Rotation numbers for quasi-periodically forced monotone circle maps

Rotation numbers have played a central role in the study of (unforced) monotone circle maps. In such a case it is possible to obtain a priori bounds of the form rho - 1/n less than or equal to (1/n) (y(n) - y(0)) less than or equal to rho + 1/n, where (1/n) (y(n) - y(0)) is an estimate of the rotation number obtained from an orbit of length n with initial condition y(0), and rho is the true rotation number. This allows rotation numbers to be computed reliably and efficiently. Although Herman has proved that quasi-periodically forced circle maps also possess a well-defined rotation number, independent of initial condition, the analogous bound does not appear to hold. In particular, two of the authors have recently given numerical evidence that there exist quasi-periodically forced circle maps for which y(n) - y(0) - rho(n) is not bounded. This renders the estimation of rotation numbers for quasi-periodically forced circle maps much more problematical. In this paper, a new characterization of the rotation number is derived for quasi-periodically forced circle maps based upon integrating iterates of an arbitrary smooth curve. This satisfies analogous bounds to above and permits us to develop improved numerical techniques for computing the rotation number. Additionally, the boundedness of y(n) - y(0) - rho(n) is considered. It is shown that if this quantity is bounded (both above and below) for one orbit, then it is bounded for all orbits. Conversely, if for any orbit y(n) - y(0) - rho(n) is unbounded either above or below, then there is a residual set of orbits for which y(n) - y(0) - rho(n) is unbounded both above and below. In proving these results a min-max characterization of the rotation number is also presented. The performance of an algorithm based on this is evaluated, and on the whole it is found to be inferior to the integral based method.