[Paper Review] The Common Structure For Objects In Aperiodic Order And The Theory Of Local Matching Topology

Yasushi Nagai|arXiv (Cornell University)|Nov 12, 2018

Quasicrystal Structures and PropertiesMaterials Science8 references3 citations

TL;DR

This paper establishes a unified topological framework for aperiodic order by introducing abstract pattern spaces that generalize tilings, Delone sets, and other aperiodic objects. It defines a local matching topology and uniform structure on these spaces, proving Hausdorff separation and completeness under mild conditions, and shows that finite local complexity (FLC) implies compactness of the continuous hull—extending a key result from Euclidean to general metric spaces with group actions.

ABSTRACT

In aperiodic order, non-periodic but "ordered" objects such as tilings, Delone sets, functions and measures are investigated. In this article we depict the common structure of these objects by using the general framework of abstract pattern spaces. In particular, using the common structure we define local matching topology and uniform structure for objects such as tilings in quite a general space and a symmetry group. We prove Hausdorff property of the topology and the completeness of the uniform structure under a mild assumption. We also prove finite local complexity implies the compactness of the continuous hull and often the converse holds.

Motivation & Objective

• To unify the study of aperiodic objects such as tilings, Delone sets, and functions under a common mathematical structure.
• To define a local matching topology and uniform structure on abstract pattern spaces that generalizes existing constructions in Euclidean space.
• To prove the Hausdorff property and completeness of the local matching uniform structure under mild assumptions.
• To establish that finite local complexity (FLC) implies compactness of the continuous hull in this general setting.
• To show that compactness of the continuous hull often implies FLC, establishing a converse under additional conditions.

Proposed method

• Axiomatize the common structural features of aperiodic objects via a cutting-off operation and a group action, defining abstract pattern spaces.
• Define the local matching topology using entourages based on compact sets and neighborhoods in the symmetry group Γ.
• Define the local matching uniform structure via a basis of entourages U_{K,V} = {(P,Q) | P∧K = γ(Q∧K) for some γ ∈ V} for compact K ⊂ X and neighborhood V ⊂ Γ.
• Prove the topology is Hausdorff (Proposition 3.7) and metrizable (Corollary 3.8) under mild assumptions.
• Establish completeness of the local matching uniform structure on subspaces of abstract pattern spaces (Theorem 3.19), using a diagonalization argument.
• Use completeness and total boundedness to prove that FLC implies compactness of the continuous hull (Theorem 3.25).

Experimental results

Research questions

• RQ1 Does the local matching topology on abstract pattern spaces yield a Hausdorff topology under mild assumptions?
• RQ2 Is the local matching uniform structure complete on abstract pattern spaces when the ambient space is a proper metric space with a proper group action?
• RQ3 Does finite local complexity (FLC) imply compactness of the continuous hull in this general framework?
• RQ4 Can the compactness of the continuous hull imply FLC under additional conditions?
• RQ5 How does local derivability affect the inheritance of FLC and compactness of the continuous hull?

Key findings

• The local matching topology on abstract pattern spaces is Hausdorff under mild assumptions, ensuring topological separation.
• The local matching uniform structure is complete on subspaces of abstract pattern spaces when the group action is proper and the space is proper.
• Finite local complexity (FLC) implies compactness of the continuous hull in the local matching topology, generalizing a known result from R^d to general metric spaces.
• Compactness of the continuous hull implies FLC under the additional assumption that the set A(P∧K) is finite for each compact K ⊂ X.
• For Delone sets and tilings with finitely many types up to Γ-action, FLC is equivalent to compactness of the continuous hull.
• The result extends the classical equivalence between FLC and hull compactness beyond Euclidean space to general proper metric spaces with isometric group actions.