Combinatorics, Words and Symbolic Dynamics

Combinatorics, Words and Symbolic Dynamics Edited by Valérie Berthé and Michel Rigo

Contents

List of contributors Preface Acknowledgments

page vi 1 9

1

Preliminaries V. Berthé and M. Rigo 1.1 Words 1.2 Morphisms 1.3 Languages and machines 1.4 Symbolic dynamics

10 10 13 14 18

2

Expansions in non-integer bases M. de Vries and V. Komornik 2.1 Introduction 2.2 Greedy and lazy expansions 2.3 On the cardinality of the sets Eβ (x) 2.4 The random map Kβ and infinite Bernoulli convolutions 2.5 Lexicographic characterisations 2.6 Univoque bases 2.7 Univoque sets 2.8 A two-dimensional univoque set 2.9 Final remarks 2.10 Exercises

27 27 27 31 34 44 48 58 64 64 66

3

Medieties, end-first algorithms, and the case of Rosen continued fractions B. Rittaud 3.1 Introduction 3.2 Generalities 3.3 Examples 3.4 End-first algorithms 3.5 Medieties with k letters 3.6 An end-first algorithm for k-medieties 3.7 Exercises 3.8 Open problems

68 68 71 77 85 92 98 101 108

iv

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Repetitions in words N. Rampersad and J. Shallit 4.1 Introduction 4.2 Avoidability 4.3 Dejean’s theorem 4.4 Avoiding repetitions in arithmetic progressions 4.5 Patterns 4.6 Abelian repetitions 4.7 Enumeration 4.8 Decidability for automatic sequences 4.9 Exercises 4.10 Notes

109 109 109 122 128 131 131 142 151 153 154

5

Text redundancies G. Badkobeh, M. Crochemore, C. S. Iliopoulos and M. Kubica 5.1 Redundancy: a versatile notion 5.2 Avoiding repetitions and repeats 5.3 Finding repetitions and runs 5.4 Finding repeats 5.5 Finding covers and seeds 5.6 Palindromes

159 159 161 165 171 175 178

6

Similarity relations on words V. Halava, T. Harju and T. Kärki 6.1 Introduction 6.2 Preliminaries 6.3 Coding 6.4 Relational periods 6.5 Repetitions in relational words 6.6 Exercises and problems

183 183 184 188 194 212 218

7

Synchronised automata M.-P. Béal and D. Perrin 7.1 Introduction 7.2 Definitions ˇ y’s conjecture 7.3 Cern´ 7.4 Road colouring

220 220 221 222 236

8

Cellular automata, tilings and (un)computability J. Kari 8.1 Cellular automata 8.2 Tilings and undecidability 8.3 Undecidability concerning cellular automata 8.4 Conclusion 8.5 Exercises

248 249 267 286 300 300

9

Multidimensional shifts of finite type and sofic shfts M. Hochman 9.1 Introduction 9.2 Shifts of finite type and sofic shifts 9.3 Basic constructions and undecidability

302 302 303 311

Contents 9.4 9.5 9.6

Degrees of computability Slices and subdynamics of sofic shifts Frequencies, word growth and periodic points

v 322 331 347

10 Linearly recursive sequences and Dynkin diagrams C. Reutenauer 10.1 Introduction 10.2 SL2 -tilings of the plane 10.3 SL2 -tiling associated with a bi-infinite discrete path 10.4 Proof of Theorem 10.3.1 10.5 N-rational sequences 10.6 N-rationality of the rays in SL2 -tilings 10.7 Friezes 10.8 Dynkin diagrams 10.9 Rational frieze implies Dynkin diagram e 10.10 Rationality for Dynkin diagrams of type A and A 10.11 Further properties of SL2 -tilings 10.12 The other extended Dynkin diagrams 10.13 Problems and conjectures 10.14 Exercises

363 363 364 365 367 369 373 374 381 386 389 391 401 401 402

11 Pseudo-randomness of a random Kronecker sequence. An instance of dynamical analysis E. Cesaratto and B. Vallée 11.1 Introduction 11.2 Five parameters for Kronecker sequences 11.3 Probabilistic models 11.4 Statements of the main results 11.5 Dynamical analysis 11.6 Balanced costs 11.7 Unbalanced costs 11.8 Summary of functional analysis 11.9 Conclusion and open problems

405 405 408 419 421 427 434 440 442 445

Notation index General index

469 471

Contributors

Valérie Berthé CNRS, LIAFA, Bât. Sophie Germain, Université Paris Diderot, Paris 7 - Case 7014, F-75205 Paris Cedex 13, France. [email protected] Michel Rigo University of Liège, Department of Mathematics, B37 Quartier Polytech 1, Allée de la Découverte 12, B-4000 Liège, Belgium. [email protected] Martijn de Vries Tussen de Grachten 213, 1381DZ Weesp, the Netherlands. [email protected] Vilmos Komornik Département de mathématique, Université de Strasbourg, 7 rue René Descartes, 67084 Strasbourg Cedex, France. [email protected] Benoˆıt Rittaud Université Paris-13, Sorbonne Paris Cité, LAGA, CNRS, UMR 7539, F-93430 Villetaneuse, France. [email protected] Narad Rampersad Dept. of Mathematics and Statistics, University of Winnipeg, 515 Portage Ave., Winnipeg MB, R3B 2E9, Canada. [email protected] Jeffrey Shallit School of Computer Science, University of Waterloo, Waterloo, ON, N2L 3G1, Canada. [email protected] Golnaz Badkobeh King’s College London, London WC2R 2LS, UK. [email protected] Maxime Crochemore King’s College London, London WC2R 2LS, UK. Université Paris-Est, France [email protected] Costas S. Iliopoulos King’s College London, London WC2R 2LS, UK. [email protected] Marcin Kubica Institute of Informatics, University of Warsaw ul. Banacha 2, 02097 Warszawa, Poland. [email protected] Vesa Halava Department of Mathematics and Statistics, University of Turku, FI20014 Turku, Finland. [email protected]

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Tero Harju Department of Mathematics and Statistics, University of Turku, FI20014 Turku, Finland. [email protected] Tomi Kärki Department of Teacher Education, University of Turku, PO Box 175, FI-26101 Rauma, Finland. [email protected] Marie-Pierre Béal Université Paris-Est Marne-la-Vallée, Laboratoire d’informatique Gaspard-Monge, UMR 8049 CNRS, 5 Bd Descartes, Champs-sur-Marne, 77454 Marne-la-Vallée Cedex 2, France. [email protected] Dominique Perrin Université Paris-Est Marne-la-Vallée, Laboratoire d’informatique Gaspard-Monge, UMR 8049 CNRS, 5 Bd Descartes, Champs-sur-Marne, 77454 Marne-la-Vallée Cedex 2, France. [email protected] Jarkko Kari Department of Mathematics and Statistics, FI-20014 University of Turku, Finland. [email protected] Michael Hochman Einstein Institute of Mathematics, Edmond J. Safra Campus, Givat Ram, The Hebrew University of Jerusalem Jerusalem, 91904, Israel. [email protected] Christophe Reutenauer Département de mathématiques, Université du Québec a` Montréal, C.P. 8888, Succursale Centre-Ville Montréal, Québec H3C 3P8, Canada. [email protected] Eda Cesaratto Conicet and Univ. Nac. of Gral. Sarmiento, Instituto de Desarrollo Humano, Universidad Nacional de General Sarmiento, J. M. Gutierrez 1150 (B1613GSX), Los Polvorines, Prov. de Buenos Aires, Argentina. [email protected] Brigitte Vallée CNRS and Univ. de Caen, Informatique, Université de Caen, Bd Maréchal Juin, F-14032 Caen Cedex, France. [email protected]

Preface

Inspired by the celebrated Lothaire’s series (Lothaire, 1983, 2002, 2005) and animated by the same spirit as in the book (Berthé and Rigo, 2010), this collaborative volume aims at presenting and developing recent trends in combinatorics with applications in the study of words and in symbolic dynamics. On the one hand, some of the newest results in these areas have been selected for this volume and here benefit from a synthetic exposition. On the other hand, emphasis on the connections existing between the main topics of the book is sought. These connections arise, for instance, from numeration systems that can be associated with algorithms or dynamical systems and their corresponding expansions, from cellular automata and the computation or the realisation of a given entropy or even, from the study of friezes or, from the analysis of algorithms. This book is primarily intended to graduate students or research mathematicians and computer scientists interested in combinatorics on words, pattern avoidance, graph theory, quivers and frieze patterns, automata theory and synchronised words, tilings and theory of computation, multidimensional subshifts, discrete dynamical systems, ergodic theory and transfer operators, numeration systems, dynamical arithmetics, analytic combinatorics, continued fractions, probabilistic models. We hope that some of the chapters can serve as a useful material for lecturing at master/graduate level. Some chapters of the book can also be interesting to biologists and researchers interested in text algorithms or bio-informatics. Let us succinctly sketch the general landscape of the volume. Short abstracts of each chapter can be found below. The book can roughly be divided into four general blocks. A first one, made of Chapters 2 and 3, is devoted to numeration systems. A second block, made of Chapters 4 to 6, pertains to combinatorics of words. A third block is concerned with symbolic dynamics: in the one-dimensional setting with Chapter 7, and in the multidimensional one, with Chapters 8 and 9. The last block, made of Chapters 10 and 11, has again a combinatorial nature. Words, i.e., finite or infinite sequences of symbols taking values in a finite set, are ubiquitous in sciences. It is because of their strong representation power: they

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arise as a natural way to code elements of an infinite set using finitely many symbols. So let us start our general description with combinatorics on words. Powers, repetitions and periods are at the core of this field since its birth with the work of Thue (1906a, 1912). Thue’s work has been fruitfully extended to several important research directions. Let us mention the notion of abelian repetition (introduced by Paul Erd˝os), and the notion of fractional repetition (introduced by Françoise Dejean) leading to a famous conjecture on repetition threshold which was recently proved in 2009. We have chosen to focus on these fundamental notions in Chapters 4, 5 and 6 devoted to words. Both application-driven and theoretical viewpoints are presented. Note that the systematic study of repetitions covers a wide field of applications ranging from number theory to bio-informatics. As striking examples, let us quote the work of Novikov and Adian (1968) on the Burnside problem for groups and the work of Adamczewski and Bugeaud (2007) on the transcendence of real numbers. Chapter 4 is focused on avoidable regularities in words, which consists in avoiding some types of repetitions. This chapter also covers the use of non-effective probabilistic methods like the Lovász local lemma and introduces some decision problems about automatic sequences. Interestingly, Büchi’s theorem from 1960 and first order logic are important tools leading to decision procedures for instances of combinatorial problems that can be expressed in an extension of the Presburger arithmetic. As an example, one can get an automated certification that the Thue–Morse word is overlapfree. Chapter 5 deals with redundancies in textual data and is also built around the analysis of periodicity in words but aimed towards applications, in particular considering text algorithm, text compression, algorithms for bio-informatics, and analysis of biological sequences. It presents several methods used to detect periodicity, like the one used for compression in the Lempel–Ziv factorisation. Similarity relations on words are considered in Chapter 6 from the two perspectives of periodicity and repetition freeness. A similarity relation on words is induced by a compatibility relation on letters assumed to be a reflexive and symmetric relation. Two words are similar if they are of the same length and their corresponding letters are pairwise compatible. Similarity relations generalise the notion of partial words, i.e., words where a do-not-know symbol ⋄ may be used. As an example, the word a ⋄ ba⋄ is compatible with the word abbab. These relations can be seen as a model for inaccurate information on words. It is motivated here again both by theoretical issues and by applications arising from computer science (e.g., string matching) and molecular biology. Combinatorial problems do not only occur in this book in the framework of words but also in a more general setting of algebraic and analytic combinatorics, in particular with Chapter 10 and 11 which are combinatorial in nature. Of course, general combinatorial tools such as formal power series occur in many chapters (e.g., enumeration estimates are provided in Chapter 4). Chapter 10 belongs to algebraic combinatorics through the study of a class of sequences of natural numbers associated

Preface

3

with certain quivers (directed graphs). Quivers are commonly used in representation theory. It is possible to associate a numerical frieze with a quiver. These are integervalued sequences. They are an extension to the whole plane of the frieze patterns introduced by Coxeter (1971). These recursions, although highly non-linear, produce sometimes rational, or even N-rational, sequences. The question of rationality of the friezes is the central question which will be answered in this chapter: if friezes are rational, they must be N-rational. This chapter also introduces the notion of SL2 tilings and their applications. An SL2 -tiling of the plane is a filling of Z2 by numbers in such a way that each adjacent two by two minor is equal to 1. Chapter 11 belongs to the framework of dynamical analysis of algorithms. It focuses on the study of random Kronecker sequences through their discrepancy and their Arnold constant. It thus provides an illustration of the use of probabilistic methods in combinatorics by applying to Kronecker sequences the dynamical analysis methodology, which is a mixing between analysis of algorithms and dynamical systems relying on spectral properties of transfer operators. Recall that the use of combinatorics in the analysis of algorithms, initiated by D. E. Knuth, greatly relies on number theory, asymptotic methods and computer use. Let us also mention recent successful applications of the dynamical approach in the analysis of algorithms in connection with number theory through the analysis of the Gauss map such as illustrated, e.g., in Baladi and Vallée (2005). Combinatorics on words and symbolic dynamics are intimately related. Indeed, the coding of orbits and trajectories by words over a (finite) alphabet constitutes the basis of symbolic dynamical systems. Recall that a discrete dynamical system is a continuous map defined on a compact metric space X onto itself. It is therefore natural to code trajectories of points in the state space using a (finite) partition of X. One thus gets infinite words as codings and the corresponding dynamical systems are said to be symbolic. The study of symbolic dynamical systems in a multidimensional setting has recently given rise to striking results intertwining computational complexity, entropy, ergodic theory and topological dynamics, such as, e.g., in Hochman and Meyerovitch (2010). Theory of recursion appears to be a key tool in the study of finite type or sofic multidimensional shifts: it appears clearly that many properties of dynamical systems can thus be described in terms of recursion theory. Classical symbolic dynamics is linked with graph theory and automata theory. This is the heart of Chapter 7 with the study of the celebrated “Road Colouring Problem”. This latter problem is a classical question about synchronisation in an automaton (or a graph). A synchronising word maps every state of an automaton to the same state. An automaton is synchronised if it has a synchronising word. The Road Colouring Theorem states that every complete deterministic automaton with an aperiodic directed underlying graph has the same graph as a synchronised automaton. This result was first conjectured in (Adler et al., 1977) and has been recently solved by Trahtman (2009). The aims of Chapter 7 are to present a proof and an efficient algorithm for this problem and also to consider links existing to another famous

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ˇ y Conjecture. This conjecture asserts conjecture in automata theory, namely the Cern´ that a synchronised deterministic automaton with n states has a synchronising word of length at most (n − 1)2. Chapters 8 and 9 are articulated on multidimensional symbolic dynamics by stressing the striking fundamental differences with the one-dimensional case. One of the main features of the multidimensional case is that computations can be implemented in subshifts of finite type. Indeed, it is possible to construct for a given Turing machine T , a shift of finite type in which every configuration represents arbitrarily long computations. This yields undecidability for problems that were clearly decidable in the one-dimensional case. Striking connections with complexity, computability and decidability issues are presented. Chapter 8 focuses on cellular automata whereas Chapter 9 focuses on shifts of finite type. Both chapters are complementary and intertwined. Chapter 8 is organised around the notion of tiling linked with cellular automata. Again fundamental differences exist between one-dimensional cellular automata and multidimensional ones. These differences may be explained by the theory of tilings, like, the existence of aperiodic tilings or the fact that the so-called domino problem is undecidable. Chapter 9 is more precisely aimed at connections between combinatorial dynamical systems and effective systems, and, in particular, at aspects of multidimensional symbolic dynamics and cellular automata, including realisation theorems. Thus will be presented the characterisation of the entropy for multidimensional shifts of finite type in terms of computable real numbers proved in Hochman and Meyerovitch (2010). Let us conclude our brief presentation of the book with numeration systems. In generic way, a numeration system allows the expansion of numbers as words over an alphabet of digits. A numeration system usually is either defined by an algorithm providing expansions, or by an iterative process associated with a dynamical system. So again, words are demonstrating their representation power. Amongst the various questions related to the expansions of numbers, we have chosen to develop two focused viewpoints on numeration systems with non-integer bases. Chapter 2 deals with the possible expansions of a number that can occur when the base and the alphabet are fixed. It concentrates on the cases where such an expansion is unique. The viewpoint on numeration provided by Chapter 3 is of an arithmetic nature and relies on the notion of mediety. A mediety is a binary operation that allows to split an interval into two smaller ones and to repeat the process. Assuming that any infinite sequence of such successive intervals decreases to a single number, we get a coding of any element of an initial interval by an infinite word over a binary alphabet. Chapter 3 revisits, under the viewpoint of medieties, various classical codings and representations such as the numeration in base 2, or continued fractions, classic ones as well as Rosen ones with k-medieties. Parts of the material presented in this book where presented during the CANT school that was organised at the Centre International de Rencontres Mathématiques

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(CIRM) from 21st to 25th May 2012 in Marseille. We thank the CIRM for supporting this event that gathered more than one hundred participants from eighteen countries. We now give a short abstract of every chapter of the book. Chapter 1 is a general introduction where are presented the main notions that will occur in this book. The reader may skip this chapter in a first reading and use it as a reference if needed. Let us now move to the main contributions of this book listed by order of appearance.

