arXiv subjects

James J. Walton

Publications and source records attributed to James J. Walton.

14 recordsLinked to original sources

arXiv2026

Unique ergodicity, and not, for primitive substitutions on compact alphabets containing isolated points

We consider generalised subshifts generated by continuous substitution on compact Hausdorff alphabets. Although primitivity still implies minimality of the subshift generated by the substitution, Durand, Ormes and Petite showed, in contrast to the finite case, that primitivity no longer implies unique ergodicity, by constructing counter-examples with Cantor alphabet. Here we show that, even for the arguably simplest case of the one-point compactification of the natural numbers, primitivity is still insufficient for unique ergodicity, or even the existence of a natural length function. In previous work with Mañibo and Rust we showed that, for irreducible substitutions, unique ergodicity and existence of a natural length function follow from strong power convergence of the renormalised substitution operator $T$; sufficient criteria were also developed that can sometimes confirm this property. We show a partial converse to this: for irreducible substitutions admitting a natural length function (for instance, all irreducible constant length substitutions), unique ergodicity implies strong power convergence of $T$. We then consider the case of alphabets with only finitely many accumulation points, showing how upper bounds (and usually an exact formula) for the essential spectral radius of $T$ can be derived from associated finite substitutions, determined by the behaviour of substitution of the accumulation points. This may sometimes be used to show quasi-compactness of $T$, and thus unique ergodicity for primitive substitutions. For primitive substitutions of alphabets with at least one isolated point, we show that strong power convergence and quasi-compactness of $T$ are equivalent and, in fact, that these properties are equivalent to iteration of substitution growing words in length uniformly across all seeds in the alphabet.

math.DS↗

arXiv2026

Recognisability for generalised hierarchical pattern spaces of finite local complexity

We develop a general framework of Euclidean patterns and pattern spaces of translational finite local complexity (FLC), analogues of translational tiling spaces. The notion of a self affine substitution of tilings is extended to both individual patterns and pattern spaces, which we ask are mapped onto by a local derivation map (the analogue of a sliding block code map from Symbolic Dynamics) from its own expansion by a linear map. We prove recognisability for these, with no minimality requirements. In particular we show that, for each pattern in the space, its substitutional pre-images are translation equivalent. Sizes of all fibres are then determined by relative groups of translational periods. This answers an open question of Cortez and Solomyak, on whether non-periodic tilings necessarily have unique pre-images under substitution: they do, even for a wider notion of pattern and being substitutional. It is shown that there exists a power of the substitution under which any given pattern of the pattern space has multiple pre-images if and only if it has disconnected group of periods. This is equivalent to being periodic in the standard return discrete cases of tilings and Delone sets, but our results also cover examples with non-discrete groups of periods, such as spaces of uniformly discrete but non-relatively dense point sets.

math.DS↗

arXiv2025

When is a cut and project set substitutional?

Cut and project sets are obtained by projecting an irrational slice through a lattice to a lower dimensional subspace. Under standard conditions, the resulting pattern has no translational periods even though it retains some regularity of the lattice. Cut and project sets are one of the archetypical examples of patterns featuring aperiodic order, the other construction methods being by substitution and matching rules. Many early examples of aperiodic tilings, including the famous Penrose and Ammann--Beenker tilings, have a description from all of these methods. In this article we answer the following question, in the case of a Euclidean total space: what property of the cut and project data characterises when the resulting cut and project sets may also be defined by a substitution rule?

math.DS↗

arXiv2025

Substitutions on compact alphabets

We develop a systematic approach to continuous substitutions on compact Hausdorff alphabets. Focussing on implications of irreducibility and primitivity, we highlight important features of the topological dynamics of their (generalised) subshifts. We then reframe questions from ergodic theory in terms of spectral properties of a corresponding substitution operator. This requires an extension of standard Perron--Frobenius theory to the setting of Banach lattices. As an application, we identify computable criteria that guarantee quasi-compactness of the substitution operator. This allows unique ergodicity to be verified for several classes of examples. For instance, it follows that every primitive and constant length substitution on an alphabet with an isolated point is uniquely ergodic, a result which fails when there are no isolated points.

math.DS↗

arXiv2024

A characterisation of linear repetitivity for cut and project sets with general polytopal windows

