arXiv subjects

Michael Baake

Publications and source records attributed to Michael Baake.

At least 55 records · Page 3Linked to original sources

arXiv2020

Three variations on a theme by Fibonacci

Several variants of the classic Fibonacci inflation tiling are considered in an illustrative fashion, in one and in two dimensions, with an eye on changes or robustness of diffraction and dynamical spectra. In one dimension, we consider extension mechanisms of deterministic and of stochastic nature, while we look at direct product variations in a planar extension. For the pure point part, we systematically employ a cocycle approach that is based on the underlying renormalisation structure. It allows explicit calculations, particularly in cases where one meets regular model sets with Rauzy fractals as windows.

math.DS↗

arXiv2020

Pure point measures with sparse support and sparse Fourier--Bohr support

Fourier-transformable Radon measures are called doubly sparse when both the measure and its transform are pure point measures with sparse support. Their structure is reasonably well understood in Euclidean space, based on the use of tempered distributions. Here, we extend the theory to second countable, locally compact Abelian groups, where we can employ general cut and project schemes and the structure of weighted model combs, along with the theory of almost periodic measures. In particular, for measures with Meyer set support, we characterise sparseness of the Fourier--Bohr spectrum via conditions of crystallographic type, and derive representations of the measures in terms of trigonometric polynomials. More generally, we analyse positive definite, doubly sparse measures in a natural cut and project setting, which results in a Poisson summation type formula.

math.MG↗

arXiv2020

Notes on Markov embedding

The representation problem of finite-dimensional Markov matrices in Markov semigroups is revisited, with emphasis on concrete criteria for matrix subclasses of theoretical or practical relevance, such as equal-input, circulant, symmetric or doubly stochastic matrices. Here, we pay special attention to various algebraic properties of the embedding problem, and discuss the connection with the centraliser of a Markov matrix.

math.PR↗

arXiv2020

Ancestral lines under recombination

Solving the recombination equation has been a long-standing challenge of \emph{deterministic} population genetics. We review recent progress obtained by introducing ancestral processes, as traditionally used in the context of \emph{stochastic} models of population genetics, into the deterministic setting. With the help of an ancestral partitioning process, which is obtained by letting population size tend to infinity (without rescaling parameters or time) in an ancestral recombination graph, we obtain the solution to the recombination equation in a transparent form.

math.PR↗

arXiv2019

A brief guide to reversing and extended symmetries of dynamical systems

The reversing symmetry group is a well-studied extension of the symmetry group of a dynamical system, the latter being defined by the action of a single homeomorphism on a topological space. While it is traditionally considered in nonlinear dynamics, where the space is simple but the map is complicated, it has an interesting counterpart in symbolic dynamics, where the map is simple but the space is not. Moreover, there is an interesting extension to the case of higher-dimensional shifts, where a similar concept can be introduced via the centraliser and the normaliser of the acting group in the full automorphism group of the shift space. We recall the basic notions and review some of the known results, in a fairly informal manner, to give a first impression of the phenomena that can show up in the extension from the centraliser to the normaliser, with some emphasis on recent developments.

math.DS↗

arXiv2019

Renormalisation of pair correlation measures for primitive inflation rules and absence of absolutely continuous diffraction

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. We also discuss some consequences for the spectral measures of such systems. 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.

math.DS↗

arXiv2019

Scaling of diffraction intensities near the origin: Some rigorous results

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 behaviour known under the term hyperuniformity. Here, we consider one-dimensional systems with pure point, singular continuous and absolutely continuous diffraction spectra, which include perfectly ordered cut and project and inflation point sets as well as systems with stochastic disorder.

math.MG↗

arXiv2019

Diffraction of a model set with complex windows

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 associated control point sets can be described as regular model sets whose windows in two-dimensional internal space are Rauzy fractals with a complicated structure. Here, we calculate the resulting pure point diffraction measure via a Fourier matrix cocycle, which admits a closed formula for the Fourier transform of the Rauzy fractals, via a rapidly converging infinite product.

math.DS↗

arXiv2019

Scaling properties of the Thue--Morse measure

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 in the past, including various scaling properties and a partly heuristic multifractal analysis. Some of the difficulties emerge from the appearance of an unbounded potential in the thermodynamic formalism. It is the purpose of this article to review and prove some of the observations that were previously established via numerical or scaling arguments.