Chapter 2 by M. de Vries and V. Komornik Expansions in non-integer bases The familiar integer base expansions were extended to non-integer bases in a seminal paper of Rényi in 1957. Since then many surprising phenomena were discovered and a great number of papers were devoted to unexpected connections with probability and ergodic theory, combinatorics, symbolic dynamics, measure theory, topology and number theory. For example, although a number cannot have more than two expansions in integer bases, in non-integer bases a number has generically a continuum of expansions. Despite this generic situation, Erd˝os et al. discovered in 1990 some unexpected uniqueness phenomena which gave a new impetus to this research field. The purpose of Chapter 2 is to give an overview of parts of this rich theory. The authors present a number of elementary but powerful proofs and give many examples. Some proofs presented here are new.

Chapter 3 by B. Rittaud Medieties, end-first algorithms, and the case of Rosen continued fractions A mediety is any rule that splits a given initial interval I in two subintervals, then also these intervals into two subintervals, etc., such that any decreasing sequence of such subintervals reduces to a single element. Example of medieties are the arithmetic mean, that gives rise to the base 2-numeration system, and the mediant, from which the theory of continued fractions can be recovered. Engel continued fractions provide another example. Chapter 3 investigates some general properties of medieties, as for example the question of the numbers that are approximated the most slowly by elements of the set F of bounds of intervals defined by the mediety. For example, it is well-known that the Golden Ratio is the number the most slowly approximated by rational numbers, and this property corresponds to the case of the mediant, for which F = Q+ . This chapter also introduces some end-first algorithms, that is, algorithms that provide the coding of any element of F by the mediety starting from its end instead of its beginning. In the case of the mediant, these algorithms are related to random Fibonacci sequences. Laslty, Chapter 3 presents k-medieties, that is, medieties that split intervals into k

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subintervals. In particular, it shows that λk -Rosen continued fractions, i.e., continued fractions in which the partial quotients are integral multiples of λk := 2 cos(π /k) (for an integer k ≥ 3), derive from a mediety that generalizes the mediant in a similar way that the base k generalizes the binary numeration system.

Chapter 4 by N. Rampersad and J. Shallit Repetitions in words Avoidable repetitions in words are discussed. Chapter 4 begins by a brief overview of the avoidability of the classical patterns, such as squares, cubes, and overlaps. The authors describe the most common technique used to construct infinite words avoiding these kinds of patterns, namely, the use of iterated morphisms. They also describe a probabilistic approach to avoidability based on the Lovász local lemma. Next, they consider generalisations of the classical patterns, such as fractional powers (which leads naturally to a discussion of Dejean’s theorem), repetitions in arithmetic progressions, and abelian repetitions. Some methods for counting, or at least estimating, the number of words of a given length avoiding a pattern or set of patterns are also presented in Chapter 4. Finally, the authors briefly explain an algorithmic method for obtaining computer-assisted proofs of certain types of results on automatic sequences.

Chapter 5 by G. Badkobeh, M. Crochemore, C. S. Iliopoulos and M. Kubica Text redundancies In relation with the previous chapter, Chapter 5 deals with several types of redundancies occurring in textual data. Detecting them in texts is essential in applications like pattern matching, text compression or further to extract patterns for data mining. Considered redundancies include repetitions, word powers, maximal periodicities, repeats, palindromes, and their extension to notions of covers and seeds. Main results like lower and upper bounds as well as detection algorithms on some patterns are reported.

Chapter 6 by V. Halava, T. Harju and T. Kärki Similarity relations on words The authors of Chapter 6 consider similarity relations on words that were originally introduced in order to generalise partial word, i.e., words with a do-not-know symbol. A similarity relation on words is induced by a compatibility relation on letters assumed to be a reflexive and symmetric relation. In connection with the previous two chapters, similarity of words is studied from two perspectives that are central in combinatorics on words: periodicity and repetition freeness. In particular variations of Fine and Wilf’s theorem are stated to witness interaction properties between the

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extended notions of periodicity induced by similarity relation. Also, square-freeness of words is defined for relational words and tight bounds for the repetition thresholds are given.

Chapter 7 by M.-P. Béal and D. Perrin Synchronised automata A survey of results concerning synchronised automata is presented in Chapter 7. The ˇ y Conjecture on the minimal authors first discuss the state of art concerning the Cern´ length of synchronising words. They next describe the case of circular automata and more generally one-cluster automata. A proof of the Road Coloring Theorem is also presented in Chapter 7.

Chapter 8 by J. Kari Cellular automata, tilings and (un)computability Chapter 8 reviews some basic concepts and results on the theory of cellular automata. Algorithmic questions concerning cellular automata and tilings are also discussed. Covered topics include injectivity and surjectivity properties, the Garden-of-Eden and the Curtis–Hedlund–Lyndon theorems, as well as the balance property of surjective cellular automata. The domino problem is a classical undecidable decision problem whose variants are described in the chapter. Reductions from tiling problems to questions concerning cellular automata are also covered.

Chapter 9 by M. Hochman Multidimensional shifts of finite type and sofic shfts In Chapter 9 multidimensional shifts of finite type (SFTs) are examined from the language-theoretic and recursive-theoretic point of view, by specifically discussing the recent results of Hochman-Meyerovitch on the characterization of entropies, Simpson’s realization theorem for degrees of computability, Hochman’s characterization of “slices” of SFTs (i.e., restrictions of their language to lower-dimensional lattices), the Jeandel-Vanier characterization of sets of periods of SFTs; a variety of other related developments are also mentioned. A self-contained presentation of the basic definitions and results needed from symbolic dynamics and recursion theory is also included, but this chapter also relies on Robinson’s aperiodic tile set, which is presented in Chapter 8. Modulo this, complete proofs for many of the main results above are given.

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Chapter 10 by C. Reutenauer Linearly recursive sequences and Dynkin diagrams Following an idea of Caldero, in the realm of cluster algebras of Fomin and Zelevinsky, each acyclic quiver (i.e., a directed graph) defines a sequence of integers through a highly non-linear recursion. The Laurent phenomenon of Fomin and Zelevinsky implies that the number in the sequence are integers. It turns out that for certain quivers, the sequence satisfies, besides its non-linear defining recursion, also a linear recursion. The corresponding quivers are completely classified here: they are obtained by providing an acyclic orientation to a Dynkin graph, or an extended Dynkin graph. An important tool in order to give proofs is the concept of SL2 -tilings; this is a filling of the discrete plane by numbers in such a way that each connected two by two submatrix has determinant 1.

Chapter 11 by E. Cesaratto and B. Vallée Pseudo-randomness of a random Kronecker sequence. An instance of dynamical analysis In the last chapter of this volume, the focus is put on probabilistic features of the celebrated Kronecker sequence K (α ) formed of the fractional parts of the multiples of a real α , when α is randomly chosen in the unit interval. The authors are interested in measures of pseudo-randomness of the sequence, via five parameters (two distances, covered space, discrepancy and Arnold constant), and they then perform a probabilistic study of pseudo-randomness, in four various probabilistic settings. Indeed, the authors first deal with two “unconstrained” probabilistic settings, where α is a random real, or α is a random rational. It is well-known that the behaviour of the sequence K (α ) heavily depends on the size of digits in the continued fraction expansion of α . This is why the authors also consider two “constrained” probabilistic settings, where α is randomly chosen among the reals (or rationals) whose digits in the continued fraction expansion are bounded (by some M). The corresponding probabilistic studies are performed, exhibiting a great similarity between the rational and real settings, and the transition from the constrained model to the unconstrained model is studied when M → ∞. Paris, March 2015 V. Berthé and M. Rigo

Acknowledgments ´ The editors would like to express their gratitude to Michelangelo Bucci, Emilie Charlier, Pierre Lecomte, Julien Leroy, Milton Minervino, Eric Rowland, Manon Sutera ´ and Elise Vandomme who were kind enough to read drafts of this book and who suggested many improvements. They also would like to thank their editor Clare Dennison whose constant support has been a precious help through all this project. The editors also thank Olivier Bodini and Thomas Ferniquefor the illustration on the cover of the book. This is a cropped view of a tiling by squares with two adjacent edges connected by a red wire (kind of “half Truchet-tiles”). The presence/absence of wires is governed on the top row by the Thue–Morse sequence (from left to right) and on the left column by the Fibonacci sequence (from top to bottom). These two sequences completely determine the tiling on the whole bottom-right quarter of plane, which can thus be seen as a deterministic crisscross pattern of the Thue–Morse and Fibonacci sequences.

1 Preliminaries Valérie Berthé and Michel Rigo

1.1 Words This section is only intended to give basic definitions about words. For material not covered in this book, classical textbooks on finite or infinite words and their properties are (Lothaire, 1983), (Lothaire, 2002), (Lothaire, 2005), (Allouche and Shallit, 2003), (Queffélec, 1987). See also the chapter (Choffrut and Karhumäki, 1997) or the tutorial (Berstel and Karhumäki, 2003). The book (Rigo, 2014) can also serve as introductory lecture notes on the subject.

1.1.1 Finite words An alphabet is a finite, nonempty set. Its elements are referred to as symbols or letters. In this book, depending on the specific context or conventions of a given chapter, alphabets will be denoted by capital letters like Σ or A. Definition 1.1.1 A (finite) word over Σ is a finite sequence of letters from Σ. The empty sequence is called the empty word and it is denoted by ε . The sets of all finite words, finite nonempty words and infinite words over Σ are denoted by Σ∗ , Σ+ and Σω , respectively. A word w = w0 w2 · · · wn where wi ∈ Σ, 0 ≤ i ≤ n, can be seen as a function w : {0, 1, . . . , n} → Σ in which w(i) = wi for all i. Definition 1.1.2 Let S be a set equipped with a single binary operation ⋆ : S × S → S. It is convenient to call this operation a multiplication over S, and the product of x, y ∈ S is usually denoted by xy. If this multiplication is associative, i.e., for all x, y, z ∈ S, (xy)z = x(yz), then the algebraic structure given by the pair (S, ⋆) is a semigroup. If, moreover, multiplication has an identity element, i.e., there exists some element 1 ∈ S such that, for all x ∈ S, x1 = x = 1x, then (S, ⋆) is a monoid. In addition if every element x ∈ S has an inverse, i.e., there exists y ∈ S such that xy = 1 = yx, then (S, ⋆) is a group.

Preliminaries

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Let u = u0 · · · um−1 and v = v0 · · · vn−1 be two words over Σ. The concatenation of u and v is the word w = w0 · · · wm+n−1 defined by wi = ui if 0 ≤ i < m, and wi = vi−m otherwise. We write u · v or simply uv to express the concatenation of u and v. The concatenation (or catenation) of words is an associative operation, i.e., given three words u, v and w, (uv)w = u(vw). Hence, parenthesis can be omitted. In particular, the set Σ∗ (resp., Σ+ ) equipped with the concatenation product is a monoid (resp., a semigroup). Concatenating a word w with itself k times is abbreviated by wk . In particular, 0 w = ε . Furthermore, for an integer m and a word w = w1 w2 · · · wn , where wi ∈ Σ for 1 ≤ i ≤ n, the rational power wm/n

is wq w1 w2 · · · wr , where m = qn + r for 0 ≤ r < n. For instance, we have (abbab)9/5 = abbababba.

(1.1)

The length of a word w, denoted by |w|, is the number of occurrences of the letters in w. In other words, if w = w0 w2 · · · wn−1 with wi ∈ Σ, 0 ≤ i < n, then |w| = n. In particular, the length of the empty word is zero. For a ∈ Σ and w ∈ Σ∗ , we write |w|a for the number of occurrences of a in w. Therefore, we have |w| =

∑ |w|a .

a∈Σ

A word u is a factor of a word v (resp., a prefix, or a suffix), if there exist words x and y such that v = xuy (resp., v = uy, or v = xu). A factor (resp., prefix, suffix) u of a word v is called proper if u 6= v and u 6= ε . Thus, for example, if w = concatenation, then con is a prefix, ate is a factor, and nation is a suffix. The mirror (sometimes called reversal) of a word u = u0 · · · um−1 is the word u˜ = um−1 · · · u0 . It can be defined inductively on the length of the word by ε˜ = ε and au = ua f ˜ for a ∈ Σ and u ∈ Σ∗ . Notice that for u, v ∈ Σ∗ , uv e = v˜u. ˜ A palindrome is a word u such that u˜ = u. For instance, the palindromes of length at most 3 in {0, 1}∗ are ε , 0, 1, 00, 11, 000, 010, 101, 111.

