Interactions between dynamics, arithmetics and combinatorics: the good, the bad, and the ugly

Berthé, Valérie; Ferenczi, Sébastien; Zamboni, Luca Q.

doi:10.1090/conm/385/07205

Cited by 38 publications

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“…, α k which are rearranged by I according to π. It was Rauzy [29] who first saw interval exchange transformations as a possible framework for generalizing the well-known interaction between circle rotations on one hand and Sturmian sequences on the other via the continued fraction algorithm (see, for example, the survey [7] [4]. But, in contrast with the Sturmian case, though this induction gives access to the symbolic dynamics of the trajectories, it is difficult to describe them explicitly and indeed the Rauzy induction was never actually used for that purpose.…”

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confidence: 99%

Structure of K-interval exchange transformations: Induction, trajectories, and distance theorems

Ferenczi

Zamboni

2010

JAMA

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Abstract. 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. PreliminariesInterval exchange transformations were originally introduced by Oseledec [27], following an idea of Arnold [2]; see also Katok and Stepin [20]. An exchange of k intervals, denoted throughout this paper by I, is given by a probability vector of k lengths (α 1 , . . . , α k ) together with a permutation π on k letters. The unit interval is partitioned into k subintervals of lengths α 1 , . . . , α k which are rearranged by I according to π. It was Rauzy [29] who first saw interval exchange transformations as a possible framework for generalizing the well-known interaction between circle rotations on one hand and Sturmian sequences on the other via the continued fraction algorithm (see, for example, the survey [7] [4]. But, in contrast with the Sturmian case, though this induction gives access to the symbolic dynamics of the trajectories, it is difficult to describe them explicitly and indeed the Rauzy induction was never actually used for that purpose. A partial description of the trajectories was given in [24], and it aroused interest by defining a different kind of induction; this da Rocha induction was then considered to be dual to the Rauzy induction: the Rauzy induction builds the points where the orbits of the discontinuities of I approximate 0, while the da Rocha induction builds the points where the orbit of 0 approximates the discontinuities of I −1 . Note that (non-constructive) combinatorial characterizations of the language of trajectories were given recently in [16] and [5].These inductions are also a fundamental tool in the study of the space of moduli of Riemann surfaces, and the various strata of its unit tangent bundle, through what has been a basic object of interest for the last 25 years, the Teichmüller flow on a stratum; in the case of interval exchanges, we consider the Teichmüller flow for the Riemann surface obtained by glueing parallel opposite sides of a polygon. The Rauzy induction chooses an initial segment of a horizontal separatrix and follows its vertical separatrix till it i...

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confidence: 99%

Structure of K-interval exchange transformations: Induction, trajectories, and distance theorems

Ferenczi

Zamboni

2010

JAMA

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“…A useful generalization of this process to simultaneous approximation of several numbers is obtained by replacing rotations by interval exchange transformations, see for example the survey [2]; the additive form of a general process for exchanges of k intervals was given by Rauzy [15], it was then proved by Veech [17] and Masur [13] that the renormalization (or induction) map has an infinite invariant measure, and this was the source of many important results in the theory of interval exchange transformations; a multiplicative form was then derived by Zorich [18], who showed that it has a finite invariant measure. Another induction was proposed by da Rocha, it is dual to the first one in a sense precised in section 2.1 below; again, the additive form has an infinite invariant measure [11] and the multiplicative form has a finite invariant measure [3], and in both cases there is an explicit formula for the density with respect to the Lebesgue measure.…”

