On the long-range order of the Spectre tilings

TL;DR

This paper investigates the Spectre tiling, an aperiodic monotile, revealing its pure point spectrum, topological conjugacy within a family of tilings, and its relation to cut-and-project schemes, highlighting its long-range order and unique properties.

Contribution

It provides a detailed analysis of the Spectre tiling's dynamical and topological properties, establishing its pure point spectrum and connection to Rauzy fractal cut-and-project schemes, which was previously unexplored.

Findings

Spectre tilings have pure point dynamical spectrum.

All tilings in the family are topologically conjugate up to rescaling and rotation.

The diffraction measure of Spectre tilings is pure point.

Abstract

The Spectre is an aperiodic monotile for the Euclidean plane that is truly chiral in the sense that it tiles the plane without any need for a reflected tile. The topological and dynamical properties of the Spectre tilings are very similar to those of the Hat tilings. Specifically, the Spectre sits within a complex $2$-dimensional family of tilings, most of which involve two shapes rather than one. All tilings in the family give topologically conjugate dynamics, up to an overall rescaling and rotation. They all have pure point dynamical spectrum with continuous eigenfunctions and may be obtained from a $4:2$ dimensional cut-and-project scheme with regular windows of Rauzy fractal type. The diffraction measure of any Spectre tiling is pure point as well. For fixed scale and orientation, varying the shapes is MLD equivalent to merely varying the projection direction. These properties all follow from the first \v{C}ech cohomology being as small as it possibly could be, leaving no room for shape changes that alter the dynamics.