Papers by Sébastien Ferenczi
Eigenvalues and Simplicity of Interval Exchange Transformations
In this paper we consider a class of d-interval exchange transformations, which we call the symme... more In this paper we consider a class of d-interval exchange transformations, which we call the symmetric class. For this 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
COMBINATORIAL STRUCTURE OF SYMMETRIC k-INTERVAL EXCHANGE TRANSFORMATIONS
We describe a combinatorial algorithm for generating the symbolic sequences which code the orbits... more We describe a combinatorial algorithm for generating the symbolic sequences which code the orbits of points under an interval exchange transformation on k intervals, using the "symmetric" permutation i 7→ k −i+1; this algorithm is based on a new induction, described by two classes of graphs. As a consequence we obtain a complete characterization of those sequences of subword complexity
Journal D Analyse Mathematique, Oct 1, 2010
We define a new induction algorithm for k-interval exchange transformations associated to the "sy... more We define a new induction algorithm for k-interval exchange transformations associated to the "symmetric" permutation i → k − i + 1. Acting as a multi-dimensional continued fraction algorithm, it defines a sequence of generalized partial quotients given by an infinite path in a graph whose vertices, or states, are certain trees we call trees of relations. This induction is self-dual for the duality between the usual Rauzy induction and the da Rocha induction. We use it to describe those words obtained by coding orbits of points under a symmetric interval exchange, in terms of the generalized partial quotients associated with the vector of lengths of the k intervals. As a consequence, we improve a bound of Boshernitzan in a generalization of the three-distances theorem for rotations. However, a variant of our algorithm, applied to a class of interval exchange transformations with a different permutation, shows that the former bound is optimal outside the hyperelliptic class of permutations.
Languages of k-interval exchange transformations
ABSTRACT
Combinatorial trees arising in the study of interval exchange transformations
European Journal of Combinatorics, 2011
ABSTRACT In this paper we study a new class of combinatorial objects that we call trees of relati... more ABSTRACT In this paper we study a new class of combinatorial objects that we call trees of relations equipped with an operation that we call induction. These trees were first introduced in Ferenczi and Zamboni (2010) [3] in the context of interval exchange transformations but they may be studied independently from a purely combinatorial point of view. They possess a variety of interesting combinatorial properties and have already been linked to a number of different areas including ergodic theory and number theory—see Ferenczi and Zamboni (2010, in press) [3,4]. In a recent sequel to this paper, Marsh and Schroll have established interesting connections to the theory of cluster algebras and polygonal triangulations: Marsh and Schroll (2010) [5]. For each tree of relations G, we let Γ(G) denote the smallest set of trees of relations containing G and invariant under induction. The induction mapping allows us to endow Γ(G) with the structure of a connected directed graph, which we call the graph of graphs. We investigate the structure of Γ(G) and define a circular order based on the tree structure which turns out to be a complete invariant for the induction mapping. This gives a complete characterization of Γ(G) which allows us to compute its cardinality in terms of Catalan numbers. We show that the circular order also defines an abstract secondary structure similar to one occurring in genetics in the study of RNA.
Annales de l’institut Fourier, 2001
We describe an algorithm for generating the symbolic sequences which code the orbits of points un... more We describe an algorithm for generating the symbolic sequences which code the orbits of points under an interval exchange transformation on three intervals. The algorithm has two components. The first is an arithmetic division algorithm applied to the lengths of the intervals. This arithmetic construction was originally introduced by the authors in an earlier paper and may be viewed as a two-dimensional generalization of the regular continued fraction. The second component is a combinatorial algorithm which generates the bispecial factors of the associated symbolic subshift as a function of the arithmetic expansion. As a consequence we obtain a complete characterization of those sequences of block complexity ¾Ò · ½ which are natural codings of orbits of three-interval exchange transformations, thereby answering an old question of Rauzy.
Ergodic Theory and Dynamical Systems, 2005
We show that among three-interval exchange transformations there exists a dichotomy: T has minima... more We show that among three-interval exchange transformations there exists a dichotomy: T has minimal self-joinings whenever the associated subshift is linearly recurrent, and is rigid otherwise. We build also a family of simple rigid three-interval exchange transformations, which is a step towards an old question of Veech, and a family of rigid three-interval exchange transformations which includes Katok's rank one map.
Combinatorics of Three-Interval Exchanges
Lecture Notes in Computer Science, 2001
We generalize the interaction between Sturmian infinite words and rotations of the 1-dimensional ... more We generalize the interaction between Sturmian infinite words and rotations of the 1-dimensional torus by giving a set of necessary and sufficient conditions for a language to be a natural coding of a three-interval exchange. This solves an old question of Rauzy, and allows us to give a complete combinatorial description of such languages through an algorithm of simultaneous approximation.
Imbalances in Arnoux-Rauzy sequences
... Julien Cassaigne; Sébastien Ferenczi; Luca Q. Zamboni Imbalances in Arnoux-Rauzy sequences An... more ... Julien Cassaigne; Sébastien Ferenczi; Luca Q. Zamboni Imbalances in Arnoux-Rauzy sequences Annales de l'institut Fourier, 50 no. ... properties of a rotation by angle a on the 1-dimensional torus, and the symbolic/combinatorial properties of a class of binary sequences known ...
Journal d'Analyse Mathématique, 1999
We study a particular case of the two-dimensional Steinhaus theorem, giving estimates of the poss... more We study a particular case of the two-dimensional Steinhaus theorem, giving estimates of the possible distances between points of the form ka and ka +3 on the unit circle, through an approximation algorithm of 3 by the points ka. This allows us to compute covering numbers (maximal measures of Rokhtin stacks having celxain prescribed regularity properties) for the symbolic dynamical systems associated to the rotation of argument a, acting on the partition of the circle by the points 0, 3. We can then compute topological and measure-theoretic covering numbers for exchange of three intervals; in this way, we prove that every ergodic exchange of three intervals has simple spectrum and build a hew class of three-interval exchanges which are not of rank one.
The aim of this paper is to survey possible generalizations of the well-known interaction between... more The aim of this paper is to survey possible generalizations of the well-known interaction between Sturmian sequences and rotations of T 1 , using the Euclid algorithm of continued fraction approximation. We investigate mainly similar relations between Arnoux-Rauzy sequences and rotations of T 2 , between codings of Z 2 -actions on T 1 and multidimensional Sturmian sequences, between exchanges of intervals and some sequences of linear complexity, and give an account of several tentative generalizations of continued fractions, using the notions of substitution and induction.
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Papers by Sébastien Ferenczi