An Algorithm to compute Rotation Numbers in the circle

TL;DR

This paper introduces an efficient algorithm for exactly computing rotation intervals of certain non-invertible degree one circle maps, outperforming existing methods especially for non-differentiable cases.

Contribution

It presents a novel algorithm that accurately computes rotation intervals for a natural subclass of non-invertible circle maps, including non-differentiable ones.

Findings

Algorithm computes exact rotation intervals.

Comparison with existing algorithms shows improved accuracy.

Application to non-differentiable maps demonstrates broader applicability.

Abstract

In this article we present an efficient algorithm to compute rotation intervals of circle maps of degree one. It is based on the computation of the rotation number of a monotone circle map of degree one with a constant section. The main strength of this algorithm is that it computes \emph{exactly} the rotation interval of a natural subclass of the continuous non-invertible degree one circle maps. We also compare our algorithm with other existing ones by plotting the Devil's Staircase of a one-parameter family of maps and the Arnold Tongues and rotation intervals of some special non-differentiable families, most of which were out of the reach of the existing algorithms that were centred around differentiable maps.