The cut and project method is a central construction in the theory of Aperiodic Order for generating quasicrystals with pure point diffraction. Linear repetitivity ({\bf LR}) is a form of ideal regularity of aperiodic patterns. Recently, Koivusalo and the present author characterised {\bf LR} for cut and project sets with convex polytopal windows whose supporting hyperplanes are commensurate with the lattice, the weak homogeneity property. For such cut and project sets, we show that {\bf LR} is equivalent to two properties. One is a low complexity condition, which may be determined from the cut and project data by calculating the ranks of the intersections of the projection of the lattice to the internal space with the subspaces parallel to the supporting hyperplanes of the window. The second condition is that the projection of the lattice to the internal space is Diophantine (or `badly approximable'), which loosely speaking means that the lattice points in the total space stay far from the physical space, relative to their norm. We review then extend these results to non-convex and disconnected polytopal windows, as well as windows with polytopal partitions producing cut and project sets of labelled points. Moreover, we obtain a complete characterisation of {\bf LR} in the fully general case, where weak homogeneity is not assumed. Here, the Diophantine property must be replaced with an inhomogeneous analogue. We show that cut and project schemes with internal space isomorphic to $\mathbb{R}^n \oplus G \oplus \mathbb{Z}^r$, for $G$ finite Abelian, can, up to MLD equivalence, be reduced to ones with internal space $\mathbb{R}^n$, so our results also cover cut and project sets of this form, such as the (generalised) Penrose tilings.

math.DS↗

arXiv2023

Cut and project sets with polytopal window I: complexity

We calculate the growth rate of the complexity function for polytopal cut and project sets. This generalises work of Julien where the almost canonical condition is assumed. The analysis of polytopal cut and project sets has often relied on being able to replace acceptance domains of patterns by so-called cut regions. Our results correct mistakes in the literature where these two notions are incorrectly identified. One may only relate acceptance domains and cut regions when additional conditions on the cut and project set hold. We find a natural condition, called the quasicanonical condition, guaranteeing this property and demonstrate via counterexample that the almost canonical condition is not sufficient for this. We also discuss the relevance of this condition for the current techniques used to study the algebraic topology of polytopal cut and project sets.

math.DS↗

arXiv2022

Spectral properties of substitutions on compact alphabets

We consider substitutions on compact alphabets and provide sufficient conditions for the diffraction to be pure point, absolutely continuous and singular continuous. This allows one to construct examples for which the Koopman operator on the associated function space has specific spectral components. For abelian bijective substitutions, we provide a dichotomy result regarding the spectral type of the diffraction. We also provide the first example of a substitution that has countably infinite Lebesgue spectral components and countably infinite singular continuous components. Lastly, we give a non-constant length substitution on a countably infinite alphabet that gives rise to substitutive Delone sets of infinite type. This extends the spectral theory of substitutions on finite alphabets and Delone sets of finite type with inflation symmetry.

math.DS↗

arXiv2021

Self-similarity and limit spaces of substitution tiling semigroups

We show that Kellendonk's tiling semigroup of an FLC substitution tiling is self-similar, in the sense of Bartholdi, Grigorchuk and Nekrashevych. We extend the notion of the limit space of a self-similar group to the setting of self-similar semigroups, and show that it is homeomorphic to the Anderson--Putnam complex for such substitution tilings, with natural self-map induced by the substitution. Thus, the inverse limit of the limit space, given by the limit solenoid of the self-similar semigroup, is homeomorphic to the translational hull of the tiling.

math.DS↗

arXiv2021

An aperiodic tile with edge-to-edge orientational matching rules

We present a single, connected tile which can tile the plane but only non-periodically. The tile is hexagonal with edge markings, which impose simple rules as to how adjacent tiles are allowed to meet across edges. The first of these rules is a standard matching rule, that certain decorations match across edges. The second condition is a new type of matching rule, which allows tiles to meet only when certain decorations in a particular orientation are given the opposite charge. This forces the tiles to form a hierarchy of triangles, following a central idea of the Socolar--Taylor tilings. However, the new edge-to-edge orientational matching rule forces this structure in a very different way, which allows for a surprisingly simple proof of aperiodicity. We show that the hull of all tilings satisfying our rules is uniquely ergodic and that almost all tilings in the hull belong to a minimal core of tilings generated by substitution. Identifying tilings which are charge-flips of each other, these tilings are shown to have pure point dynamical spectrum and a regular model set structure.

math.MG↗

arXiv2021

Aperiodicity, rotational tiling spaces and topological space groups

We study the rotational structures of aperiodic tilings in Euclidean space of arbitrary dimension using topological methods. Classical topological approaches to the study of aperiodic patterns have largely concentrated just on translational structures, studying an associated space, the continuous hull, here denoted $Ω_t$. In this article we consider two further spaces $Ω_r$ and $Ω_G$ (the rotational hulls) which capture the full rigid motion properties of the underlying patterns. The rotational hull $Ω_r$ is shown to be a matchbox manifold which contains $Ω_t$ as a sub-matchbox manifold. We develop new S-MLD invariants derived from the homotopical and cohomological properties of these spaces demonstrating their computational as well as theoretical utility. We compute these invariants for a variety of examples, including a class of 3-dimensional aperiodic patterns, as well as for the space of periodic tessellations of $\mathbb{R}^3$ by unit cubes. We show that the classical space group of symmetries of a periodic pattern may be recovered as the fundamental group of our space $Ω_G$. Similarly, for those patterns associated to quasicrystals, the crystallographers' aperiodic space group may be recovered as a quotient of our fundamental invariant.

