Approach to a rational rotation number in a piecewise isometric system

TL;DR

This paper investigates the behavior of a family of piecewise torus rotations as their rotation number approaches 1/4, revealing stable periodic orbits and complex dynamics in the limit.

Contribution

It introduces a novel induced map approach to analyze the system's dynamics near rational rotation numbers, demonstrating persistent stable orbits and intricate structures.

Findings

Positive measure region with dense discontinuities

Existence of stable periodic orbits with smooth parameter dependence

Irregular dynamics become negligible in the limit

Abstract

We study a parametric family of piecewise rotations of the torus, in the limit in which the rotation number approaches the rational value 1/4. There is a region of positive measure where the discontinuity set becomes dense in the limit; we prove that in this region the area occupied by stable periodic orbits remains positive. The main device is the construction of an induced map on a domain with vanishing measure; this map is the product of two involutions, and each involution preserves all its atoms. Dynamically, the composition of these involutions represents linking together two sector maps; this dynamical system features an orderly array of stable periodic orbits having a smooth parameter dependence, plus irregular contributions which become negligible in the limit.