1.1.2 Infinite words Let N denote the set {0, 1, 2, . . .}. Definition 1.1.3 An (one-sided right) infinite word is a map from N to Σ. If w is an infinite word, we often write w = a0 a1 a2 · · · , where each ai ∈ Σ. The set of all infinite words of Σ is denoted Σω (one can also find the notation ΣN ). The notions of factor, prefix or suffix introduced for finite words can be extended

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V. Berthé and M. Rigo

to infinite words. Factors and prefixes are finite words, but a suffix of an infinite word is also infinite. Definition 1.1.4 A two-sided or bi-infinite word is a map from Z to Σ. The set of all bi-infinite words is denoted ω Σω (one can also find the notation ΣZ ). Example 1.1.5 Consider the infinite word x = x0 x1 x2 · · · where the letters xi ∈ {0, . . ., 9} are given by the digits appearing in the usual decimal expansion of π − 3, +∞

π − 3 = ∑ xi 10−i−1, i=0

i.e., x = 14159265358979323846264338327950288419 · · · is an infinite word. Definition 1.1.6 An infinite word x = x0 x1 · · · is (purely) periodic if there exists a finite word u = u0 · · · uk−1 6= ε such that x = uω , i.e., for all n ≥ 0, we have xn = ur where n = dk + r with r ∈ {0, . . ., k − 1}. An infinite word x is eventually periodic (or, ultimately periodic) if there exist two finite words u, v ∈ Σ∗ , with v 6= ε such that x = uvvv · · · = uvω . Notice that purely periodic words are special cases of eventually periodic words. For any eventually periodic word x, there exist words u, v of shortest length such that x = uvω , then the integer |u| (respectively |v|) is referred to as the preperiod (respectively period) of x. An infinite word is said to be non-periodic if it is not ultimately periodic. Definition 1.1.7 The language of the infinite word x is the set of all its factors. It is denoted by L(x). The set of factors of length n occurring in x is denoted by Ln (x). Definition 1.1.8 An infinite word x is recurrent if all its factors occur infinitely often in x. It is uniformly recurrent (also called minimal), if it is recurrent and for (u) (u) (u) every factor u of x, if Tx (u) = {i1 < i2 < i3 < · · · } is the infinite set of positions where u occurs in x, then there exists a constant Cu such that, for all j ≥ 1, (u)

(u)

i j+1 − i j ≤ Cu . Definition 1.1.9 One can endow Σω with a distance d defined as follows. Let x, y be two infinite words over Σ. Let x ∧ y denote the longest common prefix of x and y. Then the distance d is given by 0, if x = y, d(x, y) := 2−|x∧y| , otherwise. It is obvious to see that, for all x, y, z ∈ Σω , d(x, y) = d(y, x), d(x, z) ≤ d(x, y) + d(y, z) and d(x, y) ≤ max(d(x, z), d(y, z)). This last property is not required to have a distance, but when it holds, the distance is said to be ultrametric. Note that we obtain an equivalent distance if we replace 2 with any real number r > 1. This notion of distance extends to ΣZ . Notice that the topology on Σω is the product topology (of the discrete topology on Σ). The space Σω is a compact Cantor set, that is, a totally disconnected compact space without isolated points. Since Σω is a

Preliminaries

13

(complete) metric space, it is therefore relevant to speak of convergent sequences of infinite words. The sequence (zn )n≥0 of infinite words over Σ converges to x ∈ Σω , if for all ε > 0, there exists N ∈ N such that, for all n ≥ N, d(zn , x) < ε . To express the fact that a sequence of finite words (wn )n≥0 over Σ converges to an infinite word y, it is assumed that Σ is extended with an extra letter c 6∈ Σ. Any finite word wn is replaced with the infinite word wn ccc · · · and if the sequence of infinite words (wn ccc · · · )n≥0 converges to y, then the sequence (wn )n≥0 is said to converge to y. Let (un )n≥0 be a sequence of non-empty finite words. If we define, for all ℓ ≥ 0, the finite word vℓ as the concatenation u0 u1 · · · uℓ , then the sequence (vℓ )ℓ≥0 of finite words converges to an infinite word. This latter word is said to be the concatenation of the elements in the infinite sequence of finite words (un )n≥0 . In particular, for a constant sequence un = u for all n ≥ 0, vℓ = uℓ+1 and the concatenation of an infinite number of copies of the finite word u is denoted by uω .

1.2 Morphisms Particular infinite words of interest can be obtained by iterating morphisms (or homomorphisms of free monoids). Morphisms are also called substitutions. A map h : Σ∗ → ∆∗ , where Σ and ∆ are alphabets, is called a morphism if h satisfies h(xy) = h(x)h(y) for all x, y ∈ Σ∗ . A morphism may be specified by providing the values h(a) for all a ∈ Σ. For example, we may define a morphism h : {0, 1, 2}∗ → {0, 1, 2}∗ by 0 7→ 01201

1 7→ 020121

(1.2)

2 7→ 0212021. This domain of a morphism is easily extended to (one-sided) infinite words. A morphism h : Σ∗ → Σ∗ such that h(a) = ax for some a ∈ Σ and x ∈ Σ∗ with hi (x) 6= ε for all i is said to be prolongable on a; we may then repeatedly iterate h to obtain the infinite fixed point hω (a) = a x h(x) h2 (x) h3 (x) · · · . This infinite word is said to be purely morphic. The morphism h given by (1.2) above is prolongable on 0, so we have the fixed point hω (0) = 01201020121021202101201020121 · · · . A morphism h is non-erasing if h(a) 6= ε for all a ∈ Σ. Otherwise it is erasing. A morphism is k-uniform if |h(a)| = k for all a ∈ Σ; it is uniform if it is k-uniform for some k. Example 1.2.1 (Thue–Morse word)

For example, if the morphism µ : {0, 1}∗ →

14


{0, 1}∗ is defined by 0 7→ 01

1 7→ 10, then µ is 2-uniform. This morphism is often referred to as the Thue–Morse morphism. The fixed point t = µ ω (0) = 0110100110010110 · · · is known as the Thue–Morse word. Example 1.2.2 (Fibonacci word) Another significant example of a purely morphic word is the Fibonacci word. It is obtained from the non-uniform morphism defined over the alphabet {0, 1} by σ : 0 7→ 01, 1 7→ 0,

σ ω (0) = (xn )n≥0 = 0100101001001010010100100101001001010010100 · · · . √ It is a Sturmian word and can be obtained as follows. Let φ = (1 + 5)/2 be the Golden Ratio. For all n ≥ 1, if ⌊(n + 1)φ ⌋ − ⌊nφ ⌋ = 2, then xn−1 = 0, otherwise xn−1 = 1.

1.3 Languages and machines Formal languages theory is mostly concerned with the study of the mathematical properties of sets of words. For an exhaustive exposition on regular languages and automata theory, see (Sakarovitch, 2003) and Perrin and Pin (2004) for the connections with infinite words. Also see the chapter (Yu, 1997), or (Sudkamp, 1997), (Hopcroft and Ullman, 1979) and the updated revision (Hopcroft et al., 2006) for general introductory books on formal languages theory.

1.3.1 Languages of finite words Let Σ be an alphabet. A subset L of Σ∗ is said to be a language. Note for instance that this terminology is consistent with the one of Definition 1.1.7. Since a language is a set of words, we can apply all the usual set operations like union, intersection or set difference: ∪, ∩ or \. The concatenation of words can be extended to define an operation on languages. If L, M are languages, LM is the language of the words obtained by concatenation of a word in L and a word in M, i.e., LM = {uv | u ∈ L, v ∈ M}. We can of course define the concatenation of a language with itself, so it permits us to introduce the power of a language. Let n ∈ N, Σ be an alphabet and L ⊆ Σ∗ be a language. The language Ln is the set of words obtained by concatenating n words in L. We set L0 := {ε }. In particular, we recall that Σn denotes the set of words of length

Preliminaries

15

n over Σ, i.e., concatenations of n letters in Σ. The (Kleene) star of the language L is defined as L∗ =

[

Li .

i≥0

Otherwise stated, L∗ contains the words that are obtained as the concatenation of an arbitrary number of words in L. Notice that the definition of Kleene star is compatible with the notation Σ∗ introduced to denote the set of finite words over Σ. We also write L≤n as a shorthand for L≤n =

n [

Li .

i=0

Note that if the empty word belongs to L, then L≤n = Ln . We recall that Σ≤n is the set of words over Σ of length at most n. More can be found in Section 6.3.1 where is introduced the notion of code. Example 1.3.1 Let L = {a, ab, aab} and M = {a, ab, ba} be two finite languages. We have L2 = {aa, aab, aaab, aba, abab, abaab, aaba, aabab, aabaab} and M 2 = {aa, aab, aba, abab, abba, baa, baab, baba}. One can notice that Card(L2 ) = (Card L)2 but Card(M 2 ) < (Card M)2 . This is due to the fact that all words in L2 have a unique factorisation as concatenation of two elements in L but this is not the case for M, where (ab)a = a(ba). We can notice that L∗ = {a}∗ ∪ {ai1 bai2 b · · · ain bain+1 | ∀n ≥ 1, i1 , . . . , in ≥ 1, in+1 ≥ 0}. Since languages are sets of (finite) words, a language can be either finite or infinite. For instance, a language L differs from 0/ or {ε } if, and only if, the language L∗ is infinite. Let L be a language, we set L+ = LL∗ . The mirror operation can also be extended from words to languages: L˜ = {u˜ | u ∈ L}. Definition 1.3.2 A language is prefix-closed (respectively suffix-closed) if it contains all prefixes (respectively suffixes) of any of its elements. A language is factorial if it contains all factors of any of its elements. Obviously, any factorial language is prefix-closed and suffix-closed. The converse does not hold. For instance, the language {an b | n > 0} is suffix-closed but not factorial. Example 1.3.3 the language

The set of words over {0, 1} containing an even number of 1’s is E = {w ∈ {0, 1}∗ | |w|1 ≡ 0 (mod 2)}

= {ε , 0, 00, 11, 000, 011, 101, 110, 0000, 0011, . . .}.

16


This language is closed under mirror, i.e., L˜ = L. Notice that the concatenation E{1}E is the language of words containing an odd number of 1’s and E ∪ E{1}E = E({ε } ∪ {1}E) = {0, 1}∗. Notice that E is neither prefix-closed, since 1001 ∈ E but 100 6∈ E, nor suffix-closed. See also Example 8.1.3 and Example 9.2.9. If a language L over Σ can be obtained by applying to some finite languages a finite number of operations of union, concatenation and Kleene star, then this language is said to be a regular language. This generation process leads to regular expressions which are well-formed expressions used to describe how a regular language is built in terms of these operations. From the definition of a regular language, the following result is immediate. ∗

Theorem 1.3.4 The class of regular languages over Σ is the smallest subset of 2Σ (for inclusion) containing the languages 0, / {a} for all a ∈ Σ and closed under union, concatenation and Kleene star. Example 1.3.5 For instance, the language L over {0, 1} whose words do not contain the factor 11 is regular. It is called the Golden mean shift, see also Example 9.2.1. This language can be described by the regular expression L = {0}∗ {1}{0, 01}∗ ∪ {0}∗. Otherwise stated, it is generated from the finite languages {0}, {0, 01} and {1} by applying union, concatenation and star operations. Its complement in Σ∗ is also regular and is described by the regular expression Σ∗ {11}Σ∗. The language E from Example 1.3.3 is also regular, we have the following regular expression {0}∗({1}{0}∗{1}{0}∗)∗ describing E.

1.3.2 Automata As we shall briefly explain in this section, the regular languages are exactly the languages recognised by finite automata. Definition 1.3.6 A finite automaton is a labelled graph given by a 5-tuple A = (Q, Σ, E, I, T ) where Q is the (finite) set of states, E ⊆ Q × Σ∗ × Q is the finite set of edges defining the transition relation, I ⊆ Q is the set of initial states and T is the set of terminal (or final) states. A path in the automaton is a sequence (q0 , u0 , q1 , u1 , . . . , qk−1 , uk−1 , qk ) such that, for all i ∈ {0, . . . , k − 1}, (qi , ui , qi+1 ) ∈ E, u0 · · · uk−1 is the label of the path. Such a path is successful if q0 ∈ I and qk ∈ T . The language L(A ) recognised (or accepted) by A is the set of labels of all successful paths in A . Any finite automaton A gives a partition of Σ∗ into L(A ) and Σ∗ \ L(A ). When depicting an automaton, initial states are marked with an incoming arrow and terminal states are marked with an outgoing arrow. A transition like (q, u, r) is represented u by a directed edge from q to r with label u, q −→ r.

17

Preliminaries

Example 1.3.7 In Figure 1.1 the automaton has two initial states p and r, three terminal states q, r and s. For instance, the word ba is recognised by the automaton. There are two successful paths corresponding to the label ba: (p, b, q, a, s) and b

a

(p, b, p, a, s). For this latter path, we can write p −→ p −→ s. On the other hand, the word baab is not recognised by the automaton. b p

b

q

a a

a

a

r

a

s b

Figure 1.1 A finite automaton.

Example 1.3.8 The automaton in Figure 1.2 recognises exactly the language E of the words having an even number of 1 from Example 1.3.3. 0

0 1 q

p 1

Figure 1.2 An automaton recognising words with an even number of 1.

Definition 1.3.9 Let A = (Q, Σ, E, I, T ) be a finite automaton. A state q ∈ Q is accessible (respectively co-accessible) if there exists a path from an initial state to q (respectively from q to some terminal state). If all states of A are both accessible and co-accessible, then A is said to be trim. Definition 1.3.10 A finite automaton A = (Q, Σ, E, I, T ) is said to be deterministic (DFA) if it has only one initial state q0 , if E is a subset of Q × Σ × Q and for each (q, a) ∈ Q × Σ there is at most one state r ∈ Q such that (q, a, r) ∈ E. In that case, E defines a partial function δA : Q × Σ → Q that is called the transition function of A . The adjective partial means that the domain of δA can be a strict subset of Q × Σ. To express that the partial transition function is total, the DFA can be said to be complete. To get a total function, one can add to Q a new ‘sink state’ s and, for all (q, a) ∈ Q × Σ such that δA is not defined, set δA (q, a) := s. This operation does

18


not alter the language recognised by A . We can extend δA to be defined on Q × Σ∗ by δA (q, ε ) = q and, for all q ∈ Q, a ∈ Σ and u ∈ Σ∗ , δA (q, au) = δA (δA (q, a), u). Otherwise stated, the language recognised by A is L(A ) = {u ∈ Σ∗ | δA (q0 , u) ∈ F} where q0 is the initial state of A . If the automaton is deterministic, it is sometimes convenient to refer to the 5-tuple A = (Q, Σ, δA , I, T ). As explained by the following result, for languages of finite words, finite automata and deterministic finite automata recognise exactly the same languages. Theorem 1.3.11 (Rabin and Scott, 1959) If L is recognised by a finite automaton A , there exists a DFA which can be effectively computed from A and recognising the same language L. A proof and more details about classical results in automata theory can be found in textbooks like (Hopcroft et al., 2006), (Sakarovitch, 2003) or (Shallit, 2008). For standard material in automata theory we shall not refer again to these references below. One important result is that the set of regular languages coincides with the set of languages recognised by finite automata. Theorem 1.3.12 (Kleene, 1956) A language is regular if, and only if, it is recognised by a (deterministic) finite automaton. Observe that if L, M are two regular languages over Σ, then L ∩ M, L ∪ M, LM and L \ M are also regular languages. In particular, a language over Σ is regular if, and only if, its complement in Σ∗ is regular. Example 1.3.13 The regular language L = {0}∗ {1}{0, 01}∗ ∪ {0}∗ from Example 1.3.5 is recognised by the DFA depicted in Figure 1.3. Notice that the state s is a sink: non-terminal state and all transitions remain in s. 0

0, 1

1 0

1

s

Figure 1.3 A DFA accepting words without factor 11.

