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Michael Baake

Publications and source records attributed to Michael Baake.

At least 19 recordsLinked to original sources
arXiv2026

Recombination in discrete and continuous time from the viewpoint of Markov embedding

The classic recombination equation, both in discrete and in continuous time, can be solved in a way that derives from the Markov chain of a partitioning process. Here, we revisit this structure from the point of view of the Markov embedding problem. In particular, we analyse when a discrete-time Markov matrix of recombination type can occur in a time-homogeneous Markov semigroup that is generated by a recombination rate matrix. En route, we also show that such rate matrices (or Markov generators) generally do not form a matrix algebra, but span a real Lie algebra.

math.PR↗
arXiv2026

Embedding of sub-stochastic matrices

The classic embedding problem for finite-dimensional Markov matrices has a natural counterpart for sub-stochastic matrices, which is analysed and discussed here. One necessary and sufficient characterisation of embeddability can be given via the unique extension of a sub-stochastic matrix to a stochastic one with one added state in conjunction with the embedding of this extension. This is then explicitly treated for $d\leqslant 3$.

math.PR↗
arXiv2026

Renormalisation techniques for inflation systems and some of their applications

Exact renormalisation techniques are important and powerful, particularly for inflation-generated systems. We review recent results in this direction. We recall the necessary notions for inflation systems and show the renormalisation principle, which allows us to obtain exact values of highly erratic functions, such as window covariograms. We apply these techniques to compute the diffraction pattern of the new monotile tilings with arbitrary precision. We also recall a recent invariant for system with pure-point spectrum, the orbit separation dimension, and its relation to renormalisation. Lastly, we recall results beyond the pure-point spectrum setting and show how renormalisation and Lyapunov exponents can be used to exclude the presence of absolutely continuous part of the spectra.

math.DS↗
arXiv2025

Embedding of reversible Markov matrices

The embeddability of reversible Markov matrices into time-homogeneous Markov semigroups is revisited, with some focus on simplifications and extensions. In particular, we do not demand irreducibility and consider weakly reversible matrices as well as reversible matrices with negative eigenvalues.

math.PR↗
arXiv2025

Pair correlations of one-dimensional model sets and monstrous covariograms of Rauzy fractals

The averaged distance structure of one-dimensional regular model sets is determined via their pair correlation functions. The latter lead to covariograms and cross covariograms of the windows, which give continuous functions in internal space. While they are simple tent-shaped, piecewise linear functions for intervals, the typical case for inflation systems leads to convolutions of Rauzy fractals, which are difficult to compute. In the presence of an inflation structure, an alternative path is possible via the exact renormalisation structures of the pair correlation functions. We introduce this approach and derive two concrete examples, which display an unexpectedly complex and wild behaviour.

math.MG↗
arXiv2025

Dynamical spectrum of power-free integers in quadratic number fields and beyond

Power-free integers and related lattice subsets give rise to interesting dynamical systems. They are revisited from a spectral perspective, in the setting of the Halmos--von Neumann theorem. With respect to the natural patch frequency measure, also known as the Mirsky measure, many of these systems have pure-point dynamical spectrum, but trivial topological point spectrum. We calculate the spectra explicitly, in additive notation, and derive their group structure, both for a large class of $\cB$-free lattice systems in $\RR^d$ and for power-free integers in quadratic number fields. Further, in all cases, the eigenfunctions can be given in closed form, via the Fourier--Bohr coefficients of generic elements and their translates, which form a subset of full Mirsky measure. Based on a simple argument via Kolmogorov's strong law of large numbers, we show how the Fourier--Bohr coefficients also provide the eigenfunctions for the unique measure of maximal entropy, and that we get phase consistency for both measures.

math.DS↗
arXiv2025

Diffraction of the Hat and Spectre tilings and some of their relatives

The diffraction spectra of the Hat and Spectre monotile tilings, which are known to be pure point, are derived and computed explicitly. This is done via model set representatives of self-similar members in the topological conjugacy classes of the Hat and the Spectre tiling, which are the CAP and the CASPr tiling, respectively. This is followed by suitable reprojections of the model sets to represent the original Hat and Spectre tilings, which also allows to calculate their Fourier--Bohr coefficients explicitly. Since the windows of the underlying model sets have fractal boundaries, these coefficients need to be computed via an exact renormalisation cocycle in internal space.

math.MG↗
arXiv2024

On the long-range order of the Spectre tilings

The Spectre is an aperiodic monotile for the Euclidean plane that is truly chiral in the sense that it tiles the plane without any need for a reflected tile. The topological and dynamical properties of the Spectre tilings are very similar to those of the Hat tilings. Specifically, the Spectre sits within a complex $2$-dimensional family of tilings, most of which involve two shapes rather than one. All tilings in the family give topologically conjugate dynamics, up to an overall rescaling and rotation. They all have pure point dynamical spectrum with continuous eigenfunctions and may be obtained from a $4:2$ dimensional cut-and-project scheme with regular windows of Rauzy fractal type. The diffraction measure of any Spectre tiling is pure point as well. For fixed scale and orientation, varying the shapes is MLD equivalent to merely varying the projection direction. These properties all follow from the first \v{C}ech cohomology being as small as it possibly could be, leaving no room for shape changes that alter the dynamics.

