A nilspace system is a generalization of a nilsystem, consisting of a compact nilspace $\text{X}$ equipped with a group of nilspace translations acting on $\text{X}$. Nilspace systems appear in different guises in several recent works and this motivates the study of these systems per se as well as their relation to more classical types of systems. In this paper we study morphisms of nilspace systems, i.e., nilspace morphisms with the additional property of being consistent with the actions of the given translations. A nilspace morphism does not necessarily have this property, but one of our main results shows that it factors through some other morphism which does have the property. As an application we obtain a strengthening of the inverse limit theorem for compact nilspaces, valid for nilspace systems. This is used in work of the first- and third-named authors to generalize the celebrated structure theorem of Host and Kra on characteristic factors.

We investigate the finite repetition threshold for k-letter alphabets, k ≥ 4, that is the smallest number r for which there exists an infinite r+-free word containing a finite number of r-powers. We show that there exists an infinite Dejean word on a 4-letter alphabet (i.e. a word without factors of exponent more than 7/5 ) containing only two 7/5 -powers. For a 5-letter alphabet, we show that there exists an infinite Dejean word containing only 60 5/4 -powers, and we conjecture that this number can be lowered to 45. Finally we show that the finite repetition threshold for k letters is equal to the repetition threshold for k letters, for every k ≥ 6.

Among the various ways to construct a characteristic Sturmian word, one of the most used consists in defining an infinite sequence of prefixes that are standard. Nevertheless in any characteristic word c, some standard words occur that are not prefixes of c. We characterize all standard words occurring in any characteristic word (and so in any Sturmian word) using firstly morphisms, then standard prefixes and finally palindromes.

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.

We present a description of asymptotic behaviour of languages of bi-infinite words obtained by iterating morphisms defined on free monoids.

A morphism f is k-power-free if and only if f(w) is k-power-free whenever w is a k-power-free word.A morphism f is k-power-free up to m if and onlyif f(w) isk-power-free whenever w is a k-power-free word of length at most m.Given an integer k ≥ 2,we prove that a binary morphism is k-power-freeif and only if it is k-power-free up to k 2.This bound becomes linear for primitive morphisms:a binary primitive morphism is k-power-freeif and only if it is k-power-free up to 2k+1