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Superdensity and super-micro-uniformity in non-integrable flat systems

J. Beck, W. W. L. Chen

Abstract

We show that on any non-integrable finite polysquare translation surface, superdensity, an optimal form of time-quantitative density, leads to an optimal form of time-quantitative uniformity that we call super-micro-uniformity.

Superdensity and super-micro-uniformity in non-integrable flat systems

Abstract

We show that on any non-integrable finite polysquare translation surface, superdensity, an optimal form of time-quantitative density, leads to an optimal form of time-quantitative uniformity that we call super-micro-uniformity.

Paper Structure

This paper contains 12 sections, 5 theorems, 166 equations.

Table of Contents

  1. Introduction
  2. Iteration process: step 0
  3. Case 1: density decrease
  4. Case 2: density increase
  5. Iteration process: step 1
  6. Case 1: density decrease
  7. Case 2: density increase
  8. Iteration process: general step
  9. Case 1: density decrease
  10. Case 2: density increase
  11. Iteration process: deriving a contradiction
  12. Proof of Theorem \ref{['thm1']}

Key Result

Theorem 1

Let $\mathcal{P}$ be a polysquare translation surface with $b$ atomic squares, and let $\alpha$ be a badly approximable real number. Let $L_\alpha(t)$, $t\geqslant0$, be a half-infinite geodesic with slope $\alpha$, equipped with the usual arc-length parametrization. For any positive integer $n$, le denote the visiting number of $I$ with respect to $\mathcal{X}_n$. Then for every sufficiently larg

Theorems & Definitions (10)

  • Theorem 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Remark
  • Lemma 4
  • proof