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Bartlett Washington 0250E 150 c Copyright 2015 Alan Bartlett Spectral Theory of Zd Substitutions Alan Bartlett A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy University of Washington 2015 Reading Committee: Boris Solomyak, Chair Christopher Hoffman Tatiana Toro Program Authorized to Offer Degree: Mathematics University of Washington Abstract d Spectral Theory of Z Substitutions Alan Bartlett Chair of the Supervisory Committee: Professor Boris Solomyak Mathematics In this paper, we generalize and develop results of Queff´elecallowing us to characterize the d spectrum of an aperiodic substitution in Z by describing the Fourier coefficients of mutually singular measures of pure type giving rise to the maximal spectral type of the translation operator on L2. This is done without any assumptions on primitivity or height, and provides a simple algorithm for determining singularity to Lebesgue spectrum for such substitutions, and we use this to show singularity of the spectrum for Queffe´lec's noncommutative bijective substitution, as well as the Table tiling, answering an open question of Solomyak. Moreover, d we also prove that the spectrum of any aperiodic bijective commutative Z substitution on a finite alphabet is purely singular. Finally, we show that every ergodic matrix of measures on a compact metric space can be diagonalized, which we use in the proof of the main result. TABLE OF CONTENTS Page Chapter 1: Introduction . .1 d 1.1 Dynamical Systems and Z Actions . .7 1.2 Substitution Dynamical Systems . 12 Chapter 2: Aperiodic q-Substitutions . 19 d 2.1 Arithmetic Base q in Z .............................. 19 2.2 Instructions and Configurations . 21 2.3 Aperiodicity and the Subshift . 27 Chapter 3: Spectral Theory . 35 3.1 The Correlation Vector - Σ . 36 3.2 The Spectral Hull - K ............................... 40 3.3 Queff´elec'sTheorem . 43 3.4 Aperiodic Bijective Commutative q-Substitutions . 46 Chapter 4: Computing the Spectrum . 49 4.1 Substititutions with Purely Singular Spectrum . 54 4.2 Substitutions with Lebesgue Spectral Component . 58 4.3 Substitutions with Every Spectral Component . 61 4.4 Height and Dekking's Criterion . 64 Chapter 5: Appendix . 69 5.1 Generalized Functionals on MpXq ........................ 69 5.2 Proof of Queff´elec'sTheorem . 75 i ACKNOWLEDGMENTS I would like to thank my advisor, Boris Solomyak, for his support and guidance. ii 1 Chapter 1 INTRODUCTION Substitutions are of interest to us as models of aperiodic phenomena - for example, mathematical quasicrystals, see [3]. Here, we consider higher dimensional analogues of d substitutions of constant length, or Z -substitutions which replace letters in a finite alphabet A with a rectangular block (indexed over the semirectangle r0; qq, see notation at the end of this section) of letters, and which we call q-substitutions. As usual, we study the translation d operator T on the hull XS of a substitution S, or the collection of all Z -indexed sequences whose local patterns can be produced by the substitution. Our primary interest is the spectrum (or the maximal spectral type, σmax) of the translation action, and our analysis is based on the work of Queff´elec,which describes the spectrum of one-sided primitive and aperiodic constant length substitutions of trivial height in one dimension (N-indexed sequences). Our formulation emphasizes the arithmetic properties of such substitutions and the recognizability properties afforded by aperiodic substitutions: those without any periodic sequences in their hull. Using only these two assumptions, we are able to describe the spectrum of such a substitution by identifying its discrete, singular continuous, and absolutely continuous spectral components as a sum of mutually singular measures of pure type with readily computable Fourier coefficients. This is done by using a representation consistent with the abelianization (which gives rise to the substitution matrix) and separates the analysis into two parts: the correlation measures, which depend on the position of letters in substituted blocks (which we call the configuration), and the spectral hull, a convex set depending on a coincidence matrix CS which is configuration-independent. An important notion in tiling theory is that of local rules which force global (aperiodic) d order. A result of Moz´es[19] asserts that self-similar tilings, corresponding to Z substitu- tions for d ¡ 1, can be obtained by local pattern matching rules, albeit, in general, at the cost of increasing the size of the alphabet, whereas this assertion is false for d “ 1. (We refer 2 the reader to [3] for details and more recent work in this direction.) Moreover, it is known that the mathematical diffraction spectrum is included in the dynamical spectrum (that of the translation operator on L2) as a consequence of Dworkin's argument [11] (see also [16], d [4] and references therein), and so our extension of Queff´elec'sresults to Z substitutions are of