Conservation du caractère reconnaissable par opérations arithmétiques dans un système de numération abstrait

In this thesis, I study the stability of recognizability under arithmetic operations like addition, multiplication by a constant or multiplication, in an abstract numeration system. The main result presented in this work concerns the abstract numeration system built on the bounded language a^*b^*. It shows that, in this case, multiplication by a constant preserves recognizability if and only if this constant is an odd square. I end this text by providing some partial results of a generalization to abstract numeration system built on any bounded language.