Les transformations de Chacon : combinatoire, structure géométrique, lien avec les systèmes de complexité
Bounded remainder sets
Systems of finite rank
A generalization of the self-dual induction to every interval exchange transformation
We generalize to all interval exchanges the induction algorithm defined by Ferenczi and Zamboni for a particular class. Each interval exchange corresponds to an infinite path in a graph whose vertices are certain unions of trees we call castle forests. We use it to describe those words obtained by coding trajectories and give an explicit representation of the system by Rokhlin towers. As an application, we build the first known example of a weakly mixing interval exchange outside the hyperelliptic...
Substitution dynamical systems on infinite alphabets
We give a few examples of substitutions on infinite alphabets, and the beginning of a general theory of the associated dynamical systems. In particular, the “drunken man” substitution can be associated to an ergodic infinite measure preserving system, of Krengel entropy zero, while substitutions of constant length with a positive recurrent infinite matrix correspond to ergodic finite measure preserving systems.
Three complexity functions
For an extensive range of infinite words, and the associated symbolic dynamical systems, we compute, together with the usual language complexity function counting the finite words, the minimal and maximal complexity functions we get by replacing finite words by finite patterns, or words with holes.
Three complexity functions
For an extensive range of infinite words, and the associated symbolic dynamical systems, we compute, together with the usual language complexity function counting the finite words, the minimal and maximal complexity functions we get by replacing finite words by finite patterns, or words with holes.
Rank and spectral multiplicity
For a dynamical system (X,T,μ), we investigate the connections between a metric invariant, the rank r(T), and a spectral invariant, the maximal multiplicity m(T). We build examples of systems for which the pair (m(T),r(T)) takes values (m,m) for any integer m ≥ 1 or (p-1, p) for any prime number p ≥ 3.
Rank is not a spectral invariant
Three complexity functions
For an extensive range of infinite words, and the associated symbolic dynamical systems, we compute, together with the usual language complexity function counting the finite words, the minimal and maximal complexity functions we get by replacing finite words by finite patterns, or words with holes.
Trajectories of rotations
Eigenvalues and simplicity of interval exchange transformations
For a class of -interval exchange transformations, which we call the symmetric class, we define a new self-dual induction process in which the system is successively induced on a union of sub-intervals. This algorithm gives rise to an underlying graph structure which reflects the dynamical behavior of the system, through the Rokhlin towers of the induced maps. We apply it to build a wide assortment of explicit examples on four intervals having different dynamical properties: these include the first...
Substitution dynamical systems : algebraic characterization of eigenvalues
Weak mixing and eigenvalues for Arnoux-Rauzy sequences
We define by simple conditions two wide subclasses of the so-called Arnoux-Rauzy systems; the elements of the first one share the property of (measure-theoretic) weak mixing, thus we generalize and improve a counter-example to the conjecture that these systems are codings of rotations; those of the second one have eigenvalues, which was known hitherto only for a very small set of examples.
On finite pseudorandom binary sequences IV: The Liouville function, II
On finite pseudorandom binary sequences III: The Liouville function, I
Imbalances in Arnoux-Rauzy sequences
In a 1982 paper Rauzy showed that the subshift generated by the morphism , and is a natural coding of a rotation on the two-dimensional torus , i.e., is measure-theoretically conjugate to an exchange of three fractal domains on a compact set in each domain being translated by the same vector modulo a lattice. It was believed more generally that each sequence of block complexity satisfying a combinatorial criterion known as the condition of Arnoux and Rauzy codes the orbit of a point...
Structure of three interval exchange transformations I: an arithmetic study
In this paper we describe a -dimensional generalization of the Euclidean algorithm which stems from the dynamics of -interval exchange transformations. We investigate various diophantine properties of the algorithm including the quality of simultaneous approximations. We show it verifies the following Lagrange type theorem: the algorithm is eventually periodic if and only if the parameters lie in the same quadratic extension of
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