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Language of a non-minimal billiard trajectory inside a cube

Moussa Barro, Nicolas Bédaride, Julien Cassaigne

TL;DR

The paper analyzes the language and complexity of non-minimal billiard trajectories inside the unit cube by coding orbits with three letters corresponding to opposite faces. It develops a recoding of a one-dimensional translation on $[0,1]$ using a six-interval partition and a return-word morphism, yielding precise linear in $n$ complexity for almost all base points and quadratic growth for the global language: $p(n,\theta_0)\sim \frac{4+\varphi}{6}n^2$. It then connects bispecial words to irreducible generalized diagonals to establish the global quadratic asymptotics $p(n)\sim \frac{4+\varphi}{6}n^2$ for the chosen direction $\theta_0=\frac{1}{2}\frac{1}{\varphi}\frac{1}{\varphi^2}$. The work links combinatorics on words, Sturmian structures, and billiard dynamics, and discusses robustness to variations in direction and higher-dimensional extensions.

Abstract

We consider a non-minimal billiard trajectory inside the cube. We study the language of the associated orbit when the map is coded with three letters associated to three non-parallel faces of the cube.

Language of a non-minimal billiard trajectory inside a cube

TL;DR

The paper analyzes the language and complexity of non-minimal billiard trajectories inside the unit cube by coding orbits with three letters corresponding to opposite faces. It develops a recoding of a one-dimensional translation on using a six-interval partition and a return-word morphism, yielding precise linear in complexity for almost all base points and quadratic growth for the global language: . It then connects bispecial words to irreducible generalized diagonals to establish the global quadratic asymptotics for the chosen direction . The work links combinatorics on words, Sturmian structures, and billiard dynamics, and discusses robustness to variations in direction and higher-dimensional extensions.

Abstract

We consider a non-minimal billiard trajectory inside the cube. We study the language of the associated orbit when the map is coded with three letters associated to three non-parallel faces of the cube.
Paper Structure (20 sections, 15 theorems, 9 equations, 4 figures)

This paper contains 20 sections, 15 theorems, 9 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Codings of dynamical systems
  3. Result and overview of the paper
  4. Background
  5. Combinatorics
  6. Billiard
  7. Lemmas and proof of Theorem \ref{['thm1']}
  8. Lemmas
  9. Proof of Theorem \ref{['thm1']}
  10. Proof of Theorem \ref{['thm2']}
  11. First part of the proof
  12. Second part
  13. Translation on five intervals
  14. Translation on three intervals
  15. Conclusion of the proof of Theorem \ref{['thm2']}
  16. ...and 5 more sections

Key Result

Theorem 1

Let us consider the cube $P=[0,1]^3$ coded with three letters, and the direction $\theta_0$ defined previously. For almost all $m$ in the face $X=0$, there exists a partition of $[0,1]$ in $6$ intervals $(I_i)_{1\leq i\leq 6}$ such that $f_{\theta_0}(m)=\Phi(v)$ where $v=g(y(m))$.

Figures (4)

  • Figure 1: Coding of the billiard inside the cube
  • Figure 2: Return words and partition of the face $X=0$.
  • Figure 3: Five circles on the torus with different partitions. One in dot corresponds to three intervals, and the other correspond to five intervals.
  • Figure 4: A circle on the torus with a partition in six intervals.

Theorems & Definitions (33)

  • Definition 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Definition 2
  • Definition 3
  • Lemma 1
  • Lemma 2
  • Corollary 1
  • proof
  • ...and 23 more