Related papers: Scaling of the Thue-Morse diffraction measure

We investigate a family of Riesz products and show that they can be regarded as diffraction measures of generalized Thue-Morse sequences, possibly over an infinite alphabet. These measures are closely related to the dynamical system arising…

The paradigm for singular continuous spectra in symbolic dynamics and in mathematical diffraction is provided by the Thue-Morse chain, in its realisation as a binary sequence with values in $\{\pm 1\}$. We revisit this example and derive a…

The classic Thue--Morse measure is a paradigmatic example of a purely singular continuous probability measure on the unit interval. Since it has a representation as an infinite Riesz product, many aspects of this measure have been studied…

The Thue-Morse system is a paradigm of singular continuous diffraction in one dimension. Here, we consider a planar system, constructed by a bijective block substitution rule, which is locally equivalent to the squiral inflation rule. For…

We regard the classic Thue--Morse diffraction measure as an equilibrium measure for a potential function with a logarithmic singularity over the doubling map. Our focus is on unusually fast scaling of the Birkhoff sums (superlinear) and of…

The translation action of $\RR^{d}$ on a translation bounded measure $\omega$ leads to an interesting class of dynamical systems, with a rather rich spectral theory. In general, the diffraction spectrum of $\omega$, which is the carrier of…

Here, we study some measures that can be represented by infinite Riesz products of 1-periodic functions and are related to the doubling map. We show that these measures are purely singular continuous with respect to Lebesgue measure and…

The classic middle-thirds Cantor set leads to a singular continuous measure via a distribution function that is know as the Devil's staircase. The support of the Cantor measure is a set of zero Lebesgue measure. Here, we discuss a class of…

We discuss how the diffraction theory of a single translation bounded measure or a family of such measures can be understood within the framework of unitary group representations. This allows us to prove an orthogonality feature of measures…

We consider topological dynamical systems over $\ZZ$ and, more generally, locally compact, $\sigma$-compact abelian groups. We relate spectral theory and diffraction theory. We first use a a recently developed general framework of…

We introduce a generalization of the Schur measure. And we give a combinatorial interpretation to this generalized measure and prove that a scaling limit for this measure is given by the Tracy-Widom distribution.

The scaling behaviour of the diffraction intensity near the origin is investigated for (partially) ordered systems, with an emphasis on illustrative, rigorous results. This is an established method to detect and quantify the fluctuation…

The diffraction of stochastic point sets, both Bernoulli and Markov, and of random tilings with crystallographic symmetries is investigated in rigorous terms. In particular, we derive the diffraction spectrum of 1D random tilings, of…

The well-known plastic number substitution gives rise to a ternary inflation tiling of the real line whose inflation factor is the smallest Pisot-Vijayaraghavan number. The corresponding dynamical system has pure point spectrum, and the…

We present cut and project formalism based on measures and continuous weight functions of sufficiently fast decay. The emerging measures are strongly almost periodic. The corresponding dynamical systems are compact groups and homomorphic…

Mathematical diffraction theory is concerned with the analysis of the diffraction measure of a translation bounded complex measure $\omega$. It emerges as the Fourier transform of the autocorrelation measure of $\omega$. The mathematically…

Asymptotics are derived for the scaling of the total diffraction intensity for the set of $k$-free integers near the origin, which is a measure for the degree of patch fluctuations.

For an expanding (unstable) foliation of a diffeomorphism, we use a natural dynamical averaging to construct transverse measures, which we call \emph{maximal}, describing the statistics of how the iterates of a given leaf intersect the…

We review the relation between compact asymptotic spectral measures and certain positive asymptotic morphism on locally compact spaces via asymptotic Riesz representation theorem, as introduced by Martinez and Trout [3]. Applications to…

In this paper we show that under suitable conditions on their Fourier--Bohr coefficients, the twisted Eberlein convolution of a measure with pure point diffraction spectra and a measure with continuous diffraction spectra is zero. In…