Lecture Notes in Mathematics

Volume 2273

Editors-in-Chief

Jean-Michel Morel, CMLA, ENS, Cachan, France

Bernard Teissier, IMJ-PRG, Paris, France

Series Editors

Karin Baur, University of Leeds, Leeds, UK

Michel Brion, UGA, Grenoble, France

Camillo De Lellis, IAS, Princeton, NJ, USA

Alessio Figalli, ETH Zurich, Zurich, Switzerland

Annette Huber, Albert Ludwig University, Freiburg, Germany

Davar Khoshnevisan, The University of Utah, Salt Lake City, UT, USA

Ioannis Kontoyiannis, University of Cambridge, Cambridge, UK

Angela Kunoth, University of Cologne, Cologne, Germany

Ariane Mézard, IMJ-PRG, Paris, France

Mark Podolskij, University of Luxembourg, Esch-sur-Alzette, Luxembourg

Sylvia Serfaty, NYU Courant, New York, NY, USA

Gabriele Vezzosi, UniFI, Florence, Italy

Anna Wienhard, Ruprecht Karl University, Heidelberg, Germany

More information about this series at http://www.springer.com/series/304

Shigeki Akiyama • Pierre ArnouxEditors

Substitution and TilingDynamics: Introductionto Self-inducing StructuresCIRM Jean-Morlet Chair, Fall 2017

EditorsShigeki AkiyamaInstitute of MathematicsUniversity of TsukubaTsukuba, Japan

Pierre ArnouxInstitut de Mathématiques deMarseille (I2M)Aix-Marseille UniversityMarseille Cedex 09, France

ISSN 0075-8434 ISSN 1617-9692 (electronic)Lecture Notes in MathematicsISBN 978-3-030-57665-3 ISBN 978-3-030-57666-0 (eBook)https://doi.org/10.1007/978-3-030-57666-0

Jointly published with Société Mathématique de France (SMF), Paris, France

Mathematics Subject Classification: 37B10, 05B45, 37B50, 68R15, 52C23

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature SwitzerlandAG 2020This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whetherthe whole or part of the material is concerned, specifically the rights of translation, reprinting, reuseof illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, andtransmission or information storage and retrieval, electronic adaptation, computer software, or by similaror dissimilar methodology now known or hereafter developed.The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoes not imply, even in the absence of a specific statement, that such names are exempt from the relevantprotective laws and regulations and therefore free for general use.The publisher, the authors, and the editors are safe to assume that the advice and information in this bookare believed to be true and accurate at the date of publication. Neither the publisher nor the authors orthe editors give a warranty, expressed or implied, with respect to the material contained herein or for anyerrors or omissions that may have been made. The publisher remains neutral with regard to jurisdictionalclaims in published maps and institutional affiliations.

This Springer imprint is published by the registered company Springer Nature Switzerland AG.The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

v

Preface

Tilings have been drawn and studied for centuries, in art and science, from theSumerian patterns, Roman mosaics, Alhambra wall tilings (see Fig. 1), and theattempts of Johannes Kepler to tile the plane with fivefold symmetric patterns (seeFig. 2), a goal which cannot be realized by a periodic tiling. These attempts wereone of the inspirations for the now-classic fivefold Penrose tiling, see the Forewordof Ref. [7] in Chap. 7.

The study of periodic tilings and their symmetry groups, the so-called crystal-lographic group, was developed to understand the structure of physical crystals;detailed account on the classification of periodic tilings and their symmetries canbe found in the book by Grünbaum–Shephard (See Ref. [58] in Chap. 2), which stillcontains a lot of interesting open questions and attracts many researchers. The book

Fig. 1 One of the many Alhambra wall tilings. Source: https://commons.wikimedia.org/wiki/File:Tassellatura_alhambra.jpg

vii

viii Preface

Fig. 2 Kepler Aa tiling: anattempt to tile the plane withfivefold symmetric patterns.Source: https://gallica.bnf.fr/ark:/12148/btv1b26001687/f4.item

also contains many examples of aperiodic tilings: however, they wrote at the time“Unlike the material in the earlier sections of the book, many of our assertions hereare not supported by published proofs” because the subject was in its infancy when[58] was published.

