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Tiling spaces are covering spaces over irrational tori

Darío Alatorre, Diego Rodríguez-Guzmán

TL;DR

This work develops a diffeological framework for one-dimensional tiling spaces and links them to irrational tori. It constructs two fibrational structures over $\mathbb{T}_\alpha$—a Cantor-fiber bundle in the diffeological setting and a flow-based $\mathbb{R}$-principal bundle—using return modules and irrational flows, and leverages the diffeological classification of irrational tori to transfer equivalence notions. The authors prove that strong orbit equivalence of canonical projection tilings corresponds to diffeomorphism of the base irrational tori and to $GL(2,\mathbb{Z})$-conjugacy of the slopes, and they compute $\pi_1(\Omega_\alpha)\cong \mathbb{F}_2$, enabling a structural classification of one-dimensional tiling spaces. Overall, the paper integrates diffeology, tiling theory, and Sturmian dynamics to provide new tools for understanding smooth structures and equivalences in tiling spaces, with potential implications for noncommutative geometry and aperiodic order.

Abstract

We study tiling spaces in the diffeological context. We prove some basic diffeological properties for tiling spaces and analyze two different fiber bundle structures of tiling spaces over irrational tori. We use the diffeological classification of irrational tori which captures their arithmetical escence in order to inherit the diffeological equivalence in the context of one-dimensional tiling spaces.

Tiling spaces are covering spaces over irrational tori

TL;DR

This work develops a diffeological framework for one-dimensional tiling spaces and links them to irrational tori. It constructs two fibrational structures over —a Cantor-fiber bundle in the diffeological setting and a flow-based -principal bundle—using return modules and irrational flows, and leverages the diffeological classification of irrational tori to transfer equivalence notions. The authors prove that strong orbit equivalence of canonical projection tilings corresponds to diffeomorphism of the base irrational tori and to -conjugacy of the slopes, and they compute , enabling a structural classification of one-dimensional tiling spaces. Overall, the paper integrates diffeology, tiling theory, and Sturmian dynamics to provide new tools for understanding smooth structures and equivalences in tiling spaces, with potential implications for noncommutative geometry and aperiodic order.

Abstract

We study tiling spaces in the diffeological context. We prove some basic diffeological properties for tiling spaces and analyze two different fiber bundle structures of tiling spaces over irrational tori. We use the diffeological classification of irrational tori which captures their arithmetical escence in order to inherit the diffeological equivalence in the context of one-dimensional tiling spaces.
Paper Structure (16 sections, 16 theorems, 58 equations, 2 figures)

This paper contains 16 sections, 16 theorems, 58 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Background on Tiling Spaces
  4. Tilings
  5. Tiling spaces
  6. Equivalence of tiling spaces
  7. Geometrical and combinatorial aspects of tiling spaces
  8. Background on Diffeology
  9. Diffeological Spaces
  10. Diffeological Fiber Bundles
  11. The irrational torus
  12. Diffeological Tiling Spaces
  13. The Diffeological Space $\Omega$
  14. The bundle $\tilde{p}:\Omega \to \mathbb{T}_\alpha$
  15. The structure groupoid for $\tilde{p}: \Omega \to \mathbb{T}_v$.
  16. ...and 1 more sections

Key Result

Proposition 1

Let $T$ be a $FLC$ tiling. Then $\Omega_T$ is compact and the following conditions are equivalent:

Figures (2)

  • Figure 1: The branched 1-dimensional manifolds $\Omega_0$, $\Omega_1$ and $\Omega_2$ associated to the subtitution rule $a\mapsto b$, $b\mapsto ab$. Note that this image is only for illustrative purposes but it is not precisely what is needed -- we are omitting here a technicality called forcing the border, see Remark \ref{['rmk:forcetheborder']}.
  • Figure 2: Cut and project tiling from dimension 2 to 1.

Theorems & Definitions (49)

  • Definition 1
  • Definition 2
  • Proposition 1
  • Theorem 2: AndPutsadun03inverseBBG06
  • Remark 1
  • Theorem 3: Sadun-Williams, 2001
  • Example 1
  • Definition 3
  • Definition 4
  • Remark 2
  • ...and 39 more