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Discretized Rotation with fixed initial points

Shigeki Akiyama, Bill Mance

Abstract

We prove that if $\max\{|a_0|,|a_1|\}\le 10$ and $λ\in\ ]-2,2[$, then the sequence defined by $$ 0 \le a_{n+2} +λa_{n+1}+a_n<1 $$ is periodic.

Discretized Rotation with fixed initial points

Abstract

We prove that if and , then the sequence defined by is periodic.

Paper Structure

This paper contains 5 sections, 9 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Lemmas: around $\pm 2$
  3. Proof of Theorems \ref{['Main']}, \ref{['Main2']} and \ref{['2']}
  4. Theorem \ref{['Main3']}: around $\lambda=-2$
  5. Proof of Theorem \ref{['Small']}

Key Result

Theorem 1

Assume Conjecture SRS and the periodicity (SRSA1) and (SRSA3). Then for any $(a_0,a_1)\in {\mathbb Z}^2$ and $\varepsilon>0$ there exists a finite number of non empty intervals $I_j$ such that

Theorems & Definitions (14)

  • Conjecture 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • ...and 4 more