Numeration systems and substitutions

Abstract

In mathematics, more precisely in discrete mathematics, we are interested in ways to represent integers uniquely. In this work, we will start by introducing numeration systems, such as the positional numeration system and the abstract numeration system. The last one can be used to generalise Cobham's equivalence between k-automatic sequences and the sequences obtained by iterating a uniform morphism of length k. Afterwards we will study a very famous positional numeration system namely the numeration system associated with the theta-expansion of 1 of a Parry number. Using this positional numeration system we can define a theta-automaton and the Fabre substitution. Among other results, we will prove a link between a theta-automaton and the fixed point of a conjugate of the Fabre substitution associated with the theta-expansion of 1. Then we will study the complexity of the fixed point of the Fabre substitution associated with a simple Parry number under specific conditions. Lastly, we will show an application of a few results encountered during this thesis in the field of string attractos.