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.

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Dynamical directions in numeration

This collaborative volume presents recent trends arising from the fruitful interaction between the themes of combinatorics on words, automata and formal language theory, and number theory. Presenting several important tools and concepts, the authors also reveal some of the exciting and important relationships that exist between these different fields. Topics include numeration systems, word complexity function, morphic words, Rauzy tilings and substitutive dynamical systems, Bratelli diagrams, frequencies and ergodicity, Diophantine approximation and transcendence, asymptotic properties of digital functions, decidability issues for D0L systems, matrix products and joint spectral radius. Topics are presented in a way that links them to the three main themes, but also extends them to dynamical systems and ergodic theory, fractals, tilings and spectral properties of matrices. Graduate students, research mathematicians and computer scientists working in combinatorics, theory of computation, number theory, symbolic dynamics, fractals, tilings and stringology will find much of interest in this book.

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Combinatorics, Automata and Number Theory

An S-adic expansion of an infinite word is a way of writing it as the limit of an infinite product of substitutions (i.e., morphisms of a free monoid). Such a description is related to continued fraction expansions of numbers and vectors. A fundamental example of this relation is between Sturmian sequences and regular continued fractions. We study S-adic words from different perspectives, namely word combinatorics, ergodic theory, and Diophantine approximation, by stressing the parallel with continued fraction expansions.

Beyond substitutive dynamical systems: S-adic expansions

The aim of this paper is to give an overview of recent results about tilings, discrete approximations of lines and planes, and Markov partitions for toral automorphisms.The main tool is a generalization of the notion of substitution. The simplest examples which correspond to algebraic parameters, are related to the iteration of one substitution, but we show that it is possible to treat arbitrary irrationalexamples by using multidimensional continued fractions.We give some non-trivial applications to Diophantine approximation, numeration systems and tilings, and we expose the main unsolved questions.

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Tilings, Quasicrystals, Discrete Planes, Generalized Substitutions, and Multidimensional Continued Fractions

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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On substitution invariant sturmian words : an application of Rauzy fractals

A generating method of self-affine tilings for Pisot, unimodular, irreducible substitutions, as well as the fact that the associated substitution dynamical systems are isomorphic to rotations on the torus are established in P. Arnoux and S. Ito. The aim of this paper is to extend these facts in the case where the characteristic polynomial of a substitution is non-irreducible for a special class of substitutions on five letters. Finally we show that the substitution dynamical systems for this class are isomorphic to induced transformations of rotations on the torus.

/pdf/tilings-from-some-non-irreducible-pisot-substitutions-2xl7sxr4mn.pdf

Tilings from some non-irreducible, Pisot substitutions

An irreducible Pisot substitution deflnes 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's fl-tiling as a subclass, are constructed. Related tiling and dynamical properties are stud- ied. Based on the coincidence condition deflned by Dekking (Dek), we introduce the super-coincidence condition. It is shown that the super-coincidence condition governs the tiling and dynamical proper- ties of atomic surfaces. We conjecture that every Pisot substitution satisfles the super-coincidence condition.

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Atomic surfaces, tilings and coincidences II. Reducible case

Using the reducible algebraic polynomial x x 1 1⁄4 ðx xþ 1Þ ðx x 1Þ, we study two types of tiling substitutions t and s : t generates a tiling of a plane based on five prototiles of polygons, and s generates a tiling of a Riemann surface, which consists of two copies of the plane, based on ten prototiles of parallelograms. Finally we claim that t -tiling of P equals a re-tiling of s -tiling of R through the canonical projection of the Riemann surface to the plane.

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Hiromi Ei

Papers

Tilings, Quasicrystals, Discrete Planes, Generalized Substitutions, and Multidimensional Continued Fractions

On substitution invariant sturmian words : an application of Rauzy fractals

Tilings from some non-irreducible, Pisot substitutions

Atomic surfaces, tilings and coincidences II. Reducible case

Tilings of a Riemann surface and cubic Pisot numbers