Dynamical Invariants from Asymptotic Composants

TL;DR

This paper introduces a simple algorithm to compute asymptotic composants in 1D inflation tiling spaces, demonstrating their effectiveness in distinguishing MLD classes and detecting symmetries, especially for pure-point spectrum tilings.

Contribution

It presents a new algorithm for determining asymptotic composants in 1D inflation tilings and shows their utility in classifying MLD tilings and identifying obstructions to reflection symmetry.

Findings

Asymptotic composants can distinguish many MLD classes of tilings.

The algorithm effectively computes asymptotic composants for primitive inflation tilings.

Asymptotic composants combined with OSD can classify all MLD classes for certain small inflation factors.

Abstract

Asymptotic composants and their incidence relations are powerful invariants of 1-dimensional inflation tilings spaces, which can distinguish many MLD classes of tilings. In particular, and unlike most other invariants, they can often provide obstructions to a tiling space being MLD to its reflection. We present a simple algorithm to determine these asymptotic composants for primitive inflation tiling spaces in one dimension, and illustrate how they can be used to tell different MLD classes of tilings apart. In an Appendix, we then show that the structure of asymptotic composants, together with the orbit separation dimension (OSD), can distinguish all MLD classes of inflations tilings with pure-point spectrum for a bunch of small inflation factors, which illustrates the power of these invariants.