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Franz Gähler

Publications and source records attributed to Franz Gähler.

At least 19 recordsLinked to original sources
arXiv2026

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 Čech cohomology being as small as it possibly could be, leaving no room for shape changes that alter the dynamics.

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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.

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arXiv2026

Covers of Tiling Spaces

We study the ways that one tiling space can be a finite regular cover of another. We classify all of the finite regular covers of a tiling space via its structure as an inverse limit space. If the tiling space $Ω$ can be written as an inverse limit $\varprojlim Γ_n$, then the étale fundamental group of $Ω$, which is defined via a limit of covers, is isomorphic to the inverse limit $\hat π_1(Ω) := \varprojlim \hat π_1(Γ_n)$ of the profinite completions of the fundamental groups $π_1(Γ_n)$. This isomorphism allows us to construct all covers of tiling spaces and to use those covers to distinguish spaces that have identical cohomology groups.

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arXiv2026

Dynamical Invariants from Asymptotic Composants

Asymptotic composants and their incidence relations are powerful invariants of 1-dimensional inflation tilings spaces, which can distinguish many MLD classes of tilings. In particular, and unlike most other invariants, they can often provide obstructions to a tiling space being MLD to its reflection. We present a simple algorithm to determine these asymptotic composants for primitive inflation tiling spaces in one dimension, and illustrate how they can be used to tell different MLD classes of tilings apart. In an Appendix, we then show that the structure of asymptotic composants, together with the orbit separation dimension (OSD), can distinguish all MLD classes of inflations tilings with pure-point spectrum for a bunch of small inflation factors, which illustrates the power of these invariants.

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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↗
arXiv2025

Dynamics and topology of the Hat family of tilings

The recently discovered Hat tiling admits a 4-dimensional family of shape deformations, including the 1-parameter family already known to yield alternate monotiles. The continuous hulls resulting from these tilings are all topologically conjugate dynamical systems, and hence have the same dynamics and topology. We construct and analyze a self-similar element of this family called the CAP tiling, and we use it to derive properties of the entire family. The CAP tiling has pure-point dynamical spectrum, which we compute explicitly, and comes from a natural cut-and-project scheme with 2-dimensional Euclidean internal space. All other members of the Hat family, in particular the original version constructed from 30-60-90 right triangles, are obtained via small modifications of the projection from this cut-and-project scheme.

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arXiv2025

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.

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arXiv2024

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.

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arXiv2022

Fibonacci direct product variation tilings

The direct product of two Fibonacci tilings can be described as a genuine stone inflation rule with four prototiles. This rule admits various modifications, which lead to 48 different inflation rules, known as the direct product variations. They all result in tilings that are measure-theoretically isomorphic by the Halmos--von Neumann theorem. They can be described as cut and project sets with characteristic windows in a two-dimensional Euclidean internal space. Here, we analyse and classify them further, in particular with respect to topological conjugacy.

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arXiv2014

Substitution Rules for Higher-Dimensional Paperfolding Structures

We present a general scheme how to construct a substitution rule for generating $d$-dimensional analogues of the paperfolding structures. This substitution is proven to be primitive, so that the translation action on the hull forms a strictly ergodic dynamical system. The substitution admits a coincidence in the sense of Dekking, which implies that the dynamical system has pure point spectrum. The same then holds true also for the diffraction spectrum. The substitution also allows us to give estimates on the complexity of the paperfolding structures, and to determine topological invariants like the Čech cohomology groups of the hull for dimensions $d\le2$.

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arXiv2014

Quotient cohomology of certain 1- and 2-dimensional substitution tiling spaces

The quotient cohomology of tiling spaces is a topological invariant that relates a tiling space to one of its factors, viewed as topological dynamical systems. In particular, it is a relative version of the tiling cohomology that distinguishes factors of tiling spaces. In this work, the quotient cohomologies within certain families of substitution tiling spaces in 1 and 2 dimensions are determined. Specifically, the quotient cohomologies for the family of the generalised Thue-Morse sequences and generalised chair tilings are presented.

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arXiv2013

Topology of the Random Fibonacci Tiling Space

We look at the topology of the tiling space of locally random Fibonacci substitution, which is defined as a-->ba with probability p, a-->ab with probability 1-p and b-->a for 0<p<1. We show that its Cech cohomology group is not finitely generated, in contrast to the case where random substitutions are applied globally.

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arXiv2013

Examples of substitution systems and their factors

The theory of substitution sequences and their higher-dimensional analogues is intimately connected with symbolic dynamics. By systematically studying the factors (in the sense of dynamical systems theory) of a substitution dynamical system, one can reach a better understanding of spectral and topological properties. We illustrate this point of view by means of some characteristic examples, including a rather universal substitution in one dimension as well as the squiral and the table tilings of the plane.

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arXiv2012

Substitution rules and topological properties of the Robinson tilings

A relatively simple substitution for the Robinson tilings is presented, which requires only 56 tiles up to translation. In this substitution, due to Joan M. Taylor, neighboring tiles are substituted by partially overlapping patches of tiles. We show that this overlapping substitution gives rise to a normal primitive substitution as well, implying that the Robinson tilings form a model set and thus have pure point diffraction. This substitution is used to compute the Cech cohomology of the hull of the Robinson tilings via the Anderson-Putnam method, and also the dynamical zeta function of the substitution action on the hull. The dynamical zeta function is then used to obtain a detailed description of the structure of the hull, relating it to features of the cohomology groups.

math.DS↗
arXiv2012

Hexagonal inflation tilings and planar monotiles

Aperiodic tilings with a small number of prototiles are of particular interest, both theoretically and for applications in crystallography. In this direction, many people have tried to construct aperiodic tilings that are built from a single prototile with nearest neighbour matching rules, which is then called a monotile. One strand of the search for a planar monotile has focussed on hexagonal analogues of Wang tiles. This led to two inflation tilings with interesting structural details. Both possess aperiodic local rules that define hulls with a model set structure. We review them in comparison, and clarify their relation with the classic half-hex tiling. In particular, we formulate various known results in a more comparative way, and augment them with some new results on the geometry and the topology of the underlying tiling spaces.

math.DS↗
arXiv2012

Spectral and topological properties of a family of generalised Thue-Morse sequences

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 singular continuous measures that emerge in mathematical diffraction theory and lead to somewhat similar distribution functions, yet with significant differences. Various properties of these measures are derived. In particular, these measures have supports of full Lebesgue measure and possess strictly increasing distribution functions. In this sense, they mark the opposite end of what is possible for singular continuous measures. For each member of the family, the underlying dynamical system possesses a topological factor with maximal pure point spectrum, and a close relation to a solenoid, which is the Kronecker factor of the system. The inflation action on the continuous hull is sufficiently explicit to permit the calculation of the corresponding dynamical zeta functions. This is achieved as a corollary of analysing the Anderson-Putnam complex for the determination of the cohomological invariants of the corresponding tiling spaces.

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arXiv2012

Combinatorics and topology of the Robinson tiling

We study the space of all tilings which can be obtained using the Robinson tiles (this is a two-dimensional subshift of finite type). We prove that it has a unique minimal subshift, and describe it by means of a substitution. This description allows to compute its cohomology groups, and prove that it is a model set.

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