Piecewise linear circle maps and conjugation to rigid rational rotations

Abstract Criteria for piecewise linear (PWL) circle homeomorphisms to be conjugate to a rigid rotation, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo accent="false" stretchy="false">→</mml:mo> <mml:mi>x</mml:mi> <mml:mo>+</mml:mo> <mml:mi>ω</mml:mi> <mml:mtext> </mml:mtext> <mml:mo stretchy="false">(</mml:mo> <mml:mtext>mod</mml:mtext> <mml:mtext> </mml:mtext> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> , with rational rotation number ω 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 PWL 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 PWL 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.