interest as they allow us to describe diffraction (and dynamical) properties of higher dimensional mathematical quasicrystals. We summarize the paper while highlighting the main ideas. First, we begin with a general introduction to topological dynamics, their invariant measures and spectral theory. d Then, in x1.2, we discuss preliminaries for the theory of Z -indexed substitution dynamical systems, including their invariant measures in x1.2.1 and spectral theory in x1.2.2. As usual, Michel's theorem 1.2.1 is used to identify the uniquely ergodic measure of a primitive system in terms of the Perron vector of MS ; its substitution matrix.A primitive reduced form (proposition 1.2.2) is used to represent an arbitrary substitution in terms of its primitive components, which combine in the nonprimitive case with theorem 1.2.4 of Cortez and Solomyak to describe the invariant measures as the convex hull of the invariant measures of its primitive components. Note that q-substitution dynamical systems are never weakly- mixing as Dekking's theorem 4.4.1 shows they always have nontrivial discrete spectrum; more generally, Dekking and Keane proved in [10] that primitive substitution systems cannot be strongly mixing. This should not be confused with the statement of our main result, theorem 3.3.1, which shows that the spectrum of S can be decomposed into measures d strongly-mixing for the q-shift on T ; as this is a statement about the spectrum of S (not its shift-invariant measures) relative to an entirely different dynamical system. Following the discussion on invariant measures, we discuss the essentials for the spectral theory of substitution dynamical systems where we give a very brief account of some existing results in the spectral theory of q-substitutions. In x2 we develop arithmetic and topological properties of aperiodic q-substitutions which A combine with a representation of the alphabet A in the vector space C , and of substitutions in the matrix algebra MApCq, to give us several useful tools (propositions 2.3.5, 2.3.7 and 2.3.8) relating the configuration of an aperiodic q-substitution to both its topological and measure theoretic structure. In x2.1 we develop some basic notation for q-adic arithmetic 3 d in Z , as well as a key lemma (2.1.1) which essentially lets us ignore the overlapping of superblocks T kSnpαq arising in a number of contexts; this removes an otherwise significant impediment to a detailed description of the spectrum, as tracking the discrepancies caused by overlaps destroys any hope of a detailed analysis, see the discussion at the end of x3.1. In x2.2, we give an arithmetic description (proposition 2.2.1) of q-substitutions in terms of configurations, which represents a q-substitution in terms of instructions A Ñ A arranged by location, and an equivalence of substitutions is described, relating two substitutions when there is a bijective rearrangement of their configurations which exchanges them. We also describe two constructions on substitutions: definitions 2.2.3 for cycled substitutions and 2.2.6 for the substitution product. A cycled q-substitution is obtained by permuting the associated configuration, and is related to the map T kSn: The substitution product takes two q-substitutions, S on A and S~ on A~, and produces a third q-substitution, S bS~ on AA~, and is related to the bisubstitution and the coincidence matrix CS “ MSbS of Queff´elec. In x2.3 we discuss consequences of aperiodicity beginning with a convenient criteria of Pansiot (lemma 2.3.1) allowing for a readily verifiable description of aperiodicity in the Z d case; a generalization to Z would be of significant interest, but no serious attempts by this author have been made. Next, we use a result of Moss´eand Solomyak (theorem 2.3.2) to identify aperiodicity with recognizability, a property of substitutions allowing for unique local desubstitution, and which gives rise to many useful topological properties summarized in proposition 2.3.4. Moreover, this allows us to describe the subshift as an iterated func- tion system on the maps T kS, for k in a fixed finite set. Using this perspective, we give a description of the Borel structure of XS through a collection of iterated cylinders (definition d 2.3.3) which represent the cylinders of AZ iterated through the above function system. This description is expressed in three ways: the partition formula of proposition 2.3.5 which describes how the lattice of iterated cylinders partition the subshift by degree; the mea- sure formula of proposition 2.3.7 which gives recursive identities allowing one to explicitly compute the measure of an arbitrary cylinder in XS , for an arbitrary invariant measure sup- ported by the subshift; and finally, the density of iterated indicators of proposition 2.3.8, permitting the study of the translation action on L2pµq in terms of the iterated indicator 1 1 functions T kSnrαs, and ultimately the indicators rαs via theorem 3.1.4 discussed below.
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