Renormalisation of Pair Correlation Measures for Primitive Inflation Rules and Absence of Absolutely Continuous Diffraction

Abstract

The pair correlations of primitive inflation rules are analysed via their exact renormalisation relations. We introduce the inflation displacement algebra that is generated by the Fourier matrix of the inflation and deduce various consequences of its structure. Moreover, we derive a sufficient criterion for the absence of absolutely continuous diffraction components, as well as a necessary criterion for its presence. This is achieved via estimates for the Lyapunov exponents of the Fourier matrix cocycle of the inflation rule. While we develop the theory first for the classic setting in one dimension, we also present its extension to primitive inflation rules in higher dimensions with finitely many prototiles up to translations.

Scaling of diffraction intensities near the origin: some rigorous results

“…The recognisability of our inflation rule then implies the following set of exact renormalisation equations [6,32,8],…”

“…The purpose of this article is to link the recent interest in these questions with some of the known techniques and results from rigorous diffraction theory, as started by Hof in [28] and later developed by many people; see [11] and references therein for a systematic account. Our approach will make substantial use of the exact renormalisation relations for primitive inflation rules [5,6,8,32,13], which will allow us to establish the scaling behaviour rigorously. This is in some contrast to classic methods of finite size scaling [1,24,23,2], where such a behaviour is extrapolated and only asymptotically true.…”

Three variations on a theme by Fibonacci

“…Since we know that the Bombieri-Taylor property holds for primitive inflation systems, see As the original Fibonacci system is a factor of its twisted bar-swap extension, via identifying a with a and b with b, it is clear that the dynamical spectrum of our bar-swap extension is of mixed type, which will be reflected for the generic choice of the weights u α in the diffraction measure as well. Since the extension is (measure-theoretically) 2:1, as follows from ordinary Fibonacci being an almost everywhere 2:1 factor, we know that the continuous part of the spectrum must be of pure type [5], which turns out to be singular continuous in this case, by an application of the Lyapunov exponent criterion for the absence of absolutely continuous spectral components [6,17]. The result can now be stated as follows.…”

“…In particular, B (1) = B and B (n) (0) = M n for all n ∈ N, where M is the substitution matrix from (1.1). Note that B (n) (y) defines a matrix cocycle, called the internal cocycle, which by (2.8) is related to the usual inflation cocycle by an application of the ⋆-map to the displacement matrices of the powers of the inflation rule [6,8].…”

“…For precisely four choices of the parameters, namely (0, 0, 0), (0, 1,9), (1, 0, 3) and (1,1,6), one obtains the windows of Section 4 or a global translate thereof; see Figure 4. Observe that the inflation rules (0, 1,9) and (1, 0, 3) emerge from (0, 0, 0) by a reflection in the horizontal and vertical axis, respectively, and (1,1,6) by applying both reflections. Since the original tiling hull is reflection symmetric, these four rules define the same hull.…”