Piecewise linear circle maps and conjugation to rigid rational rotations

TL;DR

This paper establishes criteria for piecewise linear circle homeomorphisms to be conjugate to rational rotations, illustrating how these conditions influence the structure and parameter dependence of such maps.

Contribution

It provides new criteria for conjugacy to rational rotations in piecewise linear circle maps and explores their implications in specific families, contrasting with smooth map behaviors.

Findings

Existence of conjugacy implies linear rotation number scaling near critical parameters.

No mode-locked intervals occur at conjugacy points in natural families.

Results connect rational and irrational rotation number cases for piecewise maps.

Abstract

Criteria for piecewise linear circle homeomorphisms to be conjugate to a rigid rotation, $x\to x+\omega~({\rm mod}~1)$, with rational rotation number $\omega$ are given. The consequences of the existence of such maps in families of maps is considered and the results are illustrated using two examples: Herman's classic family of piecewise linear maps with two linear components, and a map derived from geometric optics which has four components. These results show how results for piecewise smooth circle homeomorphisms with irrational rotation numbers have natural correspondences with the case of rational rotation numbers for piecewise linear maps. In natural families of maps the existence of a parameter value at which the map is conjugate to a rigid rotation implies linear scaling of the rotation number in a neighbourhood of the critical parameter value and no mode-locked intervals, in contrast to the behaviour of generic families of circle maps.