math.DS↗

arXiv2019

A measure theoretic result for approximation by Delone sets

With a view to establishing measure theoretic approximation properties of Delone sets, we study a setup which arises naturally in the problem of averaging almost periodic functions along exponential sequences. In this setting, we establish a full converse of the Borel-Cantelli lemma. This provides an analogue of more classical problems in the metric theory of Diophantine approximation, but with the distance to the nearest integer function replaced by distance to an arbitrary Delone set.

math.DS↗

arXiv2018

Diffraction of compatible random substitutions in one dimension

As a guiding example, the diffraction measure of a random local mixture of the two classic Fibonacci substitutions is determined and reanalysed via self-similar measures of Hutchinson type, defined by a finite family of contractions. Our revised approach yields explicit formulas for the pure point and the absolutely continuous parts, as well as a proof for the absence of singular continuous components. This approach is then extended to the family of random noble means substitutions and, as an example with an underlying 2-adic structure, to a locally randomised version of the period doubling chain. As a first step towards a more general approach, we interpret our findings in terms of a disintegration over the Kronecker factor, which is the maximal equicontinuous factor of a covering model set.

math.DS↗

arXiv2018

A natural probability measure derived from Stern's diatomic sequence

Stern's diatomic sequence with its intrinsic repetition and refinement structure between consecutive powers of $2$ gives rise to a rather natural probability measure on the unit interval. We construct this measure and show that it is purely singular continuous, with a strictly increasing, Hölder continuous distribution function. Moreover, we relate this function with the solution of the dilation equation for Stern's diatomic sequence.

math.NT↗

arXiv2018

Spectral analysis of a family of binary inflation rules

The family of primitive binary substitutions defined by $1 \mapsto 0 \mapsto 0 1^m$ with $m\in\mathbb{N}$ is investigated. The spectral type of the corresponding diffraction measure is analysed for its geometric realisation with prototiles (intervals) of natural length. Apart from the well-known Fibonacci inflation ($m=1$), the inflation rules either have integer inflation factors, but non-constant length, or are of non-Pisot type. We show that all of them have singular diffraction, either of pure point type or essentially singular continuous.

math.DS↗

arXiv2017

Reversing and extended symmetries of shift spaces

The reversing symmetry group is considered in the setting of symbolic dynamics. While this group is generally too big to be analysed in detail, there are interesting cases with some form of rigidity where one can determine all symmetries and reversing symmetries explicitly. They include Sturmian shifts as well as classic examples such as the Thue--Morse system with various generalisations or the Rudin--Shapiro system. We also look at generalisations of the reversing symmetry group to higher-dimensional shift spaces, then called the group of \emph{extended symmetries}. We develop their basic theory for faithful $\mathbb{Z}^{d}$-actions, and determine the extended symmetry group of the chair tiling shift, which can be described as a model set, and of Ledrappier's shift, which is an example of algebraic origin.

math.DS↗

arXiv2017

Geometric enumeration problems for lattices and embedded $\mathbb{Z}$-modules

In this review, we count and classify certain sublattices of a given lattice, as motivated by crystallography. We use methods from algebra and algebraic number theory to find and enumerate the sublattices according to their index. In addition, we use tools from analytic number theory to determine the asymptotic behaviour of the corresponding counting functions. Our main focus lies on similar sublattices and coincidence site lattices, the latter playing an important role in crystallography. As many results are algebraic in nature, we also generalise them to $\mathbb{Z}$-modules embedded in $\mathbb{R}^d$.

math.MG↗

arXiv2017

Diffraction of a binary non-Pisot inflation tiling

A one-parameter family of binary inflation rules in one dimension is considered. Apart from the first member, which is the well-known Fibonacci rule, no inflation factor is a unit. We identify all cases with pure point spectrum, and discuss the diffraction spectra of the other members of the family. Apart from the trivial Bragg peaks at the origin, they have purely singular continuous diffraction.

math-ph↗

arXiv2017

Geometric properties of a binary non-Pisot inflation and absence of absolutely continuous diffraction

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.

math.DS↗

arXiv2017

Averaging almost periodic functions along exponential sequences

The goal of this expository article is a fairly self-contained account of some averaging processes of functions along sequences of the form $(α^n x)^{}_{n\in\mathbb{N}}$, where $α$ is a fixed real number with $| α| > 1$ and $x\in\mathbb{R}$ is arbitrary. Such sequences appear in a multitude of situations including the spectral theory of inflation systems in aperiodic order. Due to the connection with uniform distribution theory, the results will mostly be metric in nature, which means that they hold for Lebesgue-almost every $x\in\mathbb{R}$.

math.NT↗