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Time-quantitative density of non-integrable systems

J. Beck, W. W. L. Chen

Abstract

We introduce a new method to establish time-quantitative density in flat dynamical systems. First we give a shorter and different proof of our earlier result that a half-infinite geodesic on an arbitrary finite polysquare surface P is superdense on P if the slope of the geodesic is a badly approximable number. We then adapt our method to study time-quantitative density of half-infinite geodesics on algebraic polyrectangle surfaces.

Time-quantitative density of non-integrable systems

Abstract

We introduce a new method to establish time-quantitative density in flat dynamical systems. First we give a shorter and different proof of our earlier result that a half-infinite geodesic on an arbitrary finite polysquare surface P is superdense on P if the slope of the geodesic is a badly approximable number. We then adapt our method to study time-quantitative density of half-infinite geodesics on algebraic polyrectangle surfaces.

Paper Structure

This paper contains 4 sections, 13 theorems, 207 equations.

Table of Contents

  1. Introduction
  2. Illustration of the method in the simplest case
  3. Adaptation to the regular octagon surface
  4. Extension to algebraic polyrectangles

Key Result

Theorem 1

Let $\mathcal{P}$ be an arbitrary finite polysquare surface. A half-infinite geodesic is superdense on $\mathcal{P}$ if and only if the slope of the geodesic is a badly approximable number.

Theorems & Definitions (27)

  • Theorem 1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • proof : Justification of Claim 1
  • proof : Justification of Claim 2
  • proof : Justification of Claim 3
  • Remark
  • Remark
  • ...and 17 more