1.4 Symbolic dynamics Let us introduce some basic notions in symbolic dynamics. For expository books on the subject, see (Cornfeld et al., 1982), (Kitchens, 1998), (Lind and Marcus, 1995a),

19

Preliminaries

(Perrin, 1995) and (Queffélec, 1987). For references on ergodic theory, see also e.g., (Walters, 1982).

1.4.1 Codings of dynamical systems A (discrete) dynamical system is a pair (X, T ) where T : X → X is a map acting on a convenient space X (e.g., X is a topological space or a metric space, in the usual setting, X is generally compact and T is continuous). We are interested in iterating the map T and we look at orbits (T n (x))n≥0 of points in X under the action T . The trajectory of x ∈ X is the sequence (T n (x))n≥0 . Roughly speaking, infinite words appear naturally as a convenient coding (with a priori some loss of information) of these trajectories (T n (x))n≥0 . So one can gain insight about the dynamical system by studying these words, with an interplay between combinatorics on words and dynamics. In that setting, the space X is discretised, i.e., it is partitioned into finitely many sets X1 , . . . , Xk and the trajectory of x is thus coded by the corresponding sequence of visited subsets, such as illustrated in Figure 1.4. Precisely, the coding of (T n (x))n≥0 is the word wx = w0 w1 w2 · · · over the alphabet {1, . . . , k} where wi = j if and only if T i (x) ∈ X j . Even though the infinite word wx contains less information than the original trajectory (T n (x))x≥0 , this discretised and simplified version of the original system can help us to understand the dynamics of the original system. X4 X3 T (x) X2 T 2 (x)

x X1

Figure 1.4 Trajectory of x in a space X = X1 ∪ X2 ∪ X2 ∪ X4 .

Example 1.4.1 (Rotation words) One of the simplest dynamical systems can be obtained from the coding of a rotation on a circle identified with the interval [0, 2π ). Instead of working modulo 2π , it is convenient to normalise the interval [0, 2π ) and consider instead the interval [0, 1). Hence we shall consider the map1 Rα : [0, 1) → [0, 1), x 7→ {x + α } where α is a fixed real number in [0, 1). To get a coding of this system, we consider a 1

The interval [0,1) is identified with the quotient set R/Z whose elements r + Z are in one-to-one correspondence with real numbers in this interval. It is more convenient to work with R/Z in mind to avoid discontinuity problems. In the literature, the map Rα is sometimes referred to a translation on the one-dimensional torus.

20


partition of [0, 1). For instance, take two real numbers γ1 , γ2 such that γ0 := 0 < γ1 < γ2 < 1 =: γ3 and define X0 = [0, γ1 ), X1 = [γ1 , γ2 ) and X2 = [γ2 , 1). In Figure 1.5, we have chosen α = 3/(8π ) ≃ 0.119, γ1 = 1.8/(2π ) ≃ 0.286, γ2 = 4.3/(2π ) ≃ 0.684 and x = 0.08. For a given x ∈ [0, 1), the coding of the trajectory is the word wx = γ1 R(x)

x

0

γ2

Figure 1.5 The first few points of a trajectory under a rotation R of angle α .

w0 w1 · · · where wi = j if and only if Ri (x) belongs to [γ j , γ j+1 ). In our example represented in Figure 1.5, we get wx = 001111220001112 · · ·. With such a setting, we get interesting words when the angle α of rotation is irrational. Indeed, a rational number would only produce periodic orbits.

1.4.2 Beta-expansions In this section, we consider two important examples of codings of systems connected to numeration. Let us first consider the base-b expansion of real numbers. Given a real number x ∈ [0, 1), the algorithm in Table 1.1 provides a sequence (ci )i≥0 of digits in {0, . . . , b − 1} such that x = ∑ ci b−i−1. i≥0

i ←0 y ←x REPEAT FOREVER ci ← ⌊by⌋ y ← {by} INCREMENT i END-REPEAT.

Table 1.1 An algorithm for computing the base-b expansion of x ∈ [0, 1).

21

Preliminaries In this algorithm, we iterate a map from the interval [0, 1) onto itself, i.e., Tb : [0, 1) → [0, 1), y 7→ {by}

(1.3)

and the value taken by the image determines the next digit in the expansion. The interval [0, 1) is thus split into b subintervals [ j/b, ( j + 1)/b), for j = 0, . . . , b − 1. For all i ≥ 0, if Tbi (x) belongs to the subinterval [ j/b, ( j + 1)/b), then the digit ci occurring in repb (x) is equal to j. It is indeed natural to consider such subintervals. If y belongs to [ j/b, ( j + 1)/b), then by has an integer part equal to j and the map Tb is continuous and increasing on every subinterval [ j/b, ( j + 1)/b). Note also that the range of Tb on any of these subintervals is [0, 1). So applying Tb to a point in one of these subintervals can lead to a point belonging to any of these subintervals (later on, we shall introduce some other transformation, such as e.g., β -transformations, where a restriction appears on the intervals that can be reached). So to speak, the base-b expansion of x can be derived from the trajectory of x under Tb , i.e., from the sequence (Tbn (x))n≥0 . As an example, consider the base b = 3 and the expansion of x = 3/10. The point lies in the interval [0, 1/3); thus the first digit of the expansion is 0. Then T3 (3/10) = 9/10 lies in the interval [2/3, 1); thus the second digit is 2. If we apply again T3 , we get T32 (3/10) = {27/10} = 7/10, which belongs again to [2/3, 1) giving the digit 2. Then T33 (3/10) = 1/10 giving the digit 0 and finally T34 (3/10) = 3/10. So rep3 (3/10) = (0220)ω . The map T3 is depicted in Figure 1.6 on the three intervals [0, 1/3), [1/3, 2/3) and [2/3, 1) and we make use of the diagonal to apply the map T3 iteratively. 1.0

0.8

0.6

0.4

0.2

0.2

0.4

0.6

0.8

1.0

Figure 1.6 The dynamics behind the transformation T3 .

A natural generalisation of base-b expansion, is to replace the base b with a real

22


number β > 1. In particular, the transformation Tb will be replaced by the so-called β -transformation. Note that we shall be concerned with expansions of numbers in [0, 1). If x ≥ 1, then there exists a smallest d such that x/β d belongs to [0, 1). It is therefore enough2 to concentrate on [0, 1). Definition 1.4.2 (β -expansions) We will only represent real numbers in the interval [0, 1). Let β > 1 be a real number. The representations discussed here are a direct generalisation of the base-b expansions. Every real number x ∈ [0, 1) can be written as a series +∞

x = ∑ ci β −i−1

(1.4)

i=0

where the ci belong to {0, ⌈β ⌉ − 1}. Recall that ⌈·⌉ denotes the ceiling function, i.e., ⌈x⌉ = inf{z ∈ Z | z ≥ x} . Note that if β is an integer, then ⌈β ⌉ − 1 = β − 1. For integer base b expansions that a number may have more than one representation, namely those ending with 0ω or (b − 1)ω . For a real base β , we obtain many more representations. Consider the Golden Ratio φ , which satisfies φ 2 − φ − 1 = 0 and thus 1 1 1 = n+1 + n+2 , n φ φ φ

∀n ≥ 0 .

As an example, the number 1/φ has thus infinitely many representations as a power series with negative powers of φ and coefficients 0 and 1, 1 1 1 1 1 1 1 1 1 1 = 2 + 3 = 2 + 4 + 5 = 2 + 4 + 6 + 7 = ··· . φ φ φ φ φ φ φ φ φ φ To get a canonical expansion for a real x ∈ [0, 1), we just have to replace the integer base b with β and consider the so-called β -transformation Tβ : [0, 1) → [0, 1), x 7→ {β x} in the algorithm from Table 1.1. For i = 0, 1, . . ., the idea is to remove the largest integer multiple ci of β −i−1 , and then repeat the process with the remainder and the next negative power of β to get (1.4). Note that ci is less than ⌈β ⌉ because of the greediness of the process. Otherwise, one could have removed a larger multiple of a power of β at a previous step. The corresponding infinite word c0 c1 · · · is called the β -expansion of x and is usually denoted by dβ (x). Any word d0 d1 · · · over a finite alphabet of non-negative integers satisfying +∞

x = ∑ di β −i−1 i=0

2

If the β -expansion of x/β d is d0 d1 ··· , then using an extra decimal point, the expansion of x is conveniently written d0 ··· dℓ−1 • dℓ dℓ+1 ··· . Note that the presentation in Chapter 2 is not entirely consistent with our present treatment if x belongs to [0,1/(β − 1)] \ [0,1).

Preliminaries

23

is said to be a β -representation of x. Thus, the β -expansion of x is the lexicographically maximal word amongst the β -representations of x. The greediness of the algorithm can be reformulated as follows. Lemma 1.4.3 A word d0 d1 · · · over {0, . . . , ⌈β ⌉ − 1} is the β -expansion of a real number x ∈ [0, 1) if and only if, for all j ≥ 0, +∞

∑ di β −i−1 < β − j .

i= j

Proposition 1.4.4 Let x, y be real numbers in [0, 1). We have x < y if and only if dβ (x) is lexicographically less than dβ (y). Remark 1.4.5 The map Tβ provides a classical example of a fibred system: as defined in (Schweiger, 1995), a fibred system is a pair (T, B) where T is a transformation on the set B, where B = ∪ j∈J B j is a partition of B with J finite or countable such that T|B j is injective for any j. For more on fibered systems, see also Remark 3.2.4. Note that a further example of a fibred dynamical system is provided in Chapter 11 with the Gauss map that produces partial quotients in the continued fraction expansion.

1.4.3 Subshifts The sets of infinite sequences obtained as codings of dynamical systems produce themselves dynamical systems, with the map acting on them being the shift. They are called symbolic since they are defined on words. Let S denote the following map3 defined on Σω , called the one-sided shift: S((xn )n≥0 ) = (xn+1 )n≥0 . In particular, if x = x0 x1 x2 · · · is an infinite word over Σ, then, for all n ≥ 0, its suffix xn xn+1 · · · is simply Sn (x). The map S is uniformly continuous, onto but not one-toone on Σω . This notion extends in a natural way to ΣZ . In this latter case, the shift S is one-to-one. We thus get symbolic dynamical systems. The definitions given below correspond to the one-sided shift, but they extend to the two-sided shift. Definition 1.4.6 Let x be an infinite word over the alphabet Σ. The orbit of x under the action of the (left) shift S is defined as the set O(x) = {Snx | n ∈ N}. The symbolic dynamical system associated with x is then defined as (O(x), S), where O(x) ⊆ Σω is the closure of the orbit of x. 3

Note that in Chapter 3, the notation σ will also be used for the shift.

24


In the case of bi-infinite words we similarly define O(x) = {Sn x | n ∈ Z} where the (two-sided) shift map is defined on ΣZ . The set Xx := O(x) is a closed subset of the compact set Σω , hence it is a compact space and S is a continuous map acting on it. One checks that, for every infinite word y ∈ Σω , the word y belongs to Xx if, and only if, L(y) ⊆ L(x). For a proof, see (Queffélec, 1987) or Chapter 1 of (Pytheas Fogg, 2002). Note that O(x) is finite if, and only if, x is eventually periodic. Generic examples of symbolic dynamical systems are provided by subshifts (also called shifts for short). Let Y be a closed subset of Σω that is stable under the action of the shift S. The system (Y, S) is called a subshift. If Y is a subshift, there exists a set F ⊂ Σ∗ of finite words such that an infinite word x belongs to X if, and only if, none of its factors belongs to F . A subshift X is called a subshift of finite type if one can choose the set F to be finite. The full shift is defined as (Σω , S). Subshifts are discussed in more details in Chapter 8 and Chapter 9 (see in particular Section 9.2.2). A subshift (X, S) is said to be periodic if there exist x ∈ X and an integer k such that X = {x, Sx, . . ., Sk x = x}. Otherwise it is said to be aperiodic. If (Y, S) is a subshift, then there exists a set X ⊆ Σ∗ such that for every x ∈ Σω , the infinite word x belongs to Y if, and only if, L(x) ∩ X = 0. / A subshift Y is said to ∗ be of finite type if the set X ⊆ Σ is finite. A subshift is said to be sofic if the set X is a regular language. For more on shifts of finite type, see Chapter 9, and see also Chapter 8. Example 1.4.7 The set of infinite words over {0, 1} of Example 1.3.5 which do not contain the factor 11 is a subshift of finite type, whereas the set of infinite words over {0, 1} having an even number of 1’s between two occurrences of the letter 0 is a sofic subshift which is not of finite type. See also Example 8.1.3 and Example 9.2.9. Example 1.4.8 We let Dβ denote the set of β -expansions of reals in [0, 1). From Lemma 1.4.3, we know that this set is shift-invariant, i.e., if w belongs to Dβ , then S(w) also belongs to Dβ . We easily get the following commutative diagram. x ∈ [0, 1)   β -expansiony

Tβ

−−−−→ S

Tβ (x) ∈ [0, 1)  β -expansion y

dβ (x) ∈ Dβ −−−−→ dβ (Tβ (x)) ∈ Dβ

Therefore it is natural to consider the closure of Dβ . This set denoted by Sβ is called the β -shift and we can consider the dynamical system (Sβ , S). Definition 1.4.9 Let Y be subshfit. For a word w = w0 · · · wr , the cylinder set [w]Y is the set {y ∈ Y | y0 = w0 , · · · , yr = wr }. If the context is clear, the subscript Y will be omitted. The cylinder sets are clopen (open and closed) sets and form a basis of open sets for the topology of Y . Furthermore, one checks that a clopen set is a finite union

Preliminaries

25

of cylinders. In the bi-infinite case the cylinders are the sets [u.v]Y = {y ∈ Y | yi = ui , y j = v j , −|u| ≤ i ≤ −1, 0 ≤ j ≤ |v| − 1} and the same remarks hold.

