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Cut and project schemes in the Poincaré disc: From cocompact Fuchsian groups to chaotic Delone sets

Richard A. Howat, Tony Samuel, Ayşe Yıltekin-Karataş

Abstract

A question raised by Davies et al [Phys. Rev. Lett. 131, 2023] is: "Can developing new cut and project models, where the lattice is not square or the curve is non-linear, generate better performing graded metamaterials?" In this article, we study a natural construction of such a cut and project scheme, namely, cut and project schemes in relation to cocompact Fuchsian groups acting on the Poincaré disc model of hyperbolic space. We present a condition on the fundamental domain (a hyperbolic polygon) of the group so that the resulting cut and project set $S \subset \mathbb{R}$ is a chaotic Delone set. We also investigate the set of tile lengths of $S$, namely $\mathcal{L}_{S} = \{ z - y : z,y \in S, \, z > y \; \text{and} \; (y,z) \cap S = \emptyset \}$, and show that this set is countably infinite. Finally, we apply our results to cocompact Fuchsian triangle groups and show that the resulting cut and project sets are chaotic Delone, complementing and extending the work of López et al. [Discrete Contin. Dyn. Syst. 41, 2021].

Cut and project schemes in the Poincaré disc: From cocompact Fuchsian groups to chaotic Delone sets

Abstract

A question raised by Davies et al [Phys. Rev. Lett. 131, 2023] is: "Can developing new cut and project models, where the lattice is not square or the curve is non-linear, generate better performing graded metamaterials?" In this article, we study a natural construction of such a cut and project scheme, namely, cut and project schemes in relation to cocompact Fuchsian groups acting on the Poincaré disc model of hyperbolic space. We present a condition on the fundamental domain (a hyperbolic polygon) of the group so that the resulting cut and project set is a chaotic Delone set. We also investigate the set of tile lengths of , namely , and show that this set is countably infinite. Finally, we apply our results to cocompact Fuchsian triangle groups and show that the resulting cut and project sets are chaotic Delone, complementing and extending the work of López et al. [Discrete Contin. Dyn. Syst. 41, 2021].
Paper Structure (10 sections, 12 theorems, 21 equations, 6 figures)

This paper contains 10 sections, 12 theorems, 21 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Length spectrum
  4. The geodesic flow on the unit tangent bundle
  5. Cocompact Fuchsian triangle groups
  6. Quadrilateral fundamental domain
  7. Hexagonal fundamental domain
  8. Chaotic Delone sets
  9. From hyperbolic polygons to chaotic Delone sets
  10. Applications to cocompact triangle groups

Key Result

Theorem 1.1

Let $\Gamma$ be a cocompact triangle group with signature $(m_1, m_2, m_3)$, $\mathcal{F}$ be a fundamental domain for $\Gamma$ with centre point $x$. Moreover, let $\ell$, be a geodesic whose orbit under the geodesic flow is dense in the unit tangent bundle of $\mathcal{F}$.

Figures (6)

  • Figure 1: Example of a cut and project scheme in $\mathbb{R}^{2}$.
  • Figure 2: A hyperbolic cut and project scheme for the triangle group with signature (6,6,3) and quadrilateral fundamental domain.
  • Figure 3: A quadrilateral fundamental domain for the triangle group with signature (6,6,3).
  • Figure 4: A hexagonal fundamental domain for the triangle group with signature $(6, 6, 3)$ and the side pairing elements $U_i$.
  • Figure 5: Sections of the geodesics $\mathfrak{q}_i^{\pm}$ passing through $\mathfrak{t}_{i,s_i}(\pm\epsilon)$ and intersecting the ball $\overline{B}(\mathfrak{m}_i(s_i),\epsilon)\subset B(\gamma_i(y),\rho)$.
  • ...and 1 more figures

Theorems & Definitions (28)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4: ChaoticDeloneSet
  • Lemma 3.1: Analogue to ChaoticDeloneSet
  • Lemma 3.2: Analogue to ChaoticDeloneSet
  • Lemma 3.3: Analogue to ChaoticDeloneSet
  • Lemma 3.4: Analogue to ChaoticDeloneSet
  • ...and 18 more