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A note on density of geodesics

J. Beck, W. W. L. Chen, Y. Yang

Abstract

We extend the famous result of Katok and Zemlyakov on the density of half-infinite geodesics on finite flat rational surfaces to half-infinite geodesics on a finite polycube translation $3$-manifold. We also extend this original result to establish a weak uniformity statement.

A note on density of geodesics

Abstract

We extend the famous result of Katok and Zemlyakov on the density of half-infinite geodesics on finite flat rational surfaces to half-infinite geodesics on a finite polycube translation -manifold. We also extend this original result to establish a weak uniformity statement.
Paper Structure (3 sections, 7 theorems, 27 equations)

This paper contains 3 sections, 7 theorems, 27 equations.

Table of Contents

  1. A density result
  2. When faces have gates and barriers
  3. Weak uniformity with bounded returns

Key Result

Theorem 1

Let $\mathcal{M}$ be a polycube translation $3$-manifold with $s$ atomic cubes. Then any half-infinite geodesic with a Kronecker direction $\mathbf{v}^*=(\alpha_1,\alpha_2,1)\in\mathbb{R}^3$ is dense in $\mathcal{M}$.

Theorems & Definitions (14)

  • Theorem 1
  • proof
  • Lemma 1.1
  • proof
  • Lemma 1.2
  • proof
  • Theorem 2
  • proof
  • Lemma 2.1
  • proof
  • ...and 4 more