1.4.4 Topological and measure-theoretic dynamical systems There are two main types of dynamical systems, namely topological ones and measuretheoretic ones. Definition 1.4.10 A topological dynamical system (X, T ) is defined as a compact metric space X together with a continuous map T defined onto the set X. Subshifts are examples of topological dynamical systems. A topological dynamical system (X, T ) is minimal if, for all x in X, the orbit of x, i.e., the set {T n x | n ∈ N}, is dense in X. Let us note that if (X, S) is a subshift, and if X is furthermore assumed to be minimal, then X is periodic if, and only if, X is finite. Moreover, if x is an infinite word, (Xx , S) is minimal if, and only if, x is uniformly recurrent. Indeed, w is a factor of x, we write O(x) =

[

S−n [w],

n∈N

and we conclude by a compactness argument. Two dynamical systems (X1 , T1 ) and (X2 , T2 ) are said to be topologically conjugate (or topologically isomorphic) if there exists an homeomorphism f from X1 onto X2 which conjugates T1 and T2 , that is: f ◦ T1 = T2 ◦ f . If f is only onto, then (X1 , T1 ) is said to factor onto (X2 , T2 ), (X2 , T2 ) is a factor of (X1 , T1 ), and f is called a factor map. For more on topological conjugacy, see also Section 9.2.3 in the higher-dimensional setting. See also Theorem 9.2.7. We have considered here the notion of dynamical system, that is, a map acting on a given set, in a topological context. This notion can be extended to measurable spaces: we thus get measure-theoretic dynamical systems. For more details, one can refer for instance to (Walters, 1982). Definition 1.4.11 A measure-theoretic dynamical system is defined as a system (X, T, µ , B), where B is a σ -algebra, µ a probability measure defined on B, and T : X → X is a measurable map which preserves the measure µ , i.e., for all B ∈ B, µ (T −1 (B)) = µ (B). Such a measure is said T -invariant and the map T is said to preserve the measure µ . A measure-theoretic dynamical system (X, T, µ , B) is ergodic if for every B ∈ B such that T −1 (B) = B, then B has either zero measure or full measure. Let (X, T ) be a topological dynamical system. A topological system (X, T ) always has an invariant probability measure. The case where there exists only one

26


T -invariant measure is of particular interest. A topological dynamical system (X, T ) is said to be uniquely ergodic if there exists one and only one T -invariant Borel probability measure over X. In particular, a uniquely ergodic topological dynamical system yields an ergodic measure-theoretic dynamical system. A measure-theoretic ergodic dynamical system satisfies the Birkhoff ergodic theorem, also called individual ergodic theorem. Let us recall that the abbreviation a.e. stands for “almost everywhere”: a property holds almost everywhere if the set of elements for which the property does not hold is contained in a set of zero measure. Theorem 1.4.12 (Birkhoff Ergodic Theorem) Let (X, T, µ , B) be a measure-theoretic k dynamical system. Let f ∈ L1 (X, R). Then the sequence ( n1 ∑n−1 k=0 fR◦T )n≥0 converges R ∗ ∗ 1 ∗ ∗ a.e. to a function f ∈ L (X, R). One has f ◦ T = f a.e. and X f d µ = X f d µ . Furthermore, if T is ergodic, since f ∗ is a.e. constant, one has: ∀ f ∈ L1 (X, R) ,

1 n−1 µ −a.e. f ◦ T k −−−−→ ∑ n→∞ n k=0

Z

f dµ.

X

Note that the notions of conjugacy and factor between two topological dynamical systems extends in a natural way to this context.

2 Expansions in non-integer bases Martijn de Vries and Vilmos Komornik

2.1 Introduction The familiar integer base expansions were extended to non-integer bases in a seminal paper of Rényi (1957a). Since then many surprising phenomena were discovered and a great number of papers were devoted to unexpected connections with probability and ergodic theory, combinatorics, symbolic dynamics, measure theory, topology and number theory. It has generally been believed for a long time that for any given q ∈ (1, 2) there are infinitely many expansions of the form c1 c2 c3 1 = + 2 + 3 + ··· q q q with digits ci ∈ {0, 1}. Erd˝os et al. (1991) made the startling discovery that for a continuum of bases q ∈ (1, 2) there is only one such expansion. This gave a new impetus to this research field. The purpose of this chapter is to give an overview of parts of this rich theory. We present a number of elementary but powerful proofs and we give many examples. Some proofs presented here are new. In most papers dedicated to ergodic and probabilistic questions, following Rényi (1957a) the base was denoted by β . On the other hand, following Erd˝os and his collaborators, most papers dealing with combinatorial and topological aspects use the letter q for the base. We keep ourselves to this tradition here: the base is denoted by β in the second, third and fourth section of this chapter, then the base will be denoted by q starting with Section 2.5. For some other important aspects of the theory, not discussed here, we refer the reader to the surveys Sidorov (2003b) and Komornik (2011).

2.2 Greedy and lazy expansions Following the pioneering works of Rényi (1957a) and Parry (1960), many works during the last fifty years were devoted to the study of expansions in non-integer

28

M. de Vries and V. Komornik

bases. Given a real number β ∈ (1, 2], a β -expansion or an expansion in base β (if the base β is understood from the context, we simply speak of expansions) of a real number x is a sequence (ci ) = c1 c2 · · · of digits ci ∈ {0, 1} such that ∞

ci . i β i=1

x=∑

Note that x must belong to Jβ := [0, 1/(β − 1)] ⊇ [0, 1]. If β = 2, each x ∈ Jβ = [0, 1] has only one β -expansion, except for the dyadic rationals in (0, 1) (numbers of the form x = i/2n with n ≥ 1 and 0 < i < 2n ) which have exactly two expansions. However, the situation is much more complicated if β ∈ (1, 2). For instance, it was proved by Sidorov (2003a) that almost every1 x ∈ Jβ has a continuum of expansions (see also Dajani and de Vries (2007)) and for each number x in the remaining null set the number of possible expansions may be countably infinite or may have any finite cardinality, depending on β (see Erd˝os and Joó (1992), Erd˝os et al. (1990)). Each number x ∈ Jβ has at least one β -expansion, namely the greedy β -expansion b(x, β ) = (bi ) = (bi (x)) = (bi (x, β )) introduced by Rényi (1957a) which can be defined recursively by the greedy algorithm: if the digits b1 , . . . , bn−1 have already been determined for some positive integer n (no condition for n = 1), then bn = 1 if, and only if, n−1

bi

1

∑ β i + β n ≤ x.

i=1

Loosely speaking, the greedy algorithm chooses in each step the largest possible digit. Let us show that (bi ) is an expansion of x for each x belonging to Jβ . If x = 1/(β − 1), then bi = 1 for each i ≥ 1, whence (bi ) is indeed an expansion of x. If 0 ≤ x < 1/(β − 1), then bn = 0 for some index n and for each such n we have ∞ 1 1 bi < ≤ . ∑ i n i β β β i=n+1 i=1 n

0 ≤ x− ∑

Hence there cannot be a last n such that bn = 0. Letting n → ∞ along the indices n for which bn = 0, we see that (bi ) is indeed an expansion of x. We equip the set Eβ (x) consisting of all possible β -expansions of x with the lexicographic order. The greedy expansion of a number x ∈ Jβ is obviously the largest element of Eβ (x). The lazy β -expansion (ei (x)) = (ei ) of a number x ∈ Jβ is the smallest element of Eβ (x). Note that (ci ) is an expansion of x if, and only if, (1 − ci) = (1 − c1)(1 − c2) · · · is an expansion of 1/(β − 1) − x. This means in particular that (ei (x)) = (1 − bi(1/(β − 1) − x)) for x ∈ Jβ . The lazy expansion can be obtained by applying the lazy algorithm: if e1 , . . . , en−1 are already defined for some positive integer n, then en = 0 if, and only if, n−1

x≤ 1

ei

∞

1 . i β i=n+1

∑ βi + ∑

i=1

When there is no reference to a measure where there should be, we always mean the Lebesgue measure.

Expansions in non-integer bases

29

Until Section 2.5, the base β is a fixed but arbitrary number strictly between 1 and 2, unless stated otherwise. Let the greedy map Gβ : Jβ → Jβ , be given by  β x

h if x ∈ G(0) := 0, β1 , h i Gβ (x) := β x − 1 if x ∈ G(1) := 1 , 1 . β β −1

One easily verifies that the greedy map generates the greedy expansion in the sense that bn (x) = j if, and only if, Gβn−1 (x) ∈ G( j), j = 0, 1. Similarly, the lazy expansion is generated by the lazy map Lβ : Jβ → Jβ , defined by  β x

h i if x ∈ L(0) := 0, β (β1−1) , i Lβ (x) := 1 1 β x − 1 if x ∈ L(1) := , β (β −1) β −1 .

We denote by µgβ the extended Gβ -invariant greedy measure (see Gel′ fond (1959); Parry (1960)) on Jβ which is absolutely continuous with density gβ (x) =

1 ∞ 1 h ∑ β n 1 0,Gβn (1)(x), F(β ) n=0

x ∈ Jβ ,

where F(β ) is a normalising constant so that µgβ (Jβ ) = 1. It is well known that the dynamical system (Jβ , µgβ , Gβ ) is ergodic. Define the lazy measure µℓβ on Jβ by setting µℓβ = µgβ ◦ rβ , where rβ : Jβ → Jβ is given by rβ (x) =

1 − x. β −1

Notice that rβ ◦ Gβ = Lβ ◦ rβ . Hence rβ is a continuous isomorphism between the dynamical systems (Jβ , µgβ , Gβ ) and (Jβ , µℓβ , Lβ ) and ℓβ := gβ ◦ rβ is a density of µℓ β . The normalised errors of an expansion (ci ) of x are defined by ! n c i θn ((ci )) = β n x − ∑ i , n ∈ N. i=1 β A straightforward application of Birkhoff’s ergodic theorem, see Theorem 1.4.12, yields that for µgβ -almost all x ∈ Jβ the limit 1 n−1 j ∑ Gβ (x) n→+∞ n j=0 lim

exists and equals 1 n−1 ∑ θ j ((bi (x))) = n→+∞ n j=0

Mg := lim

Z

Jβ

ygβ (y)dy.

Observe that the density gβ is positive on the interval [0, 1). Moreover, for each

30


x ∈ [1, 1/(β − 1)), Gnβ (x) ∈ [0, 1) for sufficiently large n. Since the map Gβ is nonsingular2, we may conclude that for almost every x ∈ Jβ the limit above exists, and equals Mg . Similarly, we may define for almost every x ∈ Jβ , 1 n−1 1 n−1 j θ j ((ei (x))) = lim ∑ ∑ Lβ (x) = n→+∞ n n→+∞ n j=0 j=0

Mℓ := lim

Z

Jβ

yℓβ (y)dy.

The following result, from Dajani and Kraaikamp (2002), shows that “on average” the greedy convergents ∑ni=1 bi (x)β −i approximate x better than the lazy convergents ∑ni=1 ei (x)β −i for almost every x ∈ Jβ . Theorem 2.2.1 One has Mg + Mℓ = 1/(β − 1) and Mg < Mℓ . Proof The first statement follows directly from the relation between gβ and ℓβ : Z 1/(β −1) Z 1/(β −1) 1 − x dx Mℓ = x ℓβ (x) dx = x gβ β −1 0 0 Z 1/(β −1) 1 1 = − y gβ (y) dy = − Mg . β −1 β −1 0 For the second statement, notice that by definition of Mg and the monotone convergence theorem, one has that 1 ∞ Mg = ∑ F(β ) n=0

Z Gn (1) x β 0

2 n 1 ∞ (Gβ (1)) dx = ∑ 2β n . βn F(β ) n=0

Furthermore, Z 1/(β −1)

Z

1 ∞ 1/(β −1) x ∑ r Gn (1) β n dx F(β ) n=0 0 β β 2 2 1 n 1 ∞ β −1 − rβ (Gβ (1)) = . ∑ F(β ) n=0 2β n

Mℓ =

xℓβ (x)dx =

The second statement now follows from the observation that for every n ≥ 0 one has that 2 1 2 2 n − rβ (Gβn (1)) , Gβ (1) ≤ β −1

where the ≤ sign can be replaced by a < sign for those n ≥ 0 for which Gnβ (1) 6= 0, such as n = 0 or n = 1. Motivated by Theorem 2.2.1, we call an expansion (di ) of x optimal if θn ((di )) ≤ θn ((ci )) for each n = 1, 2, . . . and for each expansion (ci ) of x. Since the greedy expansion is the lexicographically largest expansion of x, it is clear that only the greedy expansion can be optimal. The following example shows that the greedy expansion 2

Non-singularity of a map T : Jβ → Jβ means that T −1 (E) is a null set whenever E ⊆ Jβ is a null set.

Expansions in non-integer bases

31

of a number x ∈ Jβ is not always optimal. Other examples can be found in Dajani and Kraaikamp (2002). √ Example 2.2.2 Let φ := (1 + 5)/2 be the Golden Ratio and consider a base β ∈ (1, φ ). Note that 1 < β −1 + β −2 . The sequence (ci ) := 0110∞ is clearly an expansion of x := β −2 + β −3 . Applying the greedy algorithm we find that the first three digits of the greedy expansion (bi ) of x equal 100. Hence θ3 ((bi )) > θ3 ((ci )) = 0. Let P be the countable set consisting of the Golden Ratio φ = φ2 and the Multinacci numbers φ3 , φ4 , . . . ∈ (1, 2), defined by the equations 1=

1 1 + ···+ k , φk φk

k = 2, 3, . . . .

Theorem 2.2.3 (Dajani et al., 2012) We have the following dichotomy. • If β ∈ P, then each x ∈ Jβ has an optimal expansion. • If β ∈ (1, 2) \ P, then the set of numbers x ∈ Jβ with an optimal expansion is a nowhere dense null set. Remark 2.2.4 Let β ∈ (1, 2) \ P. The proof of Theorem 2.2.3 in (Dajani et al., 2012) shows that there exists a (non-degenerated) interval I contained in [0, 1), such that no number x ∈ I has an optimal expansion. Clearly a number belongs to I if its greedy expansion starts with b := b1 (x′ ) · · · bn (x′ ) for some x′ in the interior of I and for some large enough n. It was shown in (de Vries and Komornik, 2011) that such a block b occurs in the greedy expansion of every x ∈ Jβ , except for a set of Hausdorff dimension less than one. Since each tail of an optimal expansion must be optimal, it follows that the set of numbers x ∈ Jβ with an optimal expansion has in fact Hausdorff dimension less than one.

2.3 On the cardinality of the sets Eβ (x) We recall from the preceding section that Eβ (x) denotes the set of all possible β expansions of a number x ∈ Jβ . Let us first show that one does not need the continuum hypothesis in order to determine in general the cardinality of Eβ (x). Theorem 2.3.1 A number x ∈ Jβ with uncountably many expansions, necessarily has a continuum of expansions. Proof Give each coordinate {0, 1} the discrete topology and endow the set D := {0, 1}∞ with the Tychonoff product topology. One easily verifies that Eβ (x) is a closed subset of the Polish space3 D. Hence Eβ (x) is a Polish space, too. The assertion follows from the well known fact that uncountable Polish spaces have the cardinality of the continuum. 3

A Polish space is a topological space that is homeomorphic to a complete separable metric space.

