Discretized Rotation with fixed initial points

TL;DR

This paper proves that certain discretized rotation sequences with bounded initial points and parameters in (-2,2) are always periodic, contributing to the understanding of discrete dynamical systems.

Contribution

It establishes a new periodicity result for discretized rotations with fixed initial bounds and specific parameter ranges, expanding theoretical knowledge.

Findings

Sequences are periodic under given conditions

Bounded initial points lead to periodic behavior

Results apply for λ in (-2,2)

Abstract

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