Parry’s 1960 theorem and some applications in combinatorics on words

To manipulate numbers, we need to represent them. A numeration system is nothing but a set of rules that enables us to write each number as a sequence of symbols belonging to an alphabet of digits. In combinatorics on words, such a sequence is called a word. A result of Parry, dating back to 1960 and now classical, allows us to describe the numeration language, that is, the set of all admissible representations, in a numeration system based on a real number. In particular, in such a numeration system, the representation of 1 plays an crucial role. For so-called Parry numbers, the latter representation is particular: it is either finite or eventually periodic. This property gives rise to particularly rich numeration systems having nice exploitable properties. In this talk, I will present the framework of these numeration systems and show some applications that come up in my work in combinatorics on words.