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confidence: 99%

A self-dual induction for three-interval exchange transformations

Ferenczi

Rocha

2009

Dynamical Systems

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Abstract. We describe completely a new induction process on three-interval exchange transformations, under its additive and multiplicative forms; we describe the natural extension of the induction map, show that it is self-dual for the notion of duality which links the Rauzy/Zorich and da Rocha inductions, and compute a finite ergodic invariant measure (for the multiplicative form). Then we explain the connection with the published negative slope algorithm of Ferenczi, Holton and Zamboni, and use our measure-theoretic study to get new results on three-interval exchange transformations.The classical Euclid algorithm of continued fraction approximation for an irrational α can be seen as a renormalization process on the dynamics of the rotation of angle α; this has several forms: the additive form is usually given by the induction map on two coordinates, (x, y) → (x − y, y) if x > y, (x, y) → (x, y − x) if x < y; the multiplicative form, which is an acceleration of the additive one, is given by the Gauss map x → { 1 x }. A useful generalization of this process to simultaneous approximation of several numbers is obtained by replacing rotations by interval exchange transformations, see for example the survey [2]; the additive form of a general process for exchanges of k intervals was given by Rauzy [15], it was then proved by Veech [17] and Masur [13] that the renormalization (or induction) map has an infinite invariant measure, and this was the source of many important results in the theory of interval exchange transformations; a multiplicative form was then derived by Zorich [18], who showed that it has a finite invariant measure. Another induction was proposed by da Rocha, it is dual to the first one in a sense precised in section 2.1 below; again, the additive form has an infinite invariant measure [11] and the multiplicative form has a finite invariant measure [3], and in both cases there is an explicit formula for the density with respect to the Lebesgue measure.A third algorithm was proposed for k = 3 by Ferenczi, Holton and Zamboni [4] [5], as an approximation algorithm called the negative slope algorithm; it was effective in the study of three-interval exchange tranformations, leading to new results [5] [6] [7], and was recently studied for itself by Nakada and Ishimura [10]; however, it does not appear explicitely as an induction algorithm on interval-exchange transformations, and it exists only under a multiplicative form. A new induction algorithm is defined for all k by Ferenczi and Zamboni [8] (see also [9] for k = 4); this exists as yet only under an additive form, and its connection with the negative slope algorithm is not explicit. The present paper aims to fill this gap and to study completely the new algorithm for k = 3: we first discuss its definition, then give a self-contained description of the induction process under its additive form, and deduce its multiplicative form; we identify the natural extension of this induction process, and show that it is self-dual for the Rauzy/da Rocha duality...

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“…This refers to what is called Rauzy program in [43]: "find generalizations of the Sturmian/rotation interaction which would naturally generate approximation algorithms". Revisited with the Arnoux-IO formalism [24], this means to start from a n-dimensional vector u in R n and to decompose this vector with a continued fraction algorithm, u = M 1 M 2 .…”

Section: Effective Constructions and Generalizationsmentioning

confidence: 99%

Topological properties of Rauzy fractals

Siegel¹,

Thuswaldner²

2009

Mémoires Soc. Math. France

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Abstract. -Substitutions are combinatorial objects (one replaces a letter by a word) which produce sequences by iteration. They occur in many mathematical fields, roughly as soon as a repetitive process appears. In the present monograph we deal with topological and geometric properties of substitutions, i.e., we study properties of the Rauzy fractals associated to substitutions.To be more precise, let σ be a substitution over the finite alphabet A. We assume that the incidence matrix of σ is primitive and its dominant eigenvalue is a unit Pisot number (i.e., an algebraic integer greater than one whose norm is equal to one and all of whose Galois conjugates are of modulus strictly smaller than one). It is well-known that one can attach to σ a set T which is now called central tile or Rauzy fractal of σ. Such a central tile is a compact set that is the closure of its interior and decomposes in a natural way in n = #A subtiles T (1), . . . , T (n). The central tile as well as its subtiles are graph directed self-affine sets that often have fractal boundary.Pisot substitutions and central tiles are of high relevance in several branches of mathematics like tiling theory, spectral theory, Diophantine approximation, the construction of discrete planes and quasicrystals as well as in connection with numeration like generalized continued fractions and radix representations. The questions coming up in all these domains can often be reformulated in terms of questions related to the topology and geometry of the underlying central tile.After a thorough survey of important properties of unit Pisot substitutions and their associated Rauzy fractals the present monograph is devoted to the investigation of a variety of topological properties of T and its subtiles. Our approach is an algorithmic one. In particular, we dwell upon the question whether T and its subtiles induce a tiling, calculate the Hausdorff dimension of their boundary, give criteria for their connectivity and homeomorphy to a disk and derive properties of their fundamental group.The basic tools for our criteria are several classes of graphs built from the description of the tiles T (i) (1 ≤ i ≤ n) as graph directed iterated function systems and from the structure of the tilings induced by these tiles. These graphs are of interest in their own right. For instance, they can be used to construct the boundaries ∂T as well as ∂T (i) (1 ≤ i ≤ n) and all points where two, three or four different tiles of the induced tilings meet.When working with central tiles in one of the above mentioned contexts it is often useful to know such intersection properties of tiles. In this sense the present monograph also aims at providing tools for "everyday's life" when dealing with topological and geometric properties of substitutions.Many examples are given throughout the text in order to illustrate our results. Moreover, we give perspectives for further directions of research related to the topics discussed in this monograph.

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Interactions between dynamics, arithmetics and combinatorics: the good, the bad, and the ugly

Cited by 38 publications

References 94 publications

Structure of K-interval exchange transformations: Induction, trajectories, and distance theorems

Structure of K-interval exchange transformations: Induction, trajectories, and distance theorems

A self-dual induction for three-interval exchange transformations

Topological properties of Rauzy fractals

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