Geometric properties of a binary non-Pisot inflation and absence of absolutely continuous diffraction. (English) Zbl 1419.37017
Summary: One of the simplest non-Pisot substitution rules is investigated in its geometric version as a tiling with intervals of natural length as prototiles. Via a detailed renormalisation analysis of the pair correlation functions, we show that the diffraction measure cannot comprise any absolutely continuous component. This implies that the diffraction, apart from a trivial Bragg peak at the origin, is purely singular continuous. En route, we derive various geometric and algebraic properties of the underlying Delone dynamical system, which we expect to be relevant in other such systems as well.
MSC:
| 37B50 | Multi-dimensional shifts of finite type, tiling dynamics (MSC2010) |
| 37A30 | Ergodic theorems, spectral theory, Markov operators |
| 42A38 | Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type |
| 37H15 | Random dynamical systems aspects of multiplicative ergodic theory, Lyapunov exponents |
| 52C23 | Quasicrystals and aperiodic tilings in discrete geometry |
Software:
GADMM; Z_LINEAR_K; Genius; VINAS-P; DALIA; $Apart; reszeta.lib; Apart; StableBBasisNBM5; SaddleDrop; formConvReferences:
| [1] | Instytut Matematyczny PAN, 2019c STUDIA MATHEMATICA 247 (2) (2019) Geometric properties of a binary non-Pisot inflation and absence of absolutely continuous diffraction by Michael Baake (Bielefeld), Natalie Priebe Frank (Poughkeepsie, NY), Uwe Grimm (Milton Keynes) and E. Arthur Robinson Jr. (Washington, DC) Abstract. One of the simplest non-Pisot substitution rules is investigated in its geometric version as a tiling with intervals of natural length as prototiles. Via a detailed renormalisation analysis of the pair correlation functions, we show that the diffraction measure cannot comprise any absolutely continuous component. This implies that the diffraction, apart from a trivial Bragg peak at the origin, is purely singular continuous. En route, we derive various geometric and algebraic properties of the underlying Delone dynamical system, which we expect to be relevant in other such systems as well. 1. Introduction.The spectral structure of substitution dynamical systems is well studied, and many results are known; see [32] for a systematic introduction. The theory is in good shape for substitutions of constant length, both in one and in higher dimensions; see [20, 34, 22, 4, 12] as well as [3] and references therein. This is due to the fact that, for these systems, the symbolic side and the geometric realisation with tiles of natural size coincide, which also leads to a rather direct relation between the diffraction measures of the system (and its factors) on the one hand and the spectral measures on the other; see [9] and references therein. In general, the spectral theory of a substitution system and that of its geometric counterpart can differ considerably [17], in particular when the inflation multiplier fails to be a Pisot–Vijayaraghavan (PV) number [3, Def. 2.13]. In fact, beyond the substitutions of constant length, it often is simpler and ultimately more revealing to use the geometric setting with 2010 Mathematics Subject Classification: Primary 37A30, 42A38, 37B50; Secondary 37H15, 52C23. Key words and phrases: non-Pisot substitutions, tiling dynamics, singular spectrum, Lyapunov exponents. Received 13 June 2017; revised 7 March 2018. Published online 23 November 2018. DOI: 10.4064/sm170613-10-3 |
| [2] | Instytut Matematyczny PAN, 2019c 110M. Baake et al. natural tile (or interval) sizes, as suggested by Perron–Frobenius theory. We adopt this point of view below, and then speak of inflation rules to make the distinction. Our entire analysis in this paper will be in one dimension, where the tiles are just intervals. Since rather little is known when one leaves the realm of PV inflation multipliers, we present a detailed analysis of one of the simplest non-Pisot (or non-PV) inflation rules on two letters, for which we finally establish that the diffraction spectrum of the corresponding Delone sets on the real line, apart from the trivial peak at 0, is purely singular continuous. En route, we shall encounter a number of concepts and results that are described in some detail, in a way that will facilitate generalisations to related inflation rules [5] and beyond, as well as to higher dimensions, in the future. A key ingredient to our analysis is the study of the pair correlation functions via their exact renormalisation relations. The latter are analogous to those recently derived [2] for the Fibonacci inflation, where they led to a spectral purity result and then to pure point spectrum. This re-proved a known result in an independent way. In the binary non-Pisot system studied below, the situation is more complex because the spectrum is mixed, whence it remains to determine the nature of the continuous part. To the best of our knowledge, the answer is not in the literature, though the absence of absolutely continuous components is certainly expected [22, 4, 2, 14]. In anticipation of future work, we do not present the shortest path to the result, as that would mean to restrict more than necessary to methods that are limited to binary alphabets and to this particular example. Instead, we use the concrete system to investigate various concepts from [2] in this more complex case, with an eye to possible extensions and generalisations. The paper is organised as follows. In Section 2, we introduce the binary system via its symbolic substitution rule and the matching geometric inflation tiling of the real line by two types of intervals, following the general notions and results from [3]. Such a tiling is simultaneously considered as a two-component Delone set, by taking the left endpoints of the intervals as reference points. We also recall the construction of the hull and its dynamical system, together with the key properties of the latter. Section 3 introduces the pair correlation functions and derives exact renormalisation relations, which are then studied for their general solutions. This part is not strictly needed for our later analysis, but is interesting in its own right and helps to understand the differences to the cases treated in [2]. To continue, we need a reformulation of the pair correlation functions in terms of translation bounded measures and their Fourier transforms, which is provided in Section 4. This step emphasises the importance of two specific Diffraction of a non-Pisot inflation111 matrix families, whose structure will later provide some arguments needed in the exclusion of absolutely continuous diffraction. Section 5 analyses several properties of these matrix families by means of the (complex resp. real) algebras generated by them. Once again, some of these results go beyond what we need for our final goal, but highlight the algebraic structure of the problem. Section 6 returns to the correlation measures and their Fourier transforms. After splitting the transformed pair correlation measures into their spectral parts (Lebesgue decomposition), we rule out the existence of an absolutely continuous component by a suitable iterated application of the renormalisation relations in two directions. This approach employs the determination of the corresponding extremal Lyapunov exponents, some details of which are given in Appendix A. Two underlying renormalisation arguments are further explained in Appendix B, in the simpler setting of a scalar equation. Section 7 covers an application to the diffraction in the balanced weight case, where the pure point part is extinct. In particular, we illustrate one specific case of a singular continuous measure in this setting, based on a precise numerical calculation of the corresponding (continuous) distribution function. 2. Setting and preliminaries 2.1. Substitution, inflation and hull.We consider the primitive twoletter substitution 0 7→ 0111 (2.1) |
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