Publications

Lecture notes

1. Infinite Words with very Low Factor Complexity: an introduction to Combinatorics on Words. Dyadisc7 : Brazilian-chilean and french interplay for symbolic dynamics. Valparaiso, Chile, 2024. HAL.
These three-chapter lecture notes, including exercises, are intended to be accessible to graduate students and a broad audience of researchers.
• Chapter 1 is a self-contained introduction to combinatorics on words and Sturmian words. Contrary to most of the literature, the emphasis is placed on their connection with continued fractions. In particular, the dynamical characterizations of Sturmian words arise as a natural consequence of their algebraic structure.
• In Chapter 2, we discuss various generalizations of the Morse–Hedlund theorem on the d-ary alphabet, including a long-forgotten theorem by Tijdeman from 1999.
• Chapter 3 presents a new algebraic proof of Tijdeman’s theorem, obtained in collaboration with J. Cassaigne.

Papers and preprints (in chronological order)

2. A Proof of Rauzy's conjecture on abelian complexity. With L. Vivion , 2026. Arxiv.
   We prove a conjecture of Rauzy (1983), showing that there do not exist infinite ternary words with rationally independent letter frequencies and constant abelian complexity equal to 3.
3. A Normality conjecture in rational base number systems. With S. Eliahou and L. Vivion , 2025. Preprint.
   We formulate the conjecture that, in every rational base number system, every minimal and maximal word is normal. To support this conjecture and its interest, we present extensive numerical experiments and explain how its connects to several long-standing open problems in number theory.
4. Imbalances in hypercubic billiard words II: the rational dependent case. With L. Vivion. Mons Theoretical Computer Science Days, 2024. HAL.
We study the imbalances of words generated by a billiard in a hypercube, for momenta with rationally dependent entries.
5. Minimal complexities for infinite words written with d letters. With L. Vivion. Invited paper, WORDS, 2023. HAL.
We discuss the minimal subword complexity and the minimal abelian complexity functions for infinite d-ary words. This leads us to answer a question of Rauzy from 1983: cubic billiard words are a good generalization of Sturmian words for the abelian complexity.
6. Imbalances in hypercubic billiard words. With L. Vivion. Mons Theoretical Computer Science Days, 2022. HAL.
We completely describe the imbalances of words generated by a billiard in a hypercube, for momenta with rationally independent entries.
7. Natural coding of minimal rotations of the torus, induction and exduction. 2021. (Published as a chapter of my PhD thesis.) Preprint.
We introduce a topological definition of natural coding of a minimal rotation on the d-dimensional torus, inspired by the seminal works of Rauzy on the Tribonacci word. We prove that, under this definition, the property of being a natural coding of rotation is preserved by induction and exduction. We apply these results to Arnoux-Rauzy and C-adic words. This work corrects and improves a result of Cassaigne, Ferenczi and Zamboni from 2000.
8. A semi-algorithm to explore the set of imbalances in a S- adic system. 2021. (Published as a chapter of my PhD thesis.) Preprint.
We construct a semi-algorithm consisting of an ever-building family of finite automata whose states contain all the possible imbalances of S-adic words, where S is a finite set of substitutions. This semi-algorithm is behind the constructions 9 and 11.
9. A Rauzy fractal unbounded in all directions of the plane. Comptes Rendus de l'Académie des sciences, 2021. Arxiv.
   We construct an Arnoux-Rauzy word whose Rauzy fractal is not contained between any two parallel lines. We also prove that the letter frequencies of every Arnoux-Rauzy word are rationally independent.
10. Morphic words and equidistributed sequences. With A. E. Frid. Theoretical Computer Science, 2020. ArXiv.
    We study equidistributed sequences arising from pure morphic words. We provide a software tool to construct these sequences in the binary case. (Online demonstration version, souce code).
11. On imbalances in C-adic infinite words [in French], Proceedings of Mons Theoretical Computer Science Days, 2018. Book of abstracts.
    We construct C-adic words (= words associated with the Cassaigne-Selmer multidimensional continued fraction algorithm) with infinite imbalance.

In progress:

12. M. Andrieu, L. Vivion, Optimal balanceness constants of hypercubic billiard words (journal version).
13. M. Andrieu, L. Vivion, The abelian complexity of hypercubic billiard words is eventually constant .
14. M. Andrieu, L. Vivion, A new renormalization process on infinite words

Thesis manuscript:

15. Exceptional trajectories in the symbolic dynamics of multidimensional continued fraction algorithms. Aix-Marseille Université, 2021. HAL.