Dynamical Invariants from Asymptotic Composants

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.

Figures (2)

• Figure 1: For the Fibonacci inflation $\varrho = [ab,a]$, seeds for the (only) left asymptotic pair of inflation fixed points are shown at the top, with their inflations underneath. If we place the scaling centre at the position of the solid vertical line, the position of the left splitting point (dashed vertical line) remains stable under inflation. Note that we need patches of at least four tiles; patches of three tiles would not grow beyond the scaling centre, and a unique continuation would not be guaranteed. Note also that the inflation swaps the two members of the pair.
• Figure 2: Windows of the tiling spaces with inflation $\varrho_M = [aca,a,b]$ (left) and $\varrho_R = [aac,a,b]$ (right).

Theorems & Definitions (13)

• Lemma 1
• proof
• Lemma 2
• proof
• Remark 3
• Theorem 4
• Remark 5
• Corollary 6
• Remark 7
• Example 8