Melodie Andrieu - research

Research

Research interests:

All that involves both computer sciences and mathematics in general. I am particulary interested in discrete dynamical systems, and their interpretation in terms of words. What's more, i believe in experimentation in mathematics.



PhD Thesis:

Exceptional trajectories in the symbolic dynamics of multidimensional continued fraction algorithms

The continued fraction is a numeration system (i.e., a way of representing numbers by infinite sequences of integers, like the decimal expansion) based on Euclid's algorithm. This numeration system is extremely interesting: it systematically provides the best approximations of real numbers (for example: Pi, squareroot of 2, the golden ratio...) by ratios of integers - “better” meaning that these are the most precise approximations that we can hope with a limited number of digits. This property of best approximation has many applications, both in mathematics and beyond.

Since the 19th century, one has sought to generalize the continued fraction in higher dimension. Several algorithms have been proposed in order to approach two real numbers by two rational numbers with the same “small” denominator. My thesis is a part of a large program, consisting in studying these algorithms from the standpoint of the symbolic dynamical systems they generate. More precisely, I develop tools to detect and study a convergence anomaly identified for the first time in 2000: the imbalance. This anomaly is liable to affect the quality of the simultaneous approximations provided.

Supervisors: Julien Cassaigne, Pierre Arnoux.

> Download the manuscript.
> Have a look on the presentation.

Publications:

  1. M. Andrieu, A Rauzy fractal unbounded in all directions of the plane. Comptes Rendus de l'Académie des sciences, 2021. Journal page, Arxiv preprint.
    We construct an Arnoux-Rauzy word whose Rauzy fractal is not contained between two parallel lines. We also give an elementary proof that any Arnoux-Rauzy vector of letters frequencies has rationally independent entries.
  2. M. Andrieu, A. E. Frid, Morphic words and equidistributed sequences. Theoretical Computer Science, 2020 Journal page, ArXiv preprint.
    We discuss numeric sequences arising from the measure of shifts of a pure morphic infinite word, and construct them with morphisms. A software tool for that is provided (demonstration version, download).
  3. M. Andrieu, Autour du déséquilibre des mots C-adiques [text in French], Proceedings of Mons Theoretical Computer Science Days, 2018. Proceedings.
    We contruct C-adic words (= words associated with the Cassaigne-Selmer multidimensional continued fraction algorithm) with infinite imbalance.
  4. ...Preprints:
  5. M. Andrieu, Natural coding of minimal rotations of the torus, induction and exduction. Preprint.
  6. This article corrects and deepens the results of Cassaigne, Ferenczi and Zamboni of 2000. 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 careful 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.
  7. M. Andrieu, A semi-algorithm to explore the set of imbalances in a S- adic system. Preprint.
  8. We describe a semi-algorithm consisting of an automaton, or rather, an ever-building family of automata, whose states contain all the possible imbalances of S-adic words, where S is a finite set of substitutions.

Talks to come:

Selected talks: