16 citations · 26 across the 17 of their papers we have counts for
18 papers
Unique ergodicity, and not, for primitive substitutions on compact alphabets containing isolated points
James J. Walton
We consider generalised subshifts generated by continuous substitution on compact Hausdorff alphabets. Although primitivity still implies minimality of the subshift generated by th…
When is a cut and project set substitutional?
Edmund Harriss, Henna Koivusalo, James J. Walton
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 trans…
Recognisability for generalised hierarchical pattern spaces of finite local complexity
James J. Walton
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 se…
A characterisation of linear repetitivity for cut and project sets with general polytopal windows
James J. Walton
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…
Substitutions on compact alphabets
Neil Mañibo, Dan Rust, James J. Walton
We develop a systematic approach to continuous substitutions on compact Hausdorff alphabets. Focussing on implications of irreducibility and primitivity, we highlight important fea…
Self-similarity and limit spaces of substitution tiling semigroups
James J. Walton, Michael F. Whittaker
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…
Spectral properties of substitutions on compact alphabets
Neil Mañibo, Dan Rust, James J. Walton
We consider substitutions on compact alphabets and provide sufficient conditions for the diffraction to be pure point, absolutely continuous and singular continuous. This allows on…
Cut and project sets with polytopal window II: linear repetitivity
Henna Koivusalo, James J. Walton
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…
Cut and project sets with polytopal window I: complexity
Henna Koivusalo, James J. Walton
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 ana…
An aperiodic tile with edge-to-edge orientational matching rules
James J. Walton, Michael F. Whittaker
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…
Aperiodicity, rotational tiling spaces and topological space groups
John Hunton, James J. Walton
We study the rotational structures of aperiodic tilings in Euclidean space of arbitrary dimension using topological methods. Classical topological approaches to the study of aperio…
Statistics of patterns in typical cut and project sets
Alan Haynes, Antoine Julien, Henna Koivusalo +1
In this article pattern statistics of typical cubical cut and project sets are studied. We give estimates for the rate of convergence of appearances of patches to their asymptotic…
Cohomology of rotational tiling spaces
James J. Walton
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…
Perfectly ordered quasicrystals and the Littlewood conjecture
Alan Haynes, Henna Koivusalo, James Walton
Linearly repetitive cut and project sets are mathematical models for perfectly ordered quasicrystals. In a previous paper we presented a characterization of linearly repetitive cut…
Linear repetitivity and subadditive ergodic theorems for cut and project sets
Alan Haynes, Henna Koivusalo, James Walton
By using a combination of algebraic, geometric, and dynamical techniques, together with input from higher dimensional Diophantine approximation, we give a complete characterization…
Gaps problems and frequencies of patches in cut and project sets
Alan Haynes, Henna Koivusalo, Lorenzo Sadun +1
We establish a connection between gaps problems in Diophantine approximation and the frequency spectrum of patches in cut and project sets with special windows. Our theorems provid…
Pattern-Equivariant Homology of Finite Local Complexity Patterns
James J. Walton
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 b…
Pattern-Equivariant Homology
James J. Walton
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 cons…