If early studies were devoted to periodic tilings, our main concern in this bookis the presentation of several recent developments in the study of aperiodic tilings.One could say that the relation of aperiodic tilings to periodic tilings is similar to therelation of irrational numbers to rational numbers and similarly opens a vast rangeof new phenomena.

The modern theory of aperiodic tilings started in the beginning of the 1960s,with the proof by Berger of the undecidability of the Domino problem, usingaperiodic tilings, followed by the invention of the Penrose tiling, the Heighwaydragon, and the Knuth–Davis twindragon, soon followed by the discovery of quasi-crystals and the study of tiling spaces which are one of the basic examples ofnon-commutative geometry. Self-similar tilings, like the Fibonacci and Penrosetilings, were a fundamental element of this development. They have known manygeneralizations, and the recent years have seen a number of advances in the domain.

This book presents lecture notes delivered at the research school Tiling dynamicalsystems, one of the Morlet Chair events organized in the second semester of 2017 aspart of the program Tilings and Discrete Geometry organized by Shigeki Akiymaduring his invitation to CIRM as Morlet Chair Professor.

The simplest aperiodic tilings are defined by substitutions; there are various typesof substitutions (these various types, and their generalizations, are a central theme

Preface ix

in this book), but in the simplest form, a substitution consists of replacing eachletter of a finite alphabet by a word on this alphabet; the first one is probably theFibonacci substitution, on the alphabet {a, b}, which replaces a by ab and b by a.To this substitution, one can associate infinite words which can be infinitely recodedby first decomposing them in the words a and ab (which implies that bb does notappear) and then replacing ab by a and a by b. The set of all these words is invariantby the shift, and this defines an interesting discrete dynamical system associatedwith the substitution, a substitutive dynamical system (see Fig. 3 for the associatedself-similar tiling). Tiling dynamical systems are continuous-time analogues of thesubstitutive dynamical systems; these are tilings of the line (or the plane or space)by a finite number of shapes (the tiles) that can be either subdivided (inflation)in smaller similar tiles or regrouped (deflation) in larger similar tiles (see Fig. 4).This leads to remarkable structures, like the celebrated Penrose tiling, with “self-inducing” properties: they contain smaller and larger copies of themselves.

The main object of the research school, as well as of the whole semester,was to study these self-inducing systems and tiling dynamical systems, and their

a b a a b a b a a b a a b a b

a b a a b a b a a

Fig. 3 The Fibonacci tiling, associated with the substitution a �→ ab, b �→ a

Fig. 4 Rauzy dragon: an example of a pseudo-self-similar structure, generated by a substitutionrule, or alternatively by a fusion rule. Source: courtesy of T. Fernique

x Preface

generalizations, from many different points of views and to increase researchinteractions among related people. The semester provided excellent opportunitiesfor research discussion among people working on discrete geometry, number theory,fractal geometry, theoretical computer science, and dynamical systems.

The book consists of eight chapters, the first six of which are expanded lecturenotes, and the last two of which are selected contributions.

Chapter 1, by B. Solomyak, is an introduction to the domain of tilings. It startsby defining the fundamental notions of Delone sets and Meyer sets, the Delonesets with inflation symmetry and their number-theoretic properties (in relation withPisot and Perron numbers). It then studies the related substitution tilings and theirassociated dynamical systems, clarifying the notions of self-affine and pseudo-self-affine tilings and their properties. The last section presents some developments(infinite local complexity, pure discrete spectrum, and Fourier quasi-crystals) andsome open problems.

