Best simultaneous diophantine approximations of Pisot numbers and Rauzy fractals

Hubert, Pascal; Messaoudi, Ali

doi:10.4064/aa124-1-1

Cited by 26 publications

(25 citation statements)

References 11 publications

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“…Our induction algorithm defines a multi-dimensional continued fraction algorithm generated by the 2k − 2 non-independent parameters corresponding to the lengths of each side of β i in our k − 1 intervals. It constitutes a non-trivial generalization of the Euclid algorithm and, unlike other such algorithms arising in the context of symbolic dynamics [3,18], it is defined on a set of full measure. As in the case of the other inductions, it is described by an infinite path in a finite graph, whose vertices are the states of the inductions.…”

mentioning

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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“…Additionally, irrational numbers whose binary expansion is given by the fixed point of a substitution are all transcendental [1]. In the field of diophantine approximation, substitutions produce transcendental numbers which are very badly approximable by cubic algebraic integers [126]; the description of greedy expansions of reals in noninteger basis [5,142] by means of substitutions also results in best approximation characterizations (see [78] and [100,Chapter 10]). Representations of positive integers by number systems related to substitutions are studied in [61,62,63].…”

Section: The Role Of Substitutions In Several Branches Of Mathematicsmentioning

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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“…For instance, in the field of diophantine approximation, substitutions produce transcendental numbers which can be approximated by cubic algebraic integers only in a very bad way [81]; the description of greedy expansions of reals in noninteger base [5,89] by means of substitutions also results in best approximation characterizations [53,71]. The Cobham Theorem [41] also constitutes a strong bridge between substitutions and number theory and allows one to derive deep transcendence properties: the real numbers with continued fraction expansions given by the Thue-Morse sequence, the BaumSweet sequence [22] or the Rudin-Shapiro sequence [82] are all transcendental, the proof being based on the "substitutive" structure of these sequences [1].…”

Section: Substitutions Among Mathematics and Computer Sciencementioning

confidence: 99%

“…In number theory, diophantine properties are induced by properties of a distance function to a specific broken line [53] related to the Rauzy fractal and the size of the largest ball contained in it. Finiteness properties of digit representations in numeration systems with non-integer base are related to the fact that 0 is an inner point of the Rauzy fractal [9].…”

Section: The Geometry Of One-dimensional Substitutionsmentioning

confidence: 99%

Decidability Problems for Self-induced Systems Generated by a Substitution

Jolivet

Siegel

2015

Lecture Notes in Computer Science

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Abstract. In this talk we will survey several decidability and undecidability results on topological properties of self-affine or self-similar fractal tiles. Such tiles are obtained as fixed point of set equations governed by a graph. The study of their topological properties is known to be complex in general: we will illustrate this by undecidability results on tiles generated by multitape automata. In contrast, the class of self affine tiles called Rauzy fractals is particularly interesting. Such fractals provide geometrical representations of self-induced mathematical processes. They are associated to one-dimensional combinatorial substitutions (or iterated morphisms). They are somehow ubiquitous as self-replication processes appear naturally in several fields of mathematics. We will survey the main decidable topological properties of these specific Rauzy fractals and detail how the arithmetic properties of the substitution underlying the fractal construction make these properties decidable. We will end up this talk by discussing new questions arising in relation with continued fraction algorithm and fractal tiles generated by S-adic expansion systems.The following survey is mainly inspired by three papers from the authors and their collaborators [32,60,84].

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Best simultaneous diophantine approximations of Pisot numbers and Rauzy fractals

Cited by 26 publications

References 11 publications

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

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

Topological properties of Rauzy fractals

Decidability Problems for Self-induced Systems Generated by a Substitution

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