Quasicrystals and Geometry

In this paper we develop a general approach for investigating the asymptotic distribution of functional Xn = f((Zn+k)k∈z) of absolutely regular stochastic processes (Zn)n∈z. Such functional occur naturally as orbits of chaotic dynamical systems, and thus our results can be used to study probabilistic aspects of dynamical systems. We first prove some moment inequalities that are analogous to those for mixing sequences. With their help, several limit theorems can be proved in a rather straightforward manner. We illustrate this by re-proving a central limit theorem of Ibragimov and Linnik. Then we apply our techniques to U-statistics Matrix Equation with symmetric kernel h : R × R → R. We prove a law of large numbers, extending results of Aaronson, Burton, Dehling, Gilat, Hill and Weiss for absolutely regular processes. We also prove a central limit theorem under a different set of conditions than the known results of Denker and Keller. As our main application, we establish an invariance principle for U-processes (Un(h))h, indexed by some class of functions. We finally apply these results to study the asymptotic distribution of estimators of the fractal dimension of the attractor of a dynamical system.

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Limit theorems for functionals of mixing processes with applications to U-statistics and dimension estimation

This paper surveys different constructions and properties of some multiple tilings (that is, finite-to-one coverings) of the space that can be associated with beta-numeration and substitutions. It is indeed possible, generalizing Rauzy’s and Thurston’s constructions, to associate in a natural way either with a Pisot number β (of degree d) or with a Pisot substitution σ (on d letters) some compact basic tiles that are the closure of their interior, that have non-zero measure and a fractal boundary; they are attractors of some graph- directed Iterated Function System. We know that some translates of these prototiles under a Delone set Γ (provided by β or σ) cover Rd−1; it is conjectured that this multiple tiling is indeed a tiling (which might be either periodic or self-replicating according to the translation set Γ). This conjecture is known as the Pisot conjecture and can also be reformulated in spectral terms: the associated dynamical systems have pure discrete spectrum. We detail here the known constructions for these tilings, their main properties, some applications, and focus on some equivalent formulations of the Pisot conjecture, in the theory of quasicrystals for instance. We state in particular for Pisot substitutions a finiteness property analogous to the well-known (F) property in beta-numeration, which is a sufficient condition to get a tiling.

https://hal-lirmm.ccsd.cnrs.fr/lirmm-00105299/document

Tilings associated with beta-numeration and substitutions.

In this paper, we survey the rich theory of infinite episturmian words which generalize to any finite alphabet, in a rather resembling way, the well-known family of Sturmian words on two letters. After recalling definitions and basic properties, we consider episturmian morphisms that allow for a deeper study of these words. Some properties of factors are described, including factor complexity, palindromes, fractional powers, frequencies, and return words. We also consider lexicographical properties of episturmian words, as well as their connection to the balance property, and related notions such as finite episturmian words, Arnoux-Rauzy sequences, and "episkew words" that generalize the skew words of Morse and Hedlund.

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Episturmian words : a survey

Le but de ce survol est d'aborder definitions et proprietes concernant la numeration d'un point de vue dynamique: nous nous concentrons sur les systemes de numeration, leur compactification, et. les systemes dynamiques qui peuvent etre definis dessus. La notion de systeme de numeration fibre unifie la presentation. De nombreux exemples sont etudies. Plusieurs numerations sur les entiers naturels, relatifs, les nombres reels ou les nombres complexes sont presentees. Nous portons une attention speciale a la β-numeration ainsi qu'a ses generalisations, aux systemes de numeration abstraits, aux systemes dits "shift radix", de meme qu'aux G-echelles et aux odometres. Un paragraphe d'applications conclut ce survol.

/pdf/dynamical-directions-in-numeration-2m636zpu4b.pdf

Dynamical directions in numeration

Abstract This paper investigates representations of real numbers with an arbitrary negative base –β < –1, which we call the (–β)-expansions. They arise from the orbits of the (–β)-transformation which is a natural modification of the β-transformation. We show some fundamental properties of (–β)-expansions, each of which corresponds to a well-known fact of ordinary β-expansions. In particular, we characterize the admissible sequences of (–β)-expansions, give a necessary and sufficient condition for the (–β)-shift to be sofic, and explicitly determine the invariant measure of the (–β)-transformations.

Beta-expansions with negative bases.

