All Outputs (7)

Substitutions on compact alphabets (2025)
Journal Article
Mañibo, N., Rust, D., & Walton, J. J. (in press). Substitutions on compact alphabets. Journal of the London Mathematical Society,

We develop a systematic approach to continuous substitutions on compact Haus-dorff alphabets. Focussing on implications of irreducibility and primitivity, we highlight important features of the topological dynamics of their (generalised) subshifts. W... Read More about Substitutions on compact alphabets.

Self-similarity and limit spaces of substitution tiling semigroups (2024)
Journal Article
Walton, J., & Whittaker, M. F. (2024). Self-similarity and limit spaces of substitution tiling semigroups. Groups, Geometry, and Dynamics, 18(4), 1201-1231. https://doi.org/10.4171/ggd/807

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 semigroup... Read More about Self-similarity and limit spaces of substitution tiling semigroups.

A characterisation of linear repetitivity for cut and project sets with general polytopal windows (2024)
Journal Article
Walton, J. J. (2024). A characterisation of linear repetitivity for cut and project sets with general polytopal windows. Indagationes Mathematicae, 35(5), 1009-1056. https://doi.org/10.1016/j.indag.2024.03.003

The cut and project method is a central construction in the theory of Aperiodic Order for generating quasicrystals with pure point diffraction. Linear repetitivity (LR) is a form of ideal regularity of aperiodic patterns. Recently, Koivusalo and the... Read More about A characterisation of linear repetitivity for cut and project sets with general polytopal windows.

Spectral properties of substitutions on compact alphabets (2023)
Journal Article
Mañibo, N., Rust, D., & Walton, J. J. (2023). Spectral properties of substitutions on compact alphabets. Bulletin of the London Mathematical Society, 55(5), 2425-2445. https://doi.org/10.1112/blms.12872

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 associat... Read More about Spectral properties of substitutions on compact alphabets.

Cut and project sets with polytopal window II: Linear repetitivity (2022)
Journal Article
Koivusalo, H., & Walton, J. (2022). Cut and project sets with polytopal window II: Linear repetitivity. Transactions of the American Mathematical Society, 375(7), 5097-5149. https://doi.org/10.1090/tran/8633

In this paper we give a complete characterisation of linear repetitivity for cut and project schemes with convex polytopal windows satisfying a weak homogeneity condition. This answers a question of Lagarias and Pleasants from the 90s for a natural c... Read More about Cut and project sets with polytopal window II: Linear repetitivity.

An aperiodic tile with edge-to-edge orientational matching rules (2021)
Journal Article
Walton, J. J., & Whittaker, M. F. (2021). An aperiodic tile with edge-to-edge orientational matching rules. Journal of the Institute of Mathematics of Jussieu, 22(4), 1727-1755. https://doi.org/10.1017/S1474748021000517

We present a single, connected tile which can tile the plane but only nonperiodically. 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 standa... Read More about An aperiodic tile with edge-to-edge orientational matching rules.

Aperiodicity, rotational tiling spaces and topological space groups (2021)
Journal Article
Hunton, J., & Walton, J. J. (2021). Aperiodicity, rotational tiling spaces and topological space groups. Advances in Mathematics, 388, Article 107855. https://doi.org/10.1016/j.aim.2021.107855

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 structure... Read More about Aperiodicity, rotational tiling spaces and topological space groups.