Chapter 2, by N. P. Frank, deals with the various models of tilings, from thesimplest, discrete tilings in one dimension (symbolic systems), to continuous tilingsof the line, discrete systems in higher dimension (Zd systems), and tilings in higherdimension. It presents several ways to build systems with a hierarchical structure,either the same at all levels (substitutive systems) or with rules varying with the level(S-adic systems) and develops a new formalism, the fusion rule to build supertiles,which regroups several different ways to build recurrent tilings with interestinghierarchical properties; the properties of these tilings are studied from the dynamicaland the diffraction (Fourier analysis) perspectives, and this chapter also contains ahistorical introduction of the spectral study of tiling generated by substitution.

Chapter 3, by J.Thuswaldner, discusses in depth S-adic systems and theirgeometric realizations. S-adic symbolic systems are a natural generalization ofsubstitutive systems. They are systems which are generated by an infinite sequenceof substitutions belonging to a finite set S. The most elementary case is that ofa rotation on a circle of unit length; a rotation by an irrational quadratic numberwith periodic continued fraction has very particular diophantine properties (this is asimple example of a self-induced dynamical system) and is the geometric realizationof a substitutive system determined by this periodic continued fraction. A circlerotation by an irrational number, with an infinite nonperiodic continued fraction isinfinitely renormalizable, but not self-induced, and it is a geometric realization of acorresponding S-adic system. Is the same thing true in higher dimension, for toralrotations? In fact, it is almost always true in higher dimension. This is shown, in aprecise sense, by using properties of some generalized continued fractions.

The study of tiling dynamics is historically motivated by quasi-crystals: a realmaterial having long-range order but no translational periods. The recurrenceproperties of quasi-crystals have been studied for a long time, in relation to thediffraction spectrum and the spectral properties of the corresponding Schrödingeroperators. In pursuing these directions, topological properties of tiling dynamicsplay an essential role. They determine, for instance, the labelling of the gaps inthe spectrum. The study of the topological invariants involves elements of non-commutative geometry as developed by Connes. One of the first examples of

Preface xi

non-commutative spaces he proposed, the space of all Penrose tilings foliated by thetranslation action, stands at the beginning of the developments described in Chap. 4.In this chapter, J. Kellendonk gives an account on the construction of spaces,dynamical systems and algebras for tilings, and how their topological invariantshelp to understand the topological properties of the underlying material.

Chapter 5, by M. Rigo, deals with an apparently very different domain, thatof combinatorial games. These games provide an original introduction to thehistorical relations between tilings and language theory, logics, and computerscience. The best-known combinatorial game is certainly the Nim game, whereeach player takes out tokens from one or several piles of tokens, and the winneris the player who takes the last token. Since the game is deterministic, thereare winning and losing positions; it turns out that this set of winning positionshas a remarkable structure, which can be determined by a finite automaton. It isrelated to substitutive sequences, numeration systems, Pisot numbers, and to higherdimensional extensions associated with tilings of Nd having interesting properties.

The birth of the modern theory of tilings is associated with the famous proofby Berger of the undecidability of the Domino problem. In Chap. 6, E. Jeandel andP. Vanier revisit this proof and give us a modern view of this problem, includingthe necessary foundations on automata, Turing machines, and shift spaces. Theycarefully expose four different proofs of the undecidability result. A crucial pointof any proof of the undecidability of the Domino problem is to exhibit a set of tileswhich can tile the plane, but only in an aperiodic way, and the authors emphasize onthe various constructions of these sets of tiles.

To a tiling space, one can associate a dynamical spectrum, coming from the Rd -

action by translation, which allows to define a spectral measure; one can also definea diffraction measure, coming initially from physics, with close connections to thespectral measure; these two types of spectra already appear in the first two chapters.Chapter 7 of the book, contributed by M. Baake and U. Grimm, deals with theproperties of these measures, and in particular with the question of the existence ofa nontrivial continuous or singular component. An essential element to answer thisquestion is the study of autocorrelation and the pair correlation function. Using thistechnique, conditions are given to ensure that the diffraction measure is singular.

