Rotation number values at a discontinuity

TL;DR

This paper investigates how the rotation number behaves at discontinuities in orientation-preserving circle maps, revealing that it often takes rational values constrained by specific arithmetic relations, which are sometimes sufficient for characterization.

Contribution

The paper demonstrates that certain arithmetic relations fully characterize rotation number values at discontinuities in some cases, but not universally.

Findings

Rotation numbers at discontinuities are always rational.

These rational values follow specific arithmetic relations.

In some cases, these relations fully determine the rotation number.

Abstract

In the space of orientation-preserving circle maps that are not necessarily surjective nor injective, the rotation number does not vary continuously. Each map where one of these discontinuities occurs is itself discontinuous and we can consider the possible values of the rotation number when we modify this map only at its discontinuities. These values are always rational numbers that necessarily obey a certain arithmetic relation. In this paper we show that in several examples this relation totally characterizes the possible values of the rotation number on its discontinuities, but we also prove that in certain circumstances this relation is not sufficient for this characterization.