math.DS↗
arXiv2024

On the full centraliser of Erd\H{o}s $\cB$-free shifts

The sets of $\cB$-free integers are considered with respect to (reversing) symmetries. It is well known that, for a large class of them, the centraliser of the associated $\cB$-free shift (otherwise known as its automorphism group) is trivial. We extend this result to the full centraliser, which effectively means to show that all self-homeomorphisms of the $\cB$-free shift that commute with some power of the shift are shifts themselves. This also leads to the result that the full normaliser agrees with the normaliser for this class, which is the semi-direct product of the centraliser with the cyclic group of order two generated by reflection.

math.DS↗
arXiv2024

An alternative recursive approach to functions of simple triangular matrices

The computation of matrix functions is a well-studied problem. Of special importance are the exponential and the logarithm of a matrix, where the latter also raises existence and uniqueness questions. This is particularly relevant in the context of matrix semigroups and their generators. Here, we look at matrix functions of triangular matrices, where a recursive approach is possible when the matrix has simple spectrum. The special feature is that no knowledge of eigenvectors is required, and that the same recursion applies to the computation of multiple functions or semigroups simultaneously.

math.RA↗
arXiv2024

A multiple coupon collection process and its Markov embedding structure

The embedding problem of Markov transition matrices into continuous-time Markov semigroups is a classic problem that regained a lot of impetus and activities in recent years. We consider it here for the following generalisation of the well-known coupon collection process: from a finite set of distinct objects, a subset is drawn repeatedly according to some probability distribution, independently and with replacement, and each time united with the set of objects sampled so far. We derive and interpret properties of and explicit conditions for the resulting discrete-time Markov chain to be representable within a semigroup or a flow of a continuous-time process of the same type.

math.PR↗
arXiv2024

On the algebra of equal-input matrices in time-inhomogeneous Markov flows

Markov matrices of equal-input type constitute a widely used model class. The corresponding equal-input generators span an interesting subalgebra of the real matrices with zero row sums. Here, we summarise some of their amazing properties and discuss the corresponding Markov embedding problem, both homogeneous and inhomogeneous in time. In particular, we derive exact and explicit solutions for time-inhomogeneous Markov flows with non-commuting generator families of equal-input type and beyond.

math.PR↗
arXiv2023

A naturally appearing family of Cantorvals

The aim of this note is to show the existence of a large family of Cantorvals arising in the projection description of primitive two-letter substitutions. This provides a common and naturally occurring class of Cantorvals.

math.DS↗
arXiv2023

On $k$-free numbers in cyclotomic fields: entropy, symmetries and topological invariants

Point sets of number-theoretic origin, such as the visible lattice points or the $k$-th power free integers, have interesting geometric and spectral properties and give rise to topological dynamical systems that belong to a large class of subshifts with positive topological entropy. Among them are $\cB$-free systems in one dimension and their higher-dimensional generalisations, most prominently the $k$-free integers in algebraic number fields. Here, we extend previous work on quadratic fields to the class of cyclotomic fields. In particular, we discuss their entropy and extended symmetries, with special focus on the interplay between dynamical and number-theoretic notions.

math.DS↗
arXiv2023

On the Fibonacci tiling and its modern ramifications

In the last 30 years, the mathematical theory of aperiodic order has developed enormously. Many new tilings and properties have been discovered, few of which are covered or anticipated by the early papers and books. Here, we start from the well-known Fibonacci chain to explain some of them, with pointers to various generalisations as well as to higher-dimensional phenomena and results. This should give some entry points to the modern literature on the subject.

math.MG↗
arXiv2023

Orbit separation dimension as complexity measure for primitive inflation tilings

Orbit separation dimension (OSD), previously introduced as amorphic complexity, is a powerful complexity measure for topological dynamical systems with pure-point spectrum. Here, we develop methods and tools for it that allow a systematic application to translation dynamical systems of tiling spaces that are generated by primitive inflation rules. These systems share many nice properties that permit the explicit computation of the OSD, thus providing a rich class of examples with non-trivial OSD.

math.DS↗
arXiv2023

Embedding of Markov matrices for $d\leqslant 4$

The embedding problem of Markov matrices in Markov semigroups is a classic problem that regained a lot of impetus and activities through recent needs in phylogeny and population genetics. Here, we give an account for dimensions $d\leqslant 4$, including a complete and simplified treatment of the case $d=3$, and derive the results in a systematic fashion, with an eye on the potential applications. Further, we reconsider the setup of the corresponding problem for time-inhomogeneous Markov chains, which is needed for real-world applications because transition rates need not be constant over time. Additional cases of this more general embedding occur for any $d\geqslant 3$. We review the known case of $d=3$ and describe the setting for future work on $d=4$.

math.PR↗