32


We already mentioned that Eβ (x) has the cardinality of the continuum for almost every x ∈ Jβ . This provides a sharp contrast with the fact that almost all numbers have a unique binary expansion. We will now present a simple proof of this result using the dynamical properties of the greedy and the lazy map. Theorem 2.3.2 Almost every x ∈ Jβ has a continuum of β -expansions. Proof Let Uβ be the set of numbers x ∈ Jβ with a unique β -expansion. For ease of notation we denote in this proof the lazy and the greedy map by T0 and T1 , respectively. We also set Tu1 ···un = Tun ◦ · · · ◦ Tu1 for u1 , . . . , un ∈ {0, 1} and n = 1, 2, . . .. Since the greedy and the lazy map are non-singular, the set Nβ :=

∞ [

n=1

x ∈ Jβ | Tu1 ···un (x) ∈ Uβ for some u1 , . . . , un ∈ {0, 1}

is a null set because Uβ is a null set as follows for instance from Theorem 2.2.1. One may verify that if (ci ) is an expansion of a number x ∈ Jβ , then (cn+i ) = cn+1 cn+2 · · · is an expansion of the the number Tc1 ...cn (x) for each n ≥ 1. It follows that none of the expansions of a number x ∈ Jβ \ Nβ has a unique tail. Hence if (ci ) is an expansion of a number x ∈ Jβ \ Nβ , then there exists an index n such that x has expansions (εi ) and (ρi ) starting with c1 · · · cn−1 0 and c1 · · · cn−1 1, respectively. Similarly, there exists an index m > n such that x has expansions starting with ε1 · · · εm−1 0 and ε1 · · · εm−1 1, respectively. Continuing in this manner, we can construct recursively a full binary tree of possible expansions of x, whence Card Eβ (x) = 2ℵ0 for all x ∈ Jβ \ Nβ . Remark 2.3.3 • The set Uβ has in fact Hausdorff dimension less than one, see for instance Glendinning and Sidorov (2001); de Vries and Komornik (2011). Since the branches of the greedy and the lazy map are similarities, we have dimH (E) = dimH Gβ−1 (E) = dimH L−1 β (E) (E ⊆ Jβ ), where dimH denotes the Hausdorff dimension. The countable stability of the Hausdorff dimension yields that dimH (Uβ ) = dimH (Nβ ), whence every x ∈ Jβ has a continuum of expansions, except for a set of Hausdorff dimension less than one. A more general version of this argument can be found in Sidorov (2007), Proposition 3.8. • For each E ⊆ Jβ , we have −1 G−1 β (E) ∪ Lβ (E) =

1 1 · E ∪ · (E + 1). β β

−1 It follows that G−1 β (E) ∪ Lβ (E) is a closed null set whenever the set E is. The

results in Section 2.7 together with Exercise 2.10.8 imply that the closure Uβ of the set Uβ is a null set. Since a closed null set is nowhere dense, we conclude that the set Nβ is of the first category and therefore the set

x ∈ Jβ | Card Eβ (x) = 2ℵ0

33

Expansions in non-integer bases is also residual4 in Jβ .

Let Eβn (x) be the set of all possible prefixes of length n of sequences belonging to Eβ (x). More precisely, we set Eβn (x) := (c1 , . . . , cn ) ∈ {0, 1}n | ∃(cn+1 , cn+2 , . . .) ∈ {0, 1}∞ so that (ci ) ∈ Eβ (x)

and Nn (x, β ) := Card Eβn (x). The final result of this section strengthens Theorem 2.3.2 for β strictly between 1 and the Golden Ratio φ . Parts (i) and (ii) of Theorem 2.3.4 below are established in Erd˝os et al. (1990) and Feng and Sidorov (2011), respectively. Our proof of part (ii) is primarily based on the proof of part (i) and differs in this respect from the proof in (Feng and Sidorov, 2011). Note that the result is sharp in the sense that Card Eφ (1) = ℵ0 . One may check that Eφ (1) = {(10)n 110∞ | n = 0, 1, . . .} ∪ {(10)∞ } ∪ {(10)n 01∞ | n = 0, 1, . . .} . Theorem 2.3.4 is also sharp in a more interesting sense (see Proposition 5.5 in (Feng and Sidorov, 2011)): there exists a continuum of numbers x ∈ Jφ such that Card Eφ (x) = 2ℵ0 , yet lim

n→+∞

log2 (Nn (x, φ )) = 0. n

Theorem 2.3.4 If 1 < β < φ , then (i) Card Eβ (x) = 2ℵ0 for each x in the interior of Jβ ; (ii) there exists a positive constant c = c(β ) such that the inequality lim inf n→+∞

log2 (Nn (x, β )) ≥c n

holds for each x in the interior of Jβ . Proof (i) Since 1 < β < φ , there exists an index k such that 1 < β −2 + · · · + β −k .

(2.3.1)

Let (mi ) = m1 m2 · · · be the strictly increasing sequence consisting of all positive integers which are not multiples of k. Then for each ℓ = 1, 2, . . ., we have ∞

β −mℓ tM holds, the operator HM,(s+c/2,−b/2) is well-defined at s = σM , and the spectral decomposition at s = σM leads to the estimate HkM,(σM +c/2,−b/2) [1](x) ∼ λMk (σM + υ /2) φM,(σM +c/2,−b/2) (x) , which gives rise to the estimate E[M] [Rk ] ∼ λMk (σM + υ /2)

Z 1 0

φM,(σM +c/2,−b/2)(u) dνM (u) .

11.7.3 Step 2 in the rational model [M]

The Dirichlet series SR (s) is convergent in the half-plane ℜs > sM . For any complex number s with ℜs > sM , and close enough to the real axis, we use, as in (11.8), the dominant spectral behaviour of the operator HkM,s . This gives rises to a dominant part [M]

for the Dirichlet series SR (s), in the same vein as in (11.9). We obtain [M]

SR (s) = φM,s (0) · ΣM (s) · ΨM (s) + HM (s) where the first term collects all the dominant terms p−F(p)

ΣM (s) := ∑ λM

F(p)

(s) λM

(s + υ /2),

p

ΨM (s) := QM,s [x 7→ φM,(s+c/2,−b/2) (x)] ,

and the “remainder” term HM (s) collects the three other terms, each of them containing at least one sub-dominant term. In the present case, the admissible function intervenes in the leading term, and this is why we consider, as in (Daireaux and Vallée, 2004; Lhote and Vallée, 2008), particular admissible functions F(p) of the form F(p) = ⌊δ p⌋, with δ ∈ Q ∩ [0, 1], already defined in Section 11.3.3. Consider indeed a rational δ := B/C defined with two coprime integers B,C which satisfy B ≤ C. Then, the series ΣM is written as ΣM (s) = ℓM (s, δ ) ∑ LCp M (s, δ ) = p≥1

ℓM (s, δ , υ ) , 1 − LCM (s, δ , υ )

with

442

E. Cesaratto and B. Vallée C−1

ℓM (s, δ , υ ) =

j−⌊δ j⌋

∑ λM

⌊δ j⌋

(s)λM

(s + υ /2),

LM (s, δ , υ ) = λM1−δ (s)λMδ (s + υ /2) .

j=0

The poles of the Dirichlet series ΣM (s) are brought by the zeroes of the map s 7→ LCM (s, δ , υ ) − 1 where C is a positive integer. The map s 7→ LM (s, δ , υ ) is an extension of a map which has been already studied in (Lhote and Vallée, 2008). And the long version of this chapter (Cesaratto and Vallée, 2014) provides a proof of all the properties which are stated in Theorem 11.4.4 for the unique real zero σM (υ , δ ) of the equation LM (s, δ , υ ) − 1.

11.8 Summary of functional analysis We now recall some basic facts from functional analysis concerning the operators under study. We consider a (possibly infinite) integer M, and we denote by GM,s a generic element of the set GM,s , defined as GM,s := {HM,s }

[

{HM,(s+t,−t) ; t ∈ R} .

11.8.1 Generalities Operators in GM,s may act on functions of one or two variables. The integer q denotes the number of variables. It always equals 2 for the operator GM,s except for the (plain) operator HM,s , where it equals 1. More precisely, we consider the Banach spaces C 1 (I q ), endowed with the following norms defined from the sup-norm on I q , denoted by ||.||0 : || f ||1 = || f ||0 + || f ′ ||0

for q = 1,

||F||1 = ||F||0 + ||DF||0

for q = 2 .

We will also use another norm, the (1, τ ) norm, defined later in (11.3). There are three cases for the constraint M: the two possible cases M < ∞, M = ∞, but we are also interested in the “transition” when M → ∞. There are two important real numbers which depend on the integer M. (i) The convergence abscissa sM defines the (half)-plane ℜs > sM where the operator GM,s is well-defined: this abscissa sM equals −∞ for M < ∞ and equals 1/2 for M = ∞. (ii) The real σM is the real s for which the dominant eigenvalue λM (s) of the operator GM,s equals 1. The operators GM,s ∈ GM,s share many properties with the plain operator HM,s . We describe their main analytic properties, in particular their spectral properties, and focus on the behaviour of these operators when the parameter s equals σM .

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Pseudo-randomness of a random Kronecker sequence

11.8.2 Dominant spectral properties and spectral decomposition Each operator HM,s (when s is close to the real axis) or HM,(s,t) (when s + t is close to the real axis) is quasi-compact and has spectral dominant properties. The operator HM,s has a unique dominant eigenvalue λM (s), which is simple. This is also the dominant eigenvalue of the adjoint operator H∗M,s . The dominant eigenvector of HM,s is denoted by φM,s , the dominant eigenmeasure of the adjoint H∗M,s is denoted by QM,s , and there is a normalisation condition QM,s [φM,s ] = 1. For s close to the real axis, one has HkM,s [g](x) ∼ λMk (s)φM,s (x)QM,s [g] .

(11.1)

The operator HM,(s+t,−t) has a unique dominant eigenvalue λM (s +t, −t), which is simple and equal to λM (s). This is also the dominant eigenvalue of the adjoint operator H∗M,(s+t,−t) . The dominant eigenvector of HM,(s+t,−t) is denoted by φM,(s+t,−t) , the dominant eigenmeasure of the adjoint H∗M,(s,t) is denoted by QM,(s+t,−t) , and there is a normalisation condition QM,(s+t,−t) [φM,(s+t,−t) ] = 1. For s close to the real axis, one has HkM,(s+t,−t) [G](x, y) ∼ λMk (s)φM,(s+t,−t) (x, y)QM,(s+t,−t) [G] .

(11.2)

In Equation (11.1) or (11.2), the quasi-compactness of operators entails that the remainder term is of order |λM (s)|k O(ρ k ), with ρ < 1 and a hidden constant that is uniform with respect to s, for s close enough to the real axis, and uniform with respect to M, for M large enough. Proofs of the previous assertions on the space C 1 (I q ) closely follow Baladi (2000), given for the (constrained and unconstrained) plain transfer operator on the space of Hölder continuous functions. The “quasi-compacity” property is the first key step in the proof of the spectral decomposition. It is obtained with Hennion’s theorem. In Baladi and Vallée (2005), the central hypothesis of Hennion’s theorem (also known as Lasota-Yorke bounds) is proven to hold (for the plain transfer operator) in the space C 1 (I ). The paper Vallée (1997) establishes the spectral decomposition for the operator H(s,t) in the space of analytic functions and the paper Cesaratto et al. (2006) in the space C 1 (I 2 ). The proofs can be easily adapted to the constrained operator HM,(s,t) .

11.8.3 Special values at s = σM For any M possibly infinite, the equation λM (s) − 1 has a unique solution on the real axis, located at s = σM . The value σM is the Hausdorff dimension of the constrained set I [M] . The dominant eigenmeasure of the adjoint operator H⋆M,s at s = σM coincides with the Hausdorff measure νM of the set I [M] . For M = ∞, the eigenfunction φ1 is the Gauss density and −λ ′ (1) is the entropy Z π2 1 1 φ1 (x) = f (w)dw, and λ ′ (1) = − , Q1 [ f ] = . log 2 1 + x 6 log2 I

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The sections of the eigenfunctions φ(2,−1) and φ(3/2,−1/2) satisfy 3 log 2 · Π[φ(2,−1)](x) = (1 + x)−1 + (1 + x)−2 + (1 + x)−3 ,

2 log 2 · Π[φ(3/2,−1/2)](x) = (1 + x)−1 + (1 + x)−2 .

For M → ∞, the dominant spectral objects of the constrained operators H⋆M,s tend to the spectral objects of the unconstrained one H⋆s with a speed of order O(1/M). Gauss himself proved that φ1 is a fixed point of H1 (see e.g., Knuth (1998)). The description of eigenfunctions for the operator H(s,t) is given in (Vallée, 1997, Theorem 5). The eigenfunctions φ(2,−1) and φ(3/2,−1/2) are computed in Cesaratto et al. (2006).

11.8.4 When s is far from the real axis There are two cases, according as σ := ℜs simply satisfies σ > sM or is furthermore close to 1. For any M ≥ 2, for any σ > sM , on the vertical line ℜs = σ , with s 6= σ , one has, according to Vallée (1998) ||GM,s [F]||0 < λM (σ )||F||0 ,

for any F ∈ C 1 (I q ) .

We now consider Dolgopyat bounds when σ is close to 1. These bounds deal with the (1, τ ) norm, defined as ||F||(1,τ ) := ||F||0 +

1 ||F||1 . |τ |

(11.3)

For any τ0 > 0, there exist α , β > 0, M0 > 0, K > 0, for which, when s belongs to the part of the vertical strip |ℜs − 1| ≤ β , with τ := ℑs satisfying |τ | > τ0 > 0, the norm (1, τ ) of the n-th iterate of any operator GM,s ∈ GM,s satisfies, for M ≥ M0 ||GnM,s ||(1,τ ) ≤ K · γ n · |τ |α ,

for n ≥ 1.

This was first proved in Dolgopyat (1998) for plain transfer operators (associated with subshifts of finite type) whose adjoints fix the Lebesgue measure for s = 1. Then Baladi and Vallée (2005) extended the result to the case when the dynamical system has an infinite number of branches (the case of the Gauss map). The paper Cesaratto and Vallée (2011) provides a detailed proof of Dolgopyat–type estimates in the constrained case for large enough M and with uniform parameters (with respect to M). This result is obtained via perturbation techniques (for M → ∞) and is not a priori valid for small values of M, where sM is not close to 1. In Cesaratto et al. (2006), techniques from the paper Baladi and Vallée (2005) are adapted in order to extend Dolgopyat’s estimates to the case q = 2.


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11.8.5 Basic properties of the ζ function (τ )

The weighted operator HM,s , used to study the “middle” digit, is quite close (when τ (m) = me ) to the (possibly truncated) Riemann zeta function ζM (s − e). The plain Riemann ζ function is analytic on ℜs > 1, has a pole at s = 1, with a residue equal to 1. It is of polynomial growth (with respect to ℑs) when ℜs is close to 1. For M < ∞ and for real s < 1 with |s − 1| log M ≤ 1, the truncated ζ function satisfies (see Edwards (2001))

ζM (s) = log M + γ + O(|1 − s| log2 M) .

11.9 Conclusion and open problems Our results precisely describe the pseudo-randomness of a random Kronecker sequence K (α ) with five parameters, for two-distance truncations and in various probabilistic models (real versus rational, constrained or not constrained). Extension to three-distance truncations. We only deal here with two-distance truncations. For any three-distance truncation, and for all the four parameters, except the discrepancy, there exist general formulae (of the same type as the present ones) which express the parameter XhT i as a function of the truncation T and the four continuants qk , qk−1 , θk , θk+1 (for a truncation of index k). Any three-distance truncation may be described with two positions: the principal position µ , already used here, and another (auxiliary) position. This is why a similar study can be conducted in this case, with similar expected results (and heavier computations). The special case of the discrepancy. The situation is completely different for the discrepancy. For three-distance truncations, there does not exist a formula which expresses ∆hT i for a truncation of index k solely with T, qk , qk−1 , θk , θk+1 . However, the paper (Baxa and Schoissengeier, 1994) provides estimates that relate the discrepancy ∆hT i (α ) for an index k to the average Ak := (1/k) ∑ki=1 mi of the first k digits in the continued fraction expansion of α . We are thus led to consider constrained models of another type, which deal with the sets Ie[M] of real numbers for which each average Ak is bounded by some constant M. We may also consider their rational counterparts e [M] . In these models, we may expect a logarithmic behaviour for the mean discrepΩ N ancy. In the paper (Cesaratto and Vallée, 2006), we already studied the set Ie[M] and provided estimates for its Hausdorff dimension, with tools of dynamical analysis. Quadratic numbers. Other particular Kronecker sequences K (α ), associated with quadratic irrational numbers α , are also interesting to study from our probabilistic point of view. Such sequences are usually classified as the most random, because of their “low” worst-case discrepancy. For dynamical analysis, there is a strong parallelism between rational and irrational quadratic numbers, as it is shown for instance in (Vallée, 1998): we just replace the quasi-inverse (I − Hs )−1 by the zeta function of

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the dynamical system. It is then surely possible to conduct similar analyses among quadratic irrational numbers, with similar expected results.


