Nonlinear Rotations on a Lattice

TL;DR

This paper studies a family of invertible lattice maps modeling rotations with decreasing rotation numbers, revealing that most points are either periodic or escape to infinity, with analysis based on an interval-exchange map.

Contribution

It introduces a novel analysis of nonlinear lattice rotations using interval-exchange maps, providing insights into their long-term dynamics and escape behavior.

Findings

Most points are either periodic or escape to infinity.

The dynamics depend on parameter values.

Full density set of points exhibits these behaviors.

Abstract

We consider a prototypical two-parameter family of invertible maps of $\mathbb{Z}^2$, representing rotations with decreasing rotation number. These maps describe the dynamics inside the island chains of a piecewise affine discrete twist map of the torus, in the limit of fine discretisation. We prove that there is a set of full density of points which, depending of the parameter values, are either periodic or escape to infinity. The proof is based on the analysis of an interval-exchange map over the integers, with infinitely many intervals.