The multiplier λn of a periodic orbit of period n can be viewed as a (multiple-valued) algebraic function on the space of all complex quadratic polynomials pc(z) = z 2 + c. We provide a numerical algorithm for computing critical points of this function (i.e., points where the derivative of the multiplier with respect to the complex parameter c vanishes). We use this algorithm to compute critical points of λn up to period n = 10.

We apply set-valued numerical methods to compute an accurate enclosure of the rotation number. The described algorithm is supplemented with a method of proving the existence of periodic points, which is used to check the rationality of the rotation number. A few numerical experiments are presented to show that the implementation of interval methods produces a good enclosure of the rotation number of a circle map.

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