Geometric theory of unimodular Pisot substitutions

<abstract abstract-type="TeX"><p>We are concerned with the tiling flow <i>T</i> associated to a substitution φ over a finite alphabet. Our focus is on substitutions that are unimodular Pisot, i.e., their matrix is unimodular and has all eigenvalues strictly inside the unit circle with the exception of the Perron eigenvalue λ > 1. The motivation is provided by the (still open) conjecture asserting that <i>T</i> has pure discrete spectrum for any such φ. We develop a number of necessary and sufficient conditions for pure discrete spectrum, including: injectivity of the canonical torus map (the geometric realization), Geometric Coincidence Condition, (partial) commutation of <i>T</i> and the dual R<sup><i>d</i>-1</sup>-action, measure and tiling properties of Rauzy fractals, and concrete algorithms. Some of these are original and some have already appeared in the literature-as <i>sufficient</i> conditions only-but they all emerge from a unified approach based on the new device: the strand space <i>F</i><sub xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">φ</sub> of φ. The proof of the <i>necessity</i> hinges on determination of the discrete spectrum of <i>T</i> as that of the associated Kronecker toral flow.

[1]  Michael Baake,et al.  Digit tiling of euclidean space , 2000 .

[2]  Richard Kenyon,et al.  Arithmetic construction of sofic partitions of hyperbolic toral automorphisms , 1998, Ergodic Theory and Dynamical Systems.

[3]  C. Mauduit,et al.  Substitutions in dynamics, arithmetics, and combinatorics , 2002 .

[4]  Jean-Pierre Gazeau,et al.  From Quasicrystals to More Complex Systems , 2000 .

[5]  A. Siegel,et al.  Automate des pr'efixes-suffixes associ'e ` a une substitution primitive , 1999 .

[6]  J. Kwapisz.  A Dynamical Proof of Pisot's Theorem , 2006, Canadian Mathematical Bulletin.

[7]  Richard Kenyon.  The construction of self-similar tilings , 1995 .

[8]  Yang Wang,et al.  Self-affine tiling via substitution dynamical systems and Rauzy fractals , 2002 .

[9]  P. Arnoux,et al.  Pisot substitutions and Rauzy fractals , 2001 .

[10]  Marcy Barge,et al.  Geometric realization and coincidence for reducible non-unimodular Pisot tiling spaces with an application to $\beta $-shifts , 2006 .

[11]  Brigitte Mossé,et al.  Puissances de mots et reconnaissabilité des point fixes d'une substitution , 1992, Theor. Comput. Sci..

[12]  Boris Solomyak,et al.  Dynamics of self-similar tilings , 1997, Ergodic Theory and Dynamical Systems.

[13]  Robert V. Moody,et al.  Model Sets: A Survey , 2000 .

[14]  P. Walters.  Introduction to Ergodic Theory , 1977 .

[15]  Boris Solomyak,et al.  Two-symbol Pisot substitutions have pure discrete spectrum , 2003, Ergodic Theory and Dynamical Systems.

[16]  M. Queffélec.  Substitution dynamical systems, spectral analysis , 1987 .

[17]  G. Rauzy.  Rotations sur les groupes, nombres algébriques, et substitutions , 1988 .

[18]  Boris Solomyak,et al.  Consequences of Pure Point Diffraction Spectra for Multiset Substitution Systems , 2009, Discret. Comput. Geom..

[19]  F. M. Dekking,et al.  The spectrum of dynamical systems arising from substitutions of constant length , 1978 .

[20]  A. Livshits.  COMMUNICATIONS OF THE MOSCOW MATHEMATICAL SOCIETY: On the spectra of adic transformations of Markov compacta , 1987 .

[21]  Marcy Barge,et al.  Coincidence for substitutions of Pisot type , 2002 .

[22]  Robert F. Williams.  Classification of one dimensional attractors , 1968 .

[23]  Pierre Arnoux,et al.  Higher dimensional extensions of substitutions and their dual maps , 2001 .

[24]  Boris Solomyak,et al.  Nonperiodicity implies unique composition for self-similar translationally finite Tilings , 1998, Discret. Comput. Geom..

[25]  Randolph B. Tarrier,et al.  Groups , 1973, Algebra.

[26]  Brigitte Mosse,et al.  Properties of words and recognizability of fixed points of a substitution , 1992 .

[27]  Jörg M. Thuswaldner,et al.  Unimodular Pisot substitutions and their associated tiles , 2006 .

[28]  Nick Lord,et al.  Pisot and Salem Numbers , 1991 .

[29]  Anthony Manning,et al.  A Markov partition that reflects the geometry of a hyperbolic toral automorphism , 2002 .

[30]  Lorenzo Sadun,et al.  When shape matters: deformations of tiling spaces , 2003, Ergodic Theory and Dynamical Systems.

[31]  G. Rauzy.  Nombres algébriques et substitutions , 1982 .

[32]  A. Siegel,et al.  Pure discrete spectrum dynamical system and periodic tiling associated with a substitution , 2004 .

[33]  B. Solomyak,et al.  Pure Discrete Spectrum for One-dimensional Substitution Systems of Pisot Type , 2002, Canadian Mathematical Bulletin.

[34]  I. Putnam,et al.  Topological invariants for substitution tilings and their associated $C^\ast$-algebras , 1998, Ergodic Theory and Dynamical Systems.

[35]  R. Salem.  Algebraic numbers and Fourier analysis , 1963 .

[36]  Bernard Host,et al.  Valeurs propres des systèmes dynamiques définis par des substitutions de longueur variable , 1986, Ergodic Theory and Dynamical Systems.

[37]  A. Siegel,et al.  Geometric representation of substitutions of Pisot type , 2001 .