Discretized rotation has infinitely many periodic orbits

TL;DR

This paper proves that discretized rotations on the integer lattice, defined for parameters in (-2,2), possess infinitely many periodic orbits, revealing complex dynamical behavior in a simple discrete system.

Contribution

It establishes the existence of infinitely many periodic orbits for discretized rotations on Z^2 within a specific parameter range, a novel result in discrete dynamical systems.

Findings

Discretized rotation systems have infinitely many periodic orbits for certain parameters.

The dynamics exhibit rich periodic behavior despite their simple definition.

Abstract

For a fixed k in (-2,2), the discretized rotation on Z^2 is defined by (x,y)->(y,-[x+ky]). We prove that this dynamics has infinitely many periodic orbits.