Welcome to the home page of the **Symbolic Dynamics and Arithmetic Expansions** (SymDynAr) Project, funded by ANR and FWF.

**Project**

- ANR-23-CE40-0024-01
- FWF Principal Investigator Projects International I 6750

**Coordinators**

**Duration** 48 months, from 2024-01-01 to 2027-12-31

## Topics

In Austria as in France, research on dynamical systems, especially those related to arithmetic expansions and continued fractions, has a long tradition. The aim of our project is to bring together the expertise of both countries. Dynamical systems and their symbolic codings are at the heart of this project. We consider a wide variety of dynamical systems. This includes symbolic codings, such as substitution shifts, S-adic systems and, more generally, Cantor systems, but also continuous dynamical systems, such as those found in numeration, in continued fraction algorithms, in cut-and-project schemes for the modelling of quasicrystals and in the study of automorphisms of the torus. The focus will be on the study of Lyapunov exponents of multidimensional continued fraction algorithms. We will apply our results to arithmetic and normality, as well as to some areas of mathematical physics.

Symbolic dynamics forms an important branch of mathematics that gives rise to many new achievements in various fields. One of our aims is to exhibit symbolic systems that are conjugate to a given system and derive properties of these symbolic codings in order to gain information on the original system. Indeed, representations of these systems in terms of symbolic systems (like sofic shifts or shifts of finite type) are very useful to gain properties (eigenvalues, invariants or recurrence) of the original systems and in our case have strong relations to numeration and continued fractions. Such codings can be found e.g. by using partitions (as for instance Markov partitions) or Poincaré sections. These codings admit scale invariance that allows modeling them by renormalization schemes, which can be understood by applying the well developed theory of hyperbolic dynamics.

We focus on interval exchange transformations and systems of an arithmetic nature with continued fraction type algorithms governing the renormalization scheme, by exploiting the interplay between arithmetical functions and sequences arising from symbolic dynamical systems and symbolic expansions. We thus consider digital sequences that originate from some dynamical systems, as well as classical sequences issued from number theory and their relations, motivated by Sarnak’s conjecture, and more generally by the quantification problem for correlations between such sequences.

We structure this project into four strongly interlinked tasks. We consider

- dynamical properties of infinite interval exchange transformations in Task 1,
- renormalization with a focus on continued fractions in Task 2,
- arithmetical and spectral properties with topological dynamics and orbit equivalence in Task 3,
- normal numbers, digital sequences, correlations between such sequences and Sarnak’s conjecture from a dynamical viewpoint in Task 4.