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On balance properties of hypercubic billiard words

Nicolas Bédaride, Valérie Berthé, Antoine Julien

TL;DR

The paper addresses how balance and discrepancy manifest in hypercubic billiard words, linking symbolic dynamics with quasicrystal tilings via cut-and-project theory. It combines tiling cohomology (strong and asymptotically negligible cocycles) with bounded remainder set analysis for toral translations to study factor balance, establishing both positive and negative results. The main findings show that for $d\ge 2$ the associated billiard subshifts possess unbalanced factors, while in the cubic case no factor of length $\ge 2$ is balanced; the asymptotically negligible cohomology $H^1_{\text{an}}$ has finite rank $d$, in contrast to the non-finitely generated $H^1_{\text{strong}}$. These results illuminate the limits of balance as a marker of regularity in higher-dimensional billiard codings and reveal deep connections between symbolic dynamics, tiling cohomology, and arithmetic discrepancy theory.

Abstract

This paper studies balance properties for billiard words. Billiard words generalize Sturmian words by coding trajectories in hypercubic billiards. In the setting of aperiodic order, they also provide the simplest examples of quasicrystals, as tilings of the line obtained via cut and project sets with a cubical canonical window. By construction, the number of occurrences of each letter in a factor (i.e., a string of consecutive letters) of a hypercubic billiard word only depends on the length of the factor, up to an additive constant. In other words, the difference of the number of occurrences of each letter in factors of the same length is bounded. In contrast with the behaviour of letters, we prove the existence of words that are not balanced in billiard words: the difference of the number of occurrences of such unbalanced factors in longer factors of the same length is unbounded. The proof relies both on topological methods inspired by tiling cohomology and on arithmetic results on bounded remainder sets for toral translations.

On balance properties of hypercubic billiard words

TL;DR

The paper addresses how balance and discrepancy manifest in hypercubic billiard words, linking symbolic dynamics with quasicrystal tilings via cut-and-project theory. It combines tiling cohomology (strong and asymptotically negligible cocycles) with bounded remainder set analysis for toral translations to study factor balance, establishing both positive and negative results. The main findings show that for the associated billiard subshifts possess unbalanced factors, while in the cubic case no factor of length is balanced; the asymptotically negligible cohomology has finite rank , in contrast to the non-finitely generated . These results illuminate the limits of balance as a marker of regularity in higher-dimensional billiard codings and reveal deep connections between symbolic dynamics, tiling cohomology, and arithmetic discrepancy theory.

Abstract

This paper studies balance properties for billiard words. Billiard words generalize Sturmian words by coding trajectories in hypercubic billiards. In the setting of aperiodic order, they also provide the simplest examples of quasicrystals, as tilings of the line obtained via cut and project sets with a cubical canonical window. By construction, the number of occurrences of each letter in a factor (i.e., a string of consecutive letters) of a hypercubic billiard word only depends on the length of the factor, up to an additive constant. In other words, the difference of the number of occurrences of each letter in factors of the same length is bounded. In contrast with the behaviour of letters, we prove the existence of words that are not balanced in billiard words: the difference of the number of occurrences of such unbalanced factors in longer factors of the same length is unbounded. The proof relies both on topological methods inspired by tiling cohomology and on arithmetic results on bounded remainder sets for toral translations.

Paper Structure

This paper contains 20 sections, 18 theorems, 51 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Basic notions
  3. Words, subshifts and dynamical systems
  4. Balance and discrepancy
  5. Balance and spectral properties
  6. Bounded remainder sets
  7. Hypercubic billiards
  8. Definitions
  9. More on the cubic case
  10. Cut and project sets and tilings
  11. Cohomology and balance
  12. First cohomological definitions
  13. Strong and asymptotically negligible cocycles
  14. On the cohomology group of the billiard
  15. Examples of asymptotically negligible elements for billiards
  16. ...and 5 more sections

Key Result

Theorem 1

Let $d \geq 2$. Let $\boldsymbol{\theta}= (1,\theta_1, \ldots,\theta_d)\in {\mathbb R}^{d+1}_+$, and let $(X_{\boldsymbol{\theta}},S)$ be the subshift defined by the hypercubic billiard in $\mathbb R^{d+1}$ with direction $\boldsymbol{\theta}$. If $X_{\boldsymbol{\theta}}$ is minimal, then its langu

Figures (4)

  • Figure 1: One billiard trajectory in the square (left). One translation on the torus defined by unfolding with respect to four squares (middle). The first return on the diagonal is an exchange of two intervals (right).
  • Figure 2: The translation on the torus $\mathbb T^2$ represented by the fundamental domain $W_{\boldsymbol{\theta}}=\pi_{\boldsymbol{\theta}}[0,1]^2$ and the domain exchange $E_{\boldsymbol{\theta}}$ acting on it.
  • Figure 3: In red the line $L_{\boldsymbol{\theta}}$, and the set $\Lambda_m$ for $m=0$, which yields a tiling of the line $L_{\boldsymbol{\theta}}$, with $d=2$.
  • Figure 4: The partition ${\mathcal{P}}_{\boldsymbol{\theta}}\vee E_{\boldsymbol{\theta}}^{-1} {\mathcal{P}}_{\boldsymbol{\theta}}$ associated with two-letter factors of cubic billiard words. There are $7$ polygons, each corresponding to a factor of length 2 of the language.

Theorems & Definitions (36)

  • Theorem 1
  • Proposition 2
  • Definition 3
  • Proposition 4
  • proof
  • Lemma 5
  • proof
  • Theorem 6: Gottshalk-Hedlund's Theorem Gott.Hed.55
  • Remark 7
  • Proposition 8
  • ...and 26 more