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Uniformity of geodesic flow in non-integrable 3-manifolds

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

Abstract

Almost nothing is known concerning the extension of $3$-dimensional Kronecker--Weyl equidistribution theorem on geodesic flow from the unit torus $[0,1)^3$ to non-integrable finite polycube translation $3$-manifolds. In the special case when a finite polycube translation $3$-manifold is the cartesian product of a finite polysquare translation surface with the unit torus $[0,1)$, we have developed a splitting method with which we can make some progress. This is a somewhat restricted system, in the sense that one of the directions is integrable. We then combine this with a split-covering argument to extend our results to some other finite polycube translation $3$-manifolds which satisfy a rather special condition and where none of the $3$ directions is integrable.

Uniformity of geodesic flow in non-integrable 3-manifolds

Abstract

Almost nothing is known concerning the extension of -dimensional Kronecker--Weyl equidistribution theorem on geodesic flow from the unit torus to non-integrable finite polycube translation -manifolds. In the special case when a finite polycube translation -manifold is the cartesian product of a finite polysquare translation surface with the unit torus , we have developed a splitting method with which we can make some progress. This is a somewhat restricted system, in the sense that one of the directions is integrable. We then combine this with a split-covering argument to extend our results to some other finite polycube translation -manifolds which satisfy a rather special condition and where none of the directions is integrable.
Paper Structure (9 sections, 19 theorems, 272 equations)

This paper contains 9 sections, 19 theorems, 272 equations.

Table of Contents

  1. Introduction and the splitting method
  2. Splitting diagram and splitting permutation
  3. Proof of Lemma \ref{['lem12']}
  4. Extending ergodicity to unique ergodicity
  5. Proof of Theorem \ref{['thm2']}
  6. A stronger result
  7. Proof of Lemma \ref{['lem61']}
  8. Kronecker--Weyl polycube translation $3$-manifolds
  9. Proof of Theorem \ref{['thm4']}

Key Result

Theorem 1

Any half-infinite geodesic in the L-solid translation $3$-manifold with a Kronecker direction $\mathbf{v}\in\mathbb{R}^3$ is uniformly distributed unless it hits a singularity.

Theorems & Definitions (46)

  • Theorem 1
  • Remark
  • Remark
  • Lemma 1.1
  • proof
  • Lemma 1.2
  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • ...and 36 more