The last chapter, by P. Mercat and S. Akiyama, deals with the Pisot substitutionconjecture, one of the main open problems in the field of substitutive tilings.This conjecture, which has several forms and generalizations, says that any sub-stitution dynamical system of irreducible Pisot type has pure discrete spectrum,see Sect. 2.6.3.1 for more details (by contrast, when the inflation factor is not aPisot number, complicated behaviour such as infinite local complexity becomepossible, see Fig. 5, and also Fig. 2.6 of Chap. 2); it occurs in several chapters ofthis book. Chapter 8 contains a noteworthy simple new characterization of subshiftswith a discrete spectrum, which is basically checkable by automata computation;this gives an algorithm to verify that a given subshift of Pisot type satisfies thePisot conjecture. This characterization is shown to be an equivalence for subshiftsgenerated by irreducible Pisot substitutions and applies as well to S-adic systems.

xii Preface

Fig. 5 The Frank–Robinsontiling: an example of tilingwith a non-Pisot inflationfactor and infinite localcomplexity

We thank the lecturers for giving a series of introductory talks at CIRM as wellas providing us detailed expositions of this developing area. We believe that thisbook will give a nice access to these subjects and plenty of research directions tothe related researchers.

Tsukuba, Japan Shigeki AkiyamaMarseille, France Pierre ArnouxJanuary–July 2020

Contents

1 Delone Sets and Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Boris Solomyak1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Delone Sets of Finite Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Meyer Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Inflation Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.5 Substitution Delone Sets and Substitution Tilings . . . . . . . . . . . . . . . . . . . 12

1.5.1 Substitution Delone m-Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.5.2 Tilings and Substitution Tilings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.5.3 Representable Delone m-Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.5.4 Characterization of Expansion Maps . . . . . . . . . . . . . . . . . . . . . . . . . 181.5.5 Pseudo-Self-Affine Tilings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.6 Dynamical Systems from Delone Sets in Rd . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.6.1 Eigenvalues of Delone Set Dynamical Systems . . . . . . . . . . . . . 241.7 Concluding Remarks and Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

1.7.1 Infinite Local Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261.7.2 Pure Discrete Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261.7.3 Fourier Quasicrystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.7.4 Open Problems (Expansion Maps) . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2 Introduction to Hierarchical Tiling Dynamical Systems . . . . . . . . . . . . . . . . 33Natalie Priebe Frank2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.1.1 Outline of the Paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.1.2 Not Covered . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.2 The Fundamental Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.2.1 Motivation: Shift Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.2.2 Straightforward Generalization: Sequences in Z

d . . . . . . . . . . . 382.2.3 Straightforward Generalization: Tilings of R . . . . . . . . . . . . . . . . 382.2.4 Geometric Generalization: Tilings of Rd . . . . . . . . . . . . . . . . . . . . 39

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xiv Contents

2.3 The Dynamical Systems Viewpoint. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.3.1 Tiling Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.3.2 Notions of Equivalence for Symbolic and Tiling

Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.3.3 Repetitivity and Minimality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442.3.4 Invariant and Ergodic Measures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.4 Supertile Construction Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.4.1 Motivation: Symbolic Substitutions . . . . . . . . . . . . . . . . . . . . . . . . . . 462.4.2 Generalization: One-dimensional Self-Similar Tilings . . . . . . 482.4.3 More Tricky Generalization: Multidimensional

Constant-Length Substitutions in Zd . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.4.4 Geometric Generalization: Self-Affine, Self-Similar,and Pseudo-Self-Similar Tilings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.4.5 Fusion: A General Viewpoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542.4.6 Tiling Spaces from Supertile Methods . . . . . . . . . . . . . . . . . . . . . . . 632.4.7 Recognizability or the Unique Composition Property .. . . . . . 64

2.5 Ergodic-Theoretic and Dynamical Analysis of SupertileMethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652.5.1 Transition (a.k.a. Incidence, Substitution,