We introduce two types of real continued-fraction expansions, one of which is the real part of the complex continued-fraction expansion of HURWITZ. For the transformations associated to these expansions we shall determine the precise form of invariant measures according to the method of P. LEVY for the case of simple continued-fraction. Moreover, we shall clarify the mathematical meaning of the method of P. LEVY. § 0 Introduction In the investigation of properties of a measurable transformation given on a space, a measure invariant under the transformation, if it exists, provides a valuable clue. Hence, one often takes the following approach in such an investigation. First, one asks whether the transformation has an invariant measure possessing reasonable properties. Next, if there is sucn an invariant measure, one tries to determine its concrete form. Of course, it is, in general, difficult to obtain the precise form of an invariant measure since one has to obtain it predictly from the precise description of each transformation concerned. On the other hand, for this very reason, the derivation of the concrete form of an invariant measure, if it can be carried out, is extremely useful for the quantitative analysis of the given transformation. 159 H1Tosrn NAKADA, S11uNJI lTo and SmGERU TANAKA In this connection, we recall that there is remarkable history associated with the transformation induced by the well-known simple continued-fraction expansion. For this transformation GAuss pointed out as if it is obvious apriori that the measure having the density function of the form 1 -} 2 --1 lis invariant. Indeed, og +x if one is given the function --1-.___ _1 __ , then it is easy to prove that it is the log 2 l+:c density of a measure invariant under the simple continued-fraction transformation. However, history played a trick and left us with no clue as to how GAuss actually arrived at this function --1----1-. Much later, KuzMIN [3] and LEVY [4] log 2 l+x showed, in their respective papers, ways to arrive at the density function 1 1 2 1 1 for the invariant measure and filled this missing gap, although we og +x have no way of knowing whether the reasoning used by GAt~ss was the same as those employed by KL:ZMIN and LEVY. In this paper, we formulate and then solve a couple of problems. The first problem is to search for effective methods for determining precisely the invariant measure for simple continued-fraction transformation and other related transformations. The second problem is to clarify the mathematical structure lying behind the method used by LEVY in his derivation of the density function 1 -1-2 1 1 -. og +x With these objectives in mind, we structure this paper in the following manner: In § 1 we simplify Lf:vy's argument give in [ 4] to derive the concrete form of the invariant measure for the transformation associated with simple continued-fraction expansion. The method we employ in this section, however, is based on a rather technical and seemingly restrictive assumption, which we shall leave unexplained at that point. For this reason, we shall call the method employed in § 1 \"the method based on pure chance discovery \". In § 2 we introduce a new type of (real) continued-fraction expansion. This expansion corresponds to the real part of the complex continued-fraction expansion introduced by HuRWITZ in [2]. In § 3, we introduce still another type of real continued-fraction expansion. The relationship between these two continued-fraction expansions can be explained in the following way: Each continued-fraction expansion induces in a natural way an endomorphism on the space of infinite sequences of positive integer (symbolic space). The natural extension of one of these endomorphisms turns out to be the inverse of the natural extension of the other one. For these reason, we shall call the transformation defined in § 3 the backward transformation associated with transformation defined in § 2. In § 4, we will determine the precise form of the density function for the invariant measures for the continued-fraction transformations defined in § 2 and § 3. Our method in § 4 is different from the \"method of pure-chance discovery\" employed in § 1. We hope, in fact, that our procedure in § 4 will clarify the mathematical meaning and give justification to seemingly ad-hoc \"method of pure chance discovery \". In order to justify this claim, we shall show in § 5 that the reason why the function g(x) seems to emerge suddenly and in somewhat unnatural manner is because for the case of simple continued-fraction expansion the transformation 160 On the Invariant Measure for the Transformations Associated induced by it and its backward transformation coincide with each other, and that it is this fact which makes it difficult to explain the naturalness of the emergence of the function g(x). In concluding these introductory remarks, we would like to thank Professors TAKUJI 0NOYAMA, Yu11 ho and YmcHmo TAKAHASHI for their interest on the problem and valuable advice. § 1 The Transformation Associated with Continued-Fraction and the Invariant Measure of GAuss As it is well-known, the transformation T associated with simple continuedfraction expansions is defined as follows : For xE[O, 1),

On the invariant measure for the transformations associated with some real continued-fractions.

Nous definissons des substitutions bi-dimensionnelles ; ces substitutions engendrent des suites doubles reliees a des approximations discretes de plans irrationnels. Elles sont obtenues au moyen de l'algorithme classique de Jacobi Perron, en definissant l'induction d'une action de Z 2 par rotations sur le cercle. On donne ainsi une interpretation geometrique nouvelle de l'algorithme de Jacobi-Perron, comme application operant sur l'espace des parametres des actions de Z 2 par rotations.

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Discrete planes, ${\mathbb {Z}}^2$-actions, Jacobi-Perron algorithm and substitutions

An irreducible Pisot substitution defines a graph-directed iterated function system. The invariant sets of this iterated function system are called the atomic surfaces. In this paper, a new tiling of atomic surfaces, which contains Thurston’sβ-tiling as a subclass, is constructed. Related tiling and dynamical properties are studied. Based on the coincidence condition defined by Dekking [Dek], we introduce thesuper-coincidence condition. It is shown that the super-coincidence condition governs the tiling and dynamical properties of atomic surfaces. We conjecture that every Pisot substitution satisfies the super-coincidence condition.

Atomic surfaces, tilings and coincidence I. irreducible case

Sturmian words are infinite words that have exactly n + 1 factors of length n for every positive integer n. A Sturmian word s�,� is also defined as a coding over a two-letter alphabet of the orbit of the pointunder the action of the irrational rotation R� : x 7! x + � (mod 1). Yasutomi characterized in (34) all the pairs (�,�) such that the Sturmian word s�,� is a fixed point of some non-trivial substitution. By investigating the Rauzy frac- tals associated with invertible substitutions, we give an alternative geometric proof of Yasutomi's characterization.

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Shunji Ito

Papers

Beta-expansions with negative bases.

On the invariant measure for the transformations associated with some real continued-fractions.

Discrete planes, ${\mathbb {Z}}^2$-actions, Jacobi-Perron algorithm and substitutions

Atomic surfaces, tilings and coincidence I. irreducible case

On substitution invariant sturmian words : an application of Rauzy fractals