[Paper Review] Fusion tilings with infinite local complexity

Natalie Priebe Frank, Lorenzo Sadun|arXiv (Cornell University)|Jan 18, 2012

Quasicrystal Structures and Properties9 references17 citations

TL;DR

This paper introduces a formal framework for fusion tilings with infinite local complexity (ILC), where tile types form a compact space allowing geometric and combinatorial ILC. It establishes spectral theory for invariant measures, identifies eigenvalues via functional equations, and shows that complexity grows as $L^2/\epsilon^2$ despite a totally disconnected transversal, with convergence to the invariant measure not exponential due to accumulation of eigenvalues at 1.

ABSTRACT

We propose a formalism for tilings with infinite local complexity (ILC), and especially fusion tilings with ILC. We allow an infinite variety of tile types but require that the space of possible tile types be compact. Examples include solenoids, pinwheel tilings, tilings with fault lines, and tilings with infinitely many tile sizes, shapes, or labels. Special attention is given to tilings where the infinite local complexity comes purely from geometry (shears) or comes purely from combinatorics (labels). We examine spectral properties of the invariant measures and define a new notion of complexity that applies to ILC tilings.

Motivation & Objective

• To formalize tiling systems with infinite local complexity (ILC) where tile types form a compact metric space.
• To extend spectral theory to ILC fusion tilings, particularly analyzing invariant measures and eigenvalues of the transfer operator.
• To define a new complexity measure applicable to ILC tilings, distinguishing geometric and combinatorial sources of complexity.
• To investigate the structure of the transversal and show it is totally disconnected despite a continuum of tile sizes.
• To characterize eigenfunctions and eigenvalues of the transfer operator using functional equations and self-similarity.

Proposed method

• Define tiles as pairs of support (compact set in $\mathbb{R}^d$) and label in a compact metric space $\mathcal{L}$, with convergence under Hausdorff and label metrics.
• Construct tilings from prototiles with control points, ensuring convergence of supports and labels under limits.
• Define a metric on tilings based on patch agreement within expanding balls, leading to a compact hull $\Omega_T$.
• Introduce a transfer operator $\mathcal{T}$ acting on functions on tile lengths, with eigenvalue equation $\tilde{f}(2x)/\tilde{f}(x) = \lambda$ for $x \in [1,3]$, leading to functional equations.
• Use self-similarity and scaling to derive eigenfunctions via piecewise power laws, solving $3^\gamma = 2^\gamma + 1$ for complex $\gamma$.
• Analyze complexity $C(\epsilon, L)$ as $L^2/\epsilon^2$ by counting supertile and position choices within $\epsilon$-accuracy.

Experimental results

Research questions

• RQ1 How can infinite local complexity in fusion tilings be formally defined and characterized when tile types form a compact space?
• RQ2 What are the spectral properties of the invariant measures in ILC fusion tilings, particularly the eigenvalues and eigenfunctions of the transfer operator?
• RQ3 How does the complexity of approximating patches grow with scale $L$ and error $\epsilon$ in ILC tilings?
• RQ4 Why is the transversal of such tiling spaces totally disconnected despite a continuum of tile sizes?
• RQ5 What is the role of irrational rotations and scaling dynamics in determining the eigenvalue spectrum?

Key findings

• The eigenvalue $\lambda = 1$ corresponds to the unique invariant measure, with eigenfunction $\tilde{f}(x)$ satisfying $\tilde{f}(2x)/\tilde{f}(x) = 3\lambda/4$ on $[1,3/2]$, etc., and normalized by $\int_1^3 x\tilde{f}_n(x)dx = 1$.
• The eigenvalue equation $\left|\frac{3\lambda}{4}\right|^{\ln(3/2)}\left|\frac{9\lambda}{4}\right|^{\ln(4/3)}\left|\frac{3(3\lambda-2)}{4}\right|^{\ln(3/2)} = 1$ implies all eigenvalues lie inside or on the unit circle, with only $\lambda = 1$ on the boundary.
• For complex $\gamma$ solving $3^\gamma = 2^\gamma + 1$, the function $\tilde{f}(x) = x^{-(\gamma+1)}$ on $[1,2)$ and $3^\gamma x^{-(\gamma+1)}$ on $[2,3]$ is an eigenfunction with eigenvalue $\lambda = (3/2)^{\gamma-1}$.
• The spectrum of the transfer operator accumulates at $\lambda = 1$, implying non-exponential convergence to the invariant measure.
• Complexity $C(\epsilon, L)$ scales as $L^2/\epsilon^2$, reflecting the number of supertiles and position choices within $\epsilon$-accuracy.
• Despite a continuum of tile lengths, the transversal is totally disconnected, with clopen sets distinguishing tilings via combinatorial differences in supertile decomposition.