Abelianization, or Subdivision) Matrices . . . . . . . . . . . . . . . . . . . . 652.5.2 Primitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 662.5.3 Ergodic Measures for Fusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 702.5.4 General Result: Substitution Systems are Not

Strongly Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 712.5.5 Fusion Rules with Various Properties . . . . . . . . . . . . . . . . . . . . . . . . 71

2.6 Spectral Analysis of Supertile Methods: Dynamical Spectrum . . . . . 722.6.1 The Koopman Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 732.6.2 Eigenfunctions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 742.6.3 Pure Discrete Dynamical Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . 772.6.4 The Continuous Part of the Spectrum . . . . . . . . . . . . . . . . . . . . . . . . 80

2.7 Spectral Analysis of Supertile Methods: Diffraction Spectrum . . . . . 822.7.1 Autocorrelation for Symbolic Sequences . . . . . . . . . . . . . . . . . . . . 832.7.2 Diffraction in R

d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 842.7.3 Intensities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

2.8 Connection Between Diffraction and Dynamical Spectrum .. . . . . . . . 882.8.1 When the Diffraction is Not Pure Point . . . . . . . . . . . . . . . . . . . . . . 89

2.9 For Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

3 S-adic Sequences: A Bridge Between Dynamics, Arithmetic,and Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97Jörg M. Thuswaldner3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 983.2 The Classical Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

3.2.1 Sturmian Sequences and Their Basic Properties . . . . . . . . . . . . . 993.2.2 The Classical Continued Fraction Algorithm . . . . . . . . . . . . . . . . 103

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3.2.3 Dynamical Properties of Sturmian Sequences . . . . . . . . . . . . . . . 1073.2.4 Sturmian Sequences Code Rotations . . . . . . . . . . . . . . . . . . . . . . . . . 1103.2.5 Natural Extensions and the Geodesic Flow

on SL2(Z) \ SL2(R) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1163.3 Problems with the Generalization to Higher Dimensions . . . . . . . . . . . 122

3.3.1 Arnoux-Rauzy Sequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1223.3.2 Imbalanced Arnoux-Rauzy Sequences . . . . . . . . . . . . . . . . . . . . . . . 1243.3.3 Weak Mixing and the Existence of Eigenvalues . . . . . . . . . . . . . 127

3.4 The General Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1283.4.1 S-adic Sequences .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1293.4.2 Generalized Continued Fraction Algorithms.. . . . . . . . . . . . . . . . 132

3.5 The Importance of Primitivity and Recurrence. . . . . . . . . . . . . . . . . . . . . . . 1343.5.1 Primitivity and Minimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1353.5.2 Recurrence, Weak Convergence, and Unique Ergodicity.. . . 136

3.6 The Importance of Balance and Algebraic Irreducibility .. . . . . . . . . . . 1423.6.1 S-adic Rauzy Fractals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1433.6.2 Balance, Algebraic Irreducibility, and Strong

Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1483.7 Properties of S-adic Rauzy Fractals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

3.7.1 Set Equations for S-adic Rauzy Fractals and DualSubstitutions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

3.7.2 An S-adic Rauzy Fractal Is the Closure of Its Interior . . . . . . 1583.7.3 The Generalized Left Eigenvector .. . . . . . . . . . . . . . . . . . . . . . . . . . . 1603.7.4 An S-adic Rauzy Fractal Has a Boundary of Measure

Zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1623.8 Tilings, Coincidence Conditions, and Combinatorial Issues . . . . . . . . 165

3.8.1 Multiple Tiling and Inner Subdivision of the Subtiles . . . . . . . 1653.8.2 Coincidence Conditions and Tiling Properties .. . . . . . . . . . . . . . 1673.8.3 How to Check Geometric Coincidence and Geometric

Finiteness? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1703.9 S-adic Systems and Torus Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

3.9.1 Statement of the Conjugacy Result . . . . . . . . . . . . . . . . . . . . . . . . . . . 1753.9.2 Proof of the Conjugacy Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1763.9.3 A Metric Result. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1783.9.4 Proof of the Metric Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1793.9.5 Corollaries for Arnoux-Rauzy and Brun Systems . . . . . . . . . . . 184

