Lift-Free Approaches to Random Rotation Number and Numerical Approximation

TL;DR

This paper introduces lift-free definitions of the random rotation number for random circle homeomorphisms, proves their equivalence to the standard definition, and develops numerical algorithms for their approximation with practical error bounds.

Contribution

It presents two new lift-free definitions of the random rotation number and provides algorithms for their numerical approximation with proven error bounds.

Findings

Lift-free definitions are equivalent to the standard rotation number.

Approximation error decreases as 1/n with n iterations.

Numerical algorithms are effective across various examples.

Abstract

We study the random rotation number for random circle homeomorphisms. We introduce two new definitions of the random rotation number that can be stated without reference to any choice of lift of the dynamics to the real line, and prove that they are equivalent to the standard random rotation number. We then prove that the mean random rotation number may be approximated within an error of $1/n$ when using $n$ iterations of the dynamics. Finally, we develop numerical algorithms for approximation of the random rotation number which we test with several examples.