NUMERICAL DETERMINATION OF THE CONTINUED-FRACTION EXPANSION OF THE ROTATION NUMBER

The rotation number is an important invariant of homeomorphisms on the circle. In this paper an algorithm is described, which computes the continued fraction expansion of this number. It therefore detects rational rotation numbers and in general determines a sequence of intervals of decreasing length, containing the rotation number. It will be shown that for almost all rotation numbers the convergence is of third order. Also error analyses of some other algorithms and numerical examples are given, including a computation of resonance tongues in the Arnold family.