3.10 Concluding Remarks: Natural Extensions, Flows, and TheirPoincaré Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

4 Operators, Algebras and Their Invariants for Aperiodic Tilings . . . . . . . 193Johannes Kellendonk4.1 Tilings and the Topology of Their Hulls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

4.1.1 Basic Notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1944.1.2 Undecorated Tilings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

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4.1.3 Wang Tilings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1954.1.4 Pattern Equivariant Functions and Local Derivability .. . . . . . 196

4.2 Operators from Tilings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1984.2.1 Tight Binding Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1984.2.2 Tight Binding Operators for Wang Tilings . . . . . . . . . . . . . . . . . . . 1994.2.3 A Simple Class of One-dimensional Models . . . . . . . . . . . . . . . . 2004.2.4 Gaps and the Integrated Density of States . . . . . . . . . . . . . . . . . . . 203

4.3 Algebras for Tilings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2044.3.1 The Pattern Equivariant Approach.. . . . . . . . . . . . . . . . . . . . . . . . . . . 2044.3.2 The Groupoid Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2074.3.3 Crossed Products with Z

d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2084.3.4 Extensions and Exact Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

4.4 K-Theoretic Invariants for Tilings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2114.4.1 Short Definition of K-Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2124.4.2 Calculating K-Groups for Tiling Algebras. . . . . . . . . . . . . . . . . . . 2144.4.3 Numerical Invariants for Pattern Equivariant Operators . . . . . 216

4.5 Bulk Boundary Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2184.5.1 General Philosophy.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2184.5.2 The Toeplitz Extension as Half Space Algebra . . . . . . . . . . . . . . 2204.5.3 One-dimensional Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

5 From Combinatorial Games to Shape-Symmetric Morphisms . . . . . . . . . 227Michel Rigo5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2275.2 Notation and Conventions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2285.3 Bits of Combinatorial Game Theory.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2305.4 Automatic and Regular Sequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

5.4.1 Generalities on Sequences and Morphisms . . . . . . . . . . . . . . . . . . 2385.4.2 Iterating a Constant-Length Morphism.. . . . . . . . . . . . . . . . . . . . . . 2405.4.3 k-Kernel of a Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2435.4.4 Changing the Base. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2455.4.5 k-Regular Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

5.5 Characterizing P-Position of Wythoff’s Gamein Polynomial Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2495.5.1 From Morphic Words to Automatic Words . . . . . . . . . . . . . . . . . . 2515.5.2 Positional Numeration Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2555.5.3 Syntactic Characterization of P-Positions of

Wythoff’s Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2575.6 Extension to a Multidimensional Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

5.6.1 2-Regularity for Grundy Values of the Game of Nim . . . . . . . 2645.6.2 Grundy Values of the Game of Wythoff . . . . . . . . . . . . . . . . . . . . . . 266

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5.7 Shape-Symmetry.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2665.7.1 Abstract Numeration Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2755.7.2 Multidimensional S-Automatic Sequences . . . . . . . . . . . . . . . . . . 2805.7.3 Proof of Theorem 5.7.14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282

5.8 Games with a Finite Set of Moves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2845.9 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288

6 The Undecidability of the Domino Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293Emmanuel Jeandel and Pascal Vanier6.1 Statement and Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294

6.1.1 Definitions and Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2946.1.2 Algorithmic Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2946.1.3 Algorithms to Prove A Tileset Does Not Tile the Plane . . . . . 297

6.2 Main Ingredients in the Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3036.2.1 Turing Machines.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3046.2.2 Encoding of Turing Machines in Tilings: The Fixed

Domino Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3096.2.3 Towards the Domino Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311

6.3 The Substitutive Method 1/2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3126.3.1 Preliminary Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3126.3.2 Substitutions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3136.3.3 Grids Inside Tilings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3176.3.4 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320