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Notation index

2X (power set), 189 X s Y (X is strongly reducible to Y ), 326 X w Y (X is eakly reducible to Y ), 326 [a] (cylinder set), 305 [n]q (least positive residue), 199 [p] (cylinder determined by pattern p), 252 x ↑ y (compatible partial words), 187 ? (question mark function), 76 [1,(−1 : 1)i−1 ]λ , 96 [a0 ,... ,an ]1 , 79 [a0 ,a1 ,...]λ , 96 [a0 ,e1 : a1 ,e2 : a2 ,...]λ , 96 [(a j )mj=0 ,(a j ) j>m ], 96 [(an )n ]λ , 96 ]q1 ,q2 ,q3 ,... [, 82 hr1 ,... ,rn i (generated similarity relation), 186 AhT i (Arnold measure), 415 Bt1 ,t2 (p,q) (bound of interaction), 198

c (coding function), 74 CA (cellular automaton), 249 cD (restriction of configuration c in domain D), 249 ci,D (pattern of domain D extracted from position i in configuration c), 249 C(x), 76 d (distance on words), 12 ∆n (complexity class of real numbers), 348 DhT i , ∆hT i (discrepancy), 413 D(w) (domain of a partial word), 187

ε (the empty word), 10 ECA (elementary cellular automaton), 250 F⊥ (restriction of CA F on ⊥-finite configurations), 254 F, 71 FN , 73 FP (restriction of CA F on periodic configurations), 251 ˚ 71 F,

fσ (X) (upper frequency of σ in X), 349 fσ (X,n) (upper frequency of σ in n-cubes in X), 349 fσ (a) (frequency of σ in a), 349 FW (p,q) (extremal relational Fine and Wilf words), 209 ΓhT i (distances in a Kronecker sequence), 412 Gb (greedy map), 29 h(X) (topological entropy of X), 355 HM,s (constrained transfer operator), 421 Hs (plain transfer operator), 421 Hs (transfer operator), 407 H(s,t) (extended transfer operator), 430 (τ )

Hs (weighted transfer operator), 430 H(w) (set of holes), 187 I (closed interval), 71 ι (identity relation), 186 Iw , 73 k′ (= k − 1, for k-medieties), 92 κ (a) (Kolmogorov complexity of a), 330 K (α ) (Kronecker sequence), 405, 408 KhT i (α ) (truncated Kronecker sequence), 409

λk (for k-Rosen continued fractions), 95 Lc (X) (complementary language of a subshift), 305 Lc (X) complementary language of a set, 324 LFT (linear fractional transformation), 428 L≤n (concatenation of at most n words in L), 15 L(X) (language of a set), 324 L(X) (language of a subshift), 305 L(x) (language of a word), 12 Ln (x) (factors of length n), 12 LPF (longest previous factor occurrence), 179 LPrF (longest previous reverse factor occurrence), 179 L∗ (Kleene star), 15 Ln (power of a language), 14 M, 71

470

Notation index

µℓβ (lazy measure), 29 NFA (non-deterministic finite automaton), 265 Nn (X) (number of n-words in X), 354 NPunary (complexity class NP in unary, 360 νM (Hausdorff measure), 419 Ω (universal relation), 186 uω (concatenation), 13 ⊕, 71 ⊕i , 93

πD (higher block code with shape D), 309 Πn (complexity class of real numbers), 348 πR,e (x) (minimal external R-period), 194 πR,g (x) (minimal global R-period), 194 πR,l (x) (minimal local R-period), 194 Πsquare (set of square periods of a subshift), 360 π (w) (minimal period), 184 R (end-first tree), 85 ρ (n) (maximal number of runs in a word of length n), 170 ri (k-Rosen mediety), 95 Rk , 98 RY (restriction of R on Y ), 185

S (shift), 251 SFT (subshift of finite type), 252 Σ∗d , 303 σM (Hausdorff dimension), 419 Σn (complexity class of real numbers), 348 stab(x) (stabilizer of x), 360

τt (translation), 251 u˜ (mirror image), 11 L˜ (mirror image), 15 T (w) (w word), 75 v (value function), 74 VhT i (covered space), 412 ← − (w finite word), 80 w w (w word), 75 w⋄ (companion word), 187 x (x number), 75 X(·) (subshift defined by forbidden patterns), 303 X + (·) (subshift defined by allowing patterns), 305 XP (subshift determined by set P of forbidden patterns), 252

General index

2-Toeplitz family, 351 sequence, 351 a.e., see almost everywhere abelian cube, 131 square, 131 Aberkane, A., 154, 157 accessible state, 17 additive form (of the Euclidean algorithm), 79 function, 382 Adian, S. I., 109 adjacency matrix, 132 Adler, R. L., 236 admissible frontier, 366 pattern, 303 Akiyama, S., 65 Allouche, J.-P., 52, 116, 118, 152, 154, 158 almost everywhere, 26 Alon, N., 155 alphabet, 10 integer, 168 alteration relation, 190 aperiodic, 237 automaton, 223 graph, 237 tile set, 270 Apostolico, A., 166, 175 appearance (of a pattern), 303 approximate square, 155 arithmetic progression, 128, 155, 156 arithmetic subsequence, 128 Arnold measure, 406, 414, 415, 422, 424, 439 Arnold, V. I., 414 Arnoux, P., 71 Assem, I., 364 Aubrin, N., 334 augmenting word, 229

automatic sequence, 151 automaton, 115, 154, 221 aperiodic, 223 circular, 228 complete, 17 complete deterministic, 221 deterministic, 17, 221 equivalent, 237 irreducible, 221 level, 228, 239 one cluster, 228 strongly transitive, 228 synchronised, 221 trim, 17 Béal, M.-P., 220, 243 Baatz, M., 63 Badkobeh, G., 162, 164 Baiocchi, C., 44 Baker, S., 64 Baladi, V., 407, 443 balanced, 140 basic sequence, 433, 434, 440 basis, 376 Bean, D. A., 120, 155 Beck, J., 121, 155 Bell, J. P. , 157 Berger’s theorem, 312, 322 Berger, R., 270, 274, 312 Bergeron, F., 365 Berlinkov, M. V., 220 Berman, S., 363, 382 Bernoulli measure, 255, 259 Berstel, J., 154, 183, 187, 232, 369 β -expansion, 22, 28 greedy, 28 lazy, 28 β -representation, 23 β -shift, 24 β -transformation, 22 Birkhoff ergodic theorem, 26, 29

472 Blanchet-Sadri, F., 183 block, 124, 133 Blondel, V., 157 Boasson, L., 183, 187 Bodini, O., 9 border, 160 Borwein, P., 66 bound of t1 -t2 interaction, 198 Bousquet-Mélou, M., 370 Brandenburg, F.-J., 154, 155 Breslauer, D., 176 Brocot, A., 69 Brown, S., 154 Bruyère, V., 158 Büchi, J. R., 158 Bugeaud, Y., 71 bunch, 238 Burnside problem, 109 Burton, R., 71 Caldero, P., 363 Cantor set, 12, 56, 419, 429 Cardano, G., 102 Carpi, A., 155, 156, 220, 233 Cartan matrix, 384 ceiling function, 22 cellular automaton, 249, 346, 362 balance theorem, 258, 263 bijective, 254 CA rule controlled xor, 293 controlled-xor, 263 Game-Of-Life, 248, 261, 262, 300 majority, 259, 261 Q2R, 255, 259 rule 110, 250, 257, 259, 260, 264–266 Snake xor, 292 traffic, 250, 261 xor, 250, 260, 263 configuration, 249 asymptotic, 259 finite, 254 fixed point, 252 fully periodic, 251 Garden of Eden, 256, 301 periodic, 251 temporally periodic, 252, 301 conservation law, 256 elementary, 250 entropy, 299 equicontinuous, 299 Garden of Eden theorem, 259 injective, 254, 260, 266 invariant measure, 259 inverse automaton, 254 local rule, 250

General index neighbourhood, 250 Moore, 251 radius- 12 , 251 radius-r, 251 von Neumann, 251 nilpotent, 287, 301 orphan, 256 periodic, 295, 301 permutive, 260 preinjective, 259, 261 reversible, 254, 260 sensitive, 299 space-time diagram, 250, 296 stable state, 254 state, 249 surjective, 254, 260, 261, 263, 266 Wolfram number, 250 ˇ y’s conjecture, 222 Cern´ ˇ y, J., 222 Cern´ Cesaratto, E., 406 Chairungsee, S., 181 Chapoton, F., 401 ´ 152, 158 Charlier, E., Chen, A., 334 Chen, G., 168 child i-th left, 98 Christoffel word, 101 standard factorisation, 101 Chuquet, N., 69 clopen set, 24 cluster, 144, 228, 239 generating function, 144 method, 143, 146, 157 co-accessible state, 17 code, 190, 245 Huffman, 246 maximal, 245 prefix, 245 R-, 191 (R,S)-, 190 codewords, 245 coding, 73 codon, 188 compactness, 252 companion word, 187 compatibility relation, 185 compatible, 185 complementary language of a set, 324 of a subshift, 305 complete automaton, 17 computable function, 323, 325 real number, 347

General index concatenation, 11 configuration, 249, 303 finite, 254 fully periodic, 251 periodic, 251, 312 square period of, 312 temporally periodic, 252, 301 congruence stable, 238 stable pair, 238 conjugacy, 269 conjugate digit, 48 expansion, 48 letter, 93 of a word, 75 continued fraction expansion, 69, 79, 405, 409, 427 α -continued fraction, 79 backward, 85 beginning continuants, 409, 428, 435 depth, 409, 420, 427 digits, 406, 409, 427 ending continuants, 428, 435 partial quotients, 406 remainders, 428 Rosen, 70 Conway, J., 261, 364 Cosnard, M., 52 cost balance, 418 balanced, 418, 432, 434, 438 digit-cost, 417, 430, 437, 438 elementary, 417 extremal, 438 imbalance, 418, 425, 440, 441 ordinary, 438 standard, 438 unbalanced, 418, 421, 432, 440 cover, 159, 175 covered space, 406, 412, 422, 423, 439 Coxeter, H. S. M., 364 critical exponent, 152, 154 position, 201 Crochemore, M., 155, 162, 166, 168, 181, 212 cube, 110 abelian, 131 cubefree word, 110 Culik, II, K., 238, 270 Currie, J. D., 116, 118, 154–157, 159 Curtis, M, 307 Curtis-Hedlund-Lyndon theorem, 254 cylinder, 24, 252

473

set, 305 D’Alessandro, F., 220 Daireaux, B., 412 Dajani, K., 28, 30, 31, 71 Damanik, D., 154 Daróczy, Z., 44, 63 Daudé, H., 425 de Bruijn graph, 264 diagonal, 266 diamond, 267 pair graph, 265 de la Rue, T., 71 de Luca, A., 154, 209, 210 de Vries, M., 28, 31, 55 de Vries–Komornik constant, 55 decidability, 151 decision problem CA INJECTIVITY, 267, 293 CA NILPOTENCY, 287, 299 CA PERIODICITY, 295 CA SURJECTIVITY, 268, 296 C OMPLETION PROBLEM, 301 D OMINO PROBLEM, 270 F INITE TILING PROBLEM, 301 G ARDEN OF E DEN P ROBLEM, 301 P ERIODIC TILING PROBLEM, 286 T ILING PROBLEM WITH A SEED TILE, 273 T ILING PROBLEM, 268, 270, 281, 299 T URING MACHINE HALTING ON BLANK TAPE, 271 decoder, 245 of a prefix code, 245 Dejan, F., 2 Dejean’s conjecture, 155 Dejean’s theorem, 122, 124 Dejean, F., 123, 155, 159 Dekking, F. M., 133, 135, 154 dependency digraph, 121 DFA, 17 Di Francesco, P., 364 diagonal, 266 diagonal ray, 373 diamond, 267 directing vector of a ray, 373 Dirichlet series, 407, 432, 441 discrepancy, 405, 406, 413, 422, 424, 439 discriminant, 395 distance, 12 ultrametric, 12 distances, 405, 406, 408, 409, 412, 425, 426 Dolgopyat bounds, 433, 444 Dolgopyat, D., 433 dominating eigenvalue, 372 domino tilings, 304 Downarowicz, T., 156

474

General index

Dubuc, L., 220, 228 Durand, B., 295, 334 dynamical analysis, 407, 412, 427, 431 dynamical system conjugacy, 25 measure-theoretic, 25 symbolic, 23 topological isomorphism, 25

extension problem (tilings and SFTs), 311 extension-reduction, 314 external R-period, 194 word, 194 extremal Fine and Wilf word, 209 relational Fine and Wilf word, 209

E -admissible, 303 edge, 221 a-edge, 238 label of, 221 effective dynamical system, 346 set, 323 subset, 346 subshift, 331 effectively closed set, 323 Ehrenfeucht, A., 120, 155, 175 eigenvalues of a rational sequence, 372 Eilenberg, S., 223, 372 emptiness problem (tilings and SFTs), 311, 322 empty word, 10 end-first algorithm, 85 tree, 85 Engel, F., 70 Engel-Sierpiński mediety, 83 Entringer, R. C., 156 entropy (of a subshift), 355 enumeration overlapfree words, 142 squarefree words, 142 Eppstein,D., 220 equivalence relation, 185 equivariant, 306 Erd˝os, P., 27, 40, 64, 131 ergodic, 25 Birkhoff theorem, 26 individual ergodic theorem, 26 Euclidean algorithm, 78 additive form, 79 Euclidean dynamical system, 407, 421, 427 Evdokimov, A. A., 131 even shift, 253 eventually periodic word, 12 excess, 127 expansion, 28, 82 β -, 22 Engel-Sierpiński, 82 greedy, 28 lazy, 28 exponent of a word, 111 exponential polynomial, 372 extension (of a subshift), 306

f-factorisation, 167 factor of a word, 11 subshift, 306 proper factor of a word, 11 factor map, 306 factorial language, 15 factorisation X-factorisation, 190 f-factorisation, 167 standard, 169 Ziv-Lempel, 168 Fekete’s Lemma, 349 Feng, D.-J., 31 Fibonacci word, 14, 154 fibred system, 23, 72 fidelity relation, 190 Fife’s theorem, 115 Fife, E. D., 115, 154 final state, 16 Fine, N. J., 183 finite automaton, 17, 265 finite tiling, 301 Flajolet, P., 407, 425 Fomin, S., 363 forbidden factor, 154 fractional power, 111, 155 Fraenkel, A. S., 161 frieze, 374 of type G, 375 patterns, 364 fringe, 394 frontier, 366 full shift, 24, 252 Game-Of-Life, 248, 261, 262, 300 Garden of Eden, 256, 301 Garden of Eden theorem, 259 Garsia number, 41 Garsia, A. M., 41 Gauss map, 103, 407, 427 generating function, 143 Glendinning, P., 32, 63 global R-period, 194 Goh, T. L., 157 golden mean shift, 16, 303 Golden Ratio, 14, 31, 75