math.AT↗

arXiv2020

Cut and project sets with polytopal window II: linear repetitivity

This paper gives a complete classification of linear repetitivity (LR) for a natural class of aperiodic Euclidean cut and project schemes with convex polytopal windows. Our results cover those cut and project schemes for which the lattice projects densely into the internal space and (possibly after translation) hits each supporting hyperplane of the polytopal window. Our main result is that LR is satisfied if and only if the patterns are of low complexity (property C), and the projected lattice satisfies a Diophantine condition (property D). Property C can be checked by computation of the ranks and dimensions of linear spans of the stabiliser subgroups of the supporting hyperplanes, as investigated in Part I to this article. To define the correct Diophantine condition D, we establish new results on decomposing polytopal cut and project schemes to factors, developing concepts initiated in the work of Forrest, Hunton and Kellendonk. This means that, when C is satisfied, the window splits into components which induce a compatible splitting of the lattice. Then property D is the requirement that, for any suitable decomposition, these factors do not project close to the origin in the internal space, relative to the norm in the total space. On each factor, this corresponds to the usual notion from Diophantine Approximation of a system of linear forms being badly approximable. This extends previous work on cubical cut and project schemes to a very general class of cut and project schemes. We demonstrate our main theorem on several examples, and derive some further consequences of our main theorem, such as the equivalence LR, positivity of weights and satisfying a subadditive ergodic theorem for this class of polytopal cut and project sets.

math.DS↗

arXiv2016

Cohomology of rotational tiling spaces

A spectral sequence is defined which converges to the Čech cohomology of the Euclidean hull of a tiling of the plane with Euclidean finite local complexity. The terms of the second page are determined by the so-called ePE homology and ePE cohomology groups of the tiling, and the only potentially non-trivial boundary map has a simple combinatorial description in terms of its local patches. Using this spectral sequence, we compute the Čech cohomology of the Euclidean hull of the Penrose tilings.

math.AT↗

arXiv2016

Pattern-Equivariant Homology

Pattern-equivariant (PE) cohomology is a well-established tool with which to interpret the Čech cohomology groups of a tiling space in a highly geometric way. In this paper we consider homology groups of PE infinite chains. We establish Poincaré duality between the PE cohomology and PE homology. The Penrose kite and dart tilings are taken as our central running example, we show how through this formalism one may give highly approachable geometric descriptions of the generators of the Čech cohomology of their tiling space. These invariants are also considered in the context of rotational symmetry. Poincaré duality fails over integer coefficients for the `ePE homology groups' based upon chains which are PE with respect to orientation-preserving Euclidean motions between patches. As a result we construct a new invariant, which is of relevance to the cohomology of rotational tiling spaces. We present an efficient method of computation of the PE and ePE (co)homology groups for hierarchical tilings.

math.GN↗

arXiv2014

Pattern-Equivariant Homology of Finite Local Complexity Patterns

This thesis establishes a generalised setting with which to unify the study of finite local complexity (FLC) patterns. The abstract notion of a "pattern" is introduced, which may be seen as an analogue of the space group of isometries preserving a tiling but where, instead, one considers partial isometries preserving portions of it. These inverse semigroups of partial transformations are the suitable analogue of the space group for patterns with FLC but few global symmetries. In a similar vein we introduce the notion of a \emph{collage}, a system of equivalence relations on the ambient space of a pattern, which we show is capable of generalising many constructions applicable to the study of FLC tilings and Delone sets, such as the expression of the tiling space as an inverse limit of approximants. An invariant is constructed for our abstract patterns, the so called pattern-equivariant (PE) homology. These homology groups are defined using infinite singular chains on the ambient space of the pattern, although we show that one may define cellular versions which are isomorphic under suitable conditions. For FLC tilings these cellular PE chains are analogous to the PE cellular cochains \cite{Sadun1}. The PE homology and cohomology groups are shown to be related through Poincaré duality. An efficient and highly geometric method for the computation of the PE homology groups for hierarchical tilings is presented. The rotationally invariant PE homology groups are shown not to be a topological invariant for the associated tiling space and seem to retain extra information about global symmetries of tilings in the tiling space. We show how the PE homology groups may be incorporated into a spectral sequence converging to the Čech cohomology of the rigid hull of a tiling. These methods allow for a simple computation of the Čech cohomology of the rigid hull of the Penrose tilings.

math.GN↗