6.4 The Construction of Aanderaa and Lewis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3216.4.1 Tiling Problems on One-dimensional Objects . . . . . . . . . . . . . . . 3216.4.2 The Subshift Sp and p-Adic Numbers . . . . . . . . . . . . . . . . . . . . . . . 3276.4.3 Sp as a Toeplitz Subshift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3296.4.4 Sp × Sp as a Distance Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3316.4.5 Undecidability for Distance Shifts. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3366.4.6 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337

6.5 The Construction of Kari . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3386.5.1 Balanced Representations of Reals . . . . . . . . . . . . . . . . . . . . . . . . . . . 3386.5.2 Multiplying Balanced Representations . . . . . . . . . . . . . . . . . . . . . . . 3406.5.3 An Aperiodic Tileset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3426.5.4 Undecidability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345

6.6 The Substitutive Method 2/2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3476.6.1 The Fixed Point Theorem of Computability Theory.. . . . . . . . 3496.6.2 Simulating a Tileset with a Turing Machine . . . . . . . . . . . . . . . . . 3506.6.3 A Fixedpoint Based Tileset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3526.6.4 Undecidability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3536.6.5 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355

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7 Renormalisation of Pair Correlations and Their FourierTransforms for Primitive Block Substitutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359Michael Baake and Uwe Grimm7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3597.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361

7.2.1 Logarithmic Integrals and Mahler Measures . . . . . . . . . . . . . . . . . 3627.2.2 Radon Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3627.2.3 Riesz Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366

7.3 Uniform Distribution and Averages. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3687.4 Results in One Dimension. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3727.5 Consequences and an Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3767.6 Results in Higher Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3797.7 Binary Block Substitutions of Constant Size . . . . . . . . . . . . . . . . . . . . . . . . . 3827.8 Block Substitutions with Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3847.9 The Frank–Robinson Tiling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3897.10 Closing Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393

8 Yet Another Characterization of the Pisot Substitution Conjecture . . . 397Paul Mercat and Shigeki Akiyama8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3978.2 A Criterion for a Subshift to Have Purely Discrete Spectrum .. . . . . . 399

8.2.1 Subshift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3998.2.2 Discrete Line Associated to a Word . . . . . . . . . . . . . . . . . . . . . . . . . . 4008.2.3 Geometrical Criterion for the Pure Discreteness of

the Spectrum .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4028.2.4 An Easy Example: Generalization of Sturmian

Sequences .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4068.3 Pure Discreteness for Irreducible Unit Pisot Substitution . . . . . . . . . . . 407

8.3.1 Substitutions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4078.3.2 Topology and Main Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4088.3.3 Proof That an Inner Point Implies the Pure

Discreteness of the Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4098.4 Algebraic Coincidence Ensures an Inner Point . . . . . . . . . . . . . . . . . . . . . . 411

8.4.1 Algebraic Coincidence of Substitutive Delone Set . . . . . . . . . . 4118.4.2 Substitutive Meyer Set from Du . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4128.4.3 Algebraic Coincidence for Du . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4128.4.4 Proof of the Existence of an Inner Point . . . . . . . . . . . . . . . . . . . . . 413

8.5 Computation of the Interior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4158.5.1 Regular Languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4168.5.2 Discrete Line Associated to a Regular Language .. . . . . . . . . . . 4178.5.3 Computation of the Interior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4208.5.4 Proof of the Theorem 8.5.11 .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4218.5.5 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423

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8.6 Pure Discreteness for Various Infinite Family of Substitutions. . . . . . 4258.6.1 Proof of Pure Discreteness Using a Geometrical

Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4258.6.2 Proof of Pure Discreteness Using Automata .. . . . . . . . . . . . . . . . 432

8.7 Pure Discreteness for a S-adic System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4378.7.1 Representation of Du,a by an Automaton .. . . . . . . . . . . . . . . . . . . 4388.7.2 Proof of the Inclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4418.7.3 Pure Discreteness of the Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . 443

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449