The rotation number of a circle-map family through a rational rigid rotation is differentiable at that point under a transversality condition, with derivative equal to an explicit integral over the resonant Fourier terms.
Boyland (1986) Bifurcations of circle maps: Arnol’d tongues, bistability and rota- tion intervals, Commun
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Scaling of the rotation number for perturbations of rational rotations
The rotation number of a circle-map family through a rational rigid rotation is differentiable at that point under a transversality condition, with derivative equal to an explicit integral over the resonant Fourier terms.