General index Golod, E., 157 Goodwyn, L.W., 236 Gottschalk, W. H., 155 Goulden, I., 143 Graham, R., 156 graph aperiodic, 237 period of, 237 greedy expansion, 28 grid, 317 group, 10 growing morphism, 125 Grytczuk, J., 155, 156 ˇ 360 Gurevich, Ju. S., Haas, A., 71 Hadamard product, 371 Halava, V., 183, 210 Haluszczak, M., 155 Hancart, C., 172 hard core model, 304 Hare, K. G., 66 Harju, T., 183, 210 Hausdorff dimension, 419, 425, 443 measure, 419, 443 Hedlund, G., 155, 307 higher block code, 309 higher block presentation, 264 Hilbert curve, 289 Hippocrates of Chios, 70 Hochman, M., 334, 346, 354, 358, 362 Hohlweg, C., 169 hole, 187 Hopcroft, J. E., 265 horizontal ray, 373 Horváth, M., 27 Hubert, P., 71 Huffman code, 246 Ilie, L., 166–168 Iliopoulos, C. S., 177, 182 image, 239 in an automaton, 239 minimal, 239 independent set, 228 individual ergodic theorem, 26 infix, 119 inite-repetition threshold, 163 initial quiver, 376 integer alphabet, 168 interlacing, 371 interval stability, 60 invariant measure, 25 inverse CA, 254

isomorphism subshifts, 306 topological, 25 Jackson, D. E., 156 Jackson, D. M., 143 Janvresse, E., 71 Jeandel, E., 346, 360 Joó, I., 27, 64 Jungers, R., 157 k-automatic sequence, 151 k-power, 110 k-powerfree, 110 k-recognisable sequence, 151 k-slice (of a subshift), 333 Kärki, T., 183, 210, 216 Kallós, G., 63 Kao, J.-Y., 156 Karhumäki, J., 162, 238 Kari, J., 222, 235, 238, 270, 274, 295 Keller, B., 401 Kempton, T., 37 Keränen, V., 132 kernel repetition, 127 Kleene star, 15 Kleene, S. C., 18 Kolmogorov complexity, 330 Kolpakov, R., 167 Komatsu, T., 66 Komornik, V., 27, 44, 48, 64 Korjakov, I. O., 360 Kozik, J., 155, 156 Kraaikamp, C., 30, 31, 71 Krieger, D., 154 Kubica, M., 182 Kucherov, G., 167 Kátai, I., 44, 63 label, 221 Lai, A. C., 65 Landau theorem, 407, 433 language, 14 complementary, 305, 324 factorial, 15 finite, 15 infinite, 15 of a closed set, 324 of a subshift, 305 prefix-closed, 15 suffix-closed, 15 Laurent phenomenon, 375 polynomial, 376 lazy expansion, 28 measure, 29

475

476

General index

Lecroq, T., 172 left form (of a bound of a k-Rosen interval), 97 Lenz, D., 154 letter, 10 level of automaton, 228, 239 of state, 228, 239 Lhote, L., 412 Lind, D., 356, 361 linearisation coefficients, 391 Linek, V., 154 linked set, 226 local R-period, 194 partial period, 195 locally strongly transitive, 228 longest previous reverse factor occurrence, 179 Lorentz, R. J., 166 Loreti, P., 31, 48 Lovasz local lemma, 121, 153, 155 lower entropy dimension, 361 Lyndon tree, 169 word, 168 Lyndon, R., 307 M-balanced, 140 Main, M. G., 166 marked word, 143 matrix adjacency, 132 Mauldin, R. D., 41 maximal code, 245 McNulty, G., 120, 155 measure invariant, 25 measure-theoretic dynamical system, 25 mediant, 77 mediety, 71 k-Rosen, 95 k-mediety, 93 Engel-Sierpiński, 83 Stern-Brocot, 77 Medvedev degree, 326 Medvedev reducible, 326 merge, 371 Meyerovitch, T., 354, 358, 361 Micek, P., 155 Michaux, C., 158 Mignosi, F., 154, 209, 210 minimal dynamical system, 25 image, 239 period, 184 word, 12

minimal subshift, 331, 361 Minkowski, H., 76 Mione, L., 154 mirror, 11 Mohammad-Noori, M., 155 monoid, 10 of binary relations, 223 Moody, R., 363, 382 Moore, D., 177 Moore, E. F., 259 Mor, S. J., 154 morphism, 135 growing, 125 powerfree, 119 squarefree, 119 uniform, 124, 151 Morse, M., 154 Moser, R., 155 Moulin-Ollagnier, J., 155 Muchnik degree, 326 reducible, 326 Multinacci number, 31 multiplicative form (of a word), 75 mutation, 376 graph, 379 polynomial, 376 Myers, D., 327 Myhill, J., 259 Nakada, H., 71 nearest-neighbour SFT, 309 Nicaud, C., 220 non-deterministic polynomial complecity, 360 non-deterministic polynomial complexity in unary, 360 non-periodic word, 12 non-repetitive colouring, 155 normalised error, 29 Novikov, P. S., 109 NP (complexity class), 360 N-rational frieze, 375 sequence, 369 series, 369 number Garsia, 41 Multinacci, 31 Pisot, 40 occurrance (of a pattern), 303 Ochem, P., 155 Ollinger, N., 295 one-sided shift, 23 one-step SFT, 309 orbit of an infinite word, 23

General index origin of a ray, 373 orphan, 256, 262, 266 overlap, 113 overlapfree, 153, 154 word, 113 pair graph, 265 palindrome, 11, 159, 178, 210 gapped, 160 Pansiot recoding, 126 Pansiot, J.-J., 125, 155, 161 paperfolding word, 130 Park, K., 177 Parry, W., 27, 69 partial period, 195 word, 187 partial quotients, 79 Paterson, C. W., 188 path, 16 label, 16 successful, 16 pattern, 131, 147, 151, 156, 249, 303 alphabet, 131 appears in, 303 instance, 131 occurrs in, 303 orphan, 256, 262, 266 shape, 303 sub-pattern, 303 support, 303 variable, 131 Zimin, 131 Pavlov, R., 347, 361 Pedicini, M., 65 Pegden, W., 155 perfect set, 56 period, 12, 127, 184, 237 periodic, 151 word, 12 periodic configuration, 312 Perrin, D., 220, 232, 243 Petrov, A. N., 156 Pin, J. -E., 222 Pirillo, G., 154 Pisot number, 40 Plagne, A., 406 plane-filling property, 288 Pleasants, P. A. B., 132 positivity conjecture, 401 power series method, 146 power set, 189 powerfree morphism, 119 prefix, 11 -closed language, 15 prefix code, 245

477

synchronised, 246 Preparata, F. P., 166 preperiod, 12 primitive word, 110 principal sub-grid, 321 probabilistic method, 121 probabilistic model, 406, 412, 419 constrained probabilistic model, 406, 419, 420 rational probabilistic model, 406, 420, 432, 437 real probabilistic model, 406, 419, 431, 437 product topology, 252 progression-free set, 133 Protasov, V., 157 Prouhet, E., 154 Puglisi, S. J., 168, 170 pure period, 194 Puzyna, J., 70 Pythagoras, 68 quasi -period, 175 -periodicity, 175 -seed, 177 question mark function, 76 quiver, 363, 374 Rényi, A., 27 Rabin, 18 Rampersad, N., 152, 154–159 Ramsey theory, 109 random Fibonacci sequences, 89 rank, 233, 235, 239 of a relation, 233 of a word, 235 of an automaton, 239 Rao, M., 155, 159 rational frieze, 375 power, 11 sequence, 370 ray, 373 R-code, 191 reciprocal, 207 recurrent word, 12 recursive function, 323, 325 set, 323 recursively enumerable set, 323 reduced set, 143 reflected ratio, 102 reflexive, 185 regular language sofic shifts, 309 relational period, 194 relationally universal, 210 repeat, 159, 160, 171 repetition, 159, 165

478

General index

threshold, 122, 155, 163 repetitions, 109 representation β -, 22 Restivo, A., 113, 154 Reutenauer, C., 169, 232, 369 reversal, 11 right form (of a bound of a k-Rosen interval), 97 Riordan, O., 155 R-isomorphic, 210 Rittaud, B., 71 Robinson’s system, 320 Robinson’s tiles, 274, 298 minimal, 281 patch, 276 Robinson, R., 268, 274 Robinson, R. M., 320 Romashchenko, A., 334 Rosen continued fractions, 70 Rosen, D., 70 Rothschild, B., 156 R-overlap, 212 (R,S)-code, 190 R-square, 212 R-squarefree, 212 Ruelle, D., 429 run, 159, 167 R-universal, 210 running Turing machines on grids, 319 Rytter, W., 170, 182 Rényi, A., 69 Sée´ bold, P., 161 Saari, K., 154 Sablik, M., 334 Salemi, S., 113, 154 Sardinas, A. A., 188 Schützenberger, M.-P., 236 Schatz, J. A., 156 Scherotzke, S., 401 Schmidt, K., 361 Schmidt, T., 71 Schraudner, M., 347, 361 Sciortino, M., 154 Scott, 18 scrambled pattern, 156 seed, 159, 175 semi-adjacent minor, 392 semi-algorithm, 269 semi-decision, 323 semigroup, 10 sequence Kronecker sequence, 405, 408 parameters for Kronecker sequences, 408 pseudo-randomness of a sequence, 405 truncated Kronecker sequence, 405, 409

Series, C., 71 set Cantor, 12, 56 clopen, 24 Shallit, J., 116, 118, 152–156, 158, 162 shape (of a pattern), 303 shift, 23, 75, 251 β -shift, 24 action, 305 full, 24 one-sided, 23 two-sided, 23 shift of finite type, 303 nearest-neighbour, 309 one-step, 309 shift-invariant, 305 short repetition, 127 Shur, A., 142, 154, 155 Sidorov, N., 27, 31, 32, 63, 64 Sierpiński, W., 70 signed continuant polynomial, 393 Silva, M., 156 Simon, K., 41 Simpson, J., 156, 161, 170 Simpson, S., 327 sink, 18 SL2 -tiling of the plane, 364 slice (of a subshift), 333 sliding block code, 307 Smeets, I., 71 Smith, D., 364 Smyth, W. F., 168, 176 sofic shift, 308 Solomyak, B., 40 spanning subgraph, 213 Spencer, J., 155, 156 square, 110 abelian, 131 period, 312, 360 squarefree morphism, 119 word, 110 stability interval, 60 stable pair, 238 relation, 224 standard factorisation, 169 (of a Christoffel word), 101 state accessible, 17 co-accessible, 17 final, 16 level, 228, 239 terminal, 16 Steiner, W., 71

General index Stern, M., 69 Stern-Brocot mediety, 77 strong degree, 326 equivalence, 326 reducibility, 326 strongly irreducible subshift, 331, 361 strongly transitive automaton, 228 Sturmian word, 154 sub-pattern, 303 subaction (of a subshift), 346 subadditive function, 382 subsequence, 128 subshift, 24, 61, 252, 305 aperiodic, 24 conjugacy, 269 conjugate, 25 effective, 331 entropy, 355 even shift, 253 extension, 306 factor, 306 finite type, 24, 61, 252 isomorphism, 306 lower entropy dimension, 361 minimal, 281, 331 periodic, 24 set of square periods, 360 slice of, 333 sofic, 24 strongly irreducible, 331 upper entropy dimension, 361 substitution, 13 suffix, 11 -closed language, 15 superprimitive, 175 superstring, 160 support (of a pattern), 303 surjunctive group, 263 symbol, 10 symbol code, 311 symbolic dynamical system, 23 symbolic factor (of a subshift), 306 symmetric, 185 synchronisable pair, 221 set, 221 synchronised, 246 prefix code, 246 synchronising word, 221, 232, 246 for a code, 246 Tardos, G., 155 Tauberian theorem, 407, 433 template, 136 ancestor, 137

479

instance, 136 parent, 136 terminal state, 16 three-distance theorem, 405, 410 Thue, A., 2, 109, 110, 113, 154, 155, 159, 161 Thue-Morse morphism, 14, 111, 161 word, 14, 110, 113, 151, 153, 154, 158 Tijdeman, R., 210 tiling, 269 tiling space, 252 Toeplitz family, 351 sequence, 351 Toeplitz subshift, 351 topological conjugacy, 25 dynamical system, 25 entropy, 299, 355, 362 isomorphism, 25 Trahtman, A., 220, 224, 237 trajectory, 19 transfer operator, 407, 425, 434, 440–442 constrained, 419, 421, 429 extended, 430 middle, 437 plain, 421, 429 weighted, 430, 434, 445 transition function, 17 monoid, 223 relation, 16 transitive, 185 translate (of a pattern), 303 translation, 188, 251 transpose, 395 trim, 17 truncation boundary position, 405, 411, 414, 422, 439 boundary truncation, 411, 422 generic position, 405, 411, 414, 423, 424, 439 generic truncation, 411, 423, 424 position of the truncation, 405, 411, 417 two-distance truncation, 405, 410, 411, 414, 422 Turing degree, 331 Turing machine, 271, 316 Turing, A., 271 two-distance phenomenon, 405, 408 Ullman, J. D., 265 ultimately periodic, 12, 151, 152 uniform morphism, 124, 151 uniformly recurrent word, 12 unique ergodicity, 26 universal critical exponent, 152

480 univoque base, 48 upper entropy dimension, 361 Vallée, B., 406, 407 van der Waerden’s theorem, 128, 140, 153 van der Waerden, B. L., 156 Vanier, P., 360 variables of a frontier, 366 Vasiga, T., 154 vertical ray, 373 Vichniak, G. Y., 255 Villemaire, R., 158 ` B., 363, 382 Vinberg, E. Volkov, M. V., 222, 227 Waleń, T., 182 Wang tile, 268 4-way deterministic, 299 aperiodic tile set, 270 directed, 288 NW-deterministic, 296 plane-filling, 288 Robinson’s, 274, 298 S NAKES , 291 Wang tiles, 309 Wang, H., 270, 309, 311 Ward, T., 361 weak degree, 326 equivalence, 326 R-code, 191 reducibility, 326 Weihrauch, K., 348 Weiss, B., 236 Werckmeister, A., 70 Wilf, H. S., 183 Witkowski, M., 155, 156 Wonenburger, M., 363, 382 word, 10, 109 2-dimensional, 130, 156 associated with a point, 366 augmenting, 229 bi-infinite, 12 concatenation, 11 distance, 12 eventually periodic, 12 factor, 11 Fibonacci, 14 infinite, 11 Lyndon, 168 marked, 143 minimal, 12 mirror, 11 non-periodic, 12 overlapfree, 113 partial, 187 periodic, 12

General index preperiod, 12 primitive, 110 purely morphic, 13 recurrent, 12 reversal, 11 Sturmian, 209 synchronising, 221, 232 uniformly recurrent, 12 X-factorisation, 190 Zamboni, L. Q., 210 Zaremba’s conjecture, 108 Zelevinsky, A., 363 Zheng, X., 348 Zimin pattern, 131 Ziv-Lempel factorisation, 168, 179

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