by morphisms under the LS operator, focusing in particular on the Fibonacci word.