Gyration number and topology of the period-doubling bifurcation

The topological invariant g is defined; its systematics in the period-doubling regime are shown to be ruled by generalization of Farey-sum operation. Thus the limiting values of g can be obtained at the accumulation point of a period-doubling cascade. The main consequence of the existence of this limit is the appearance of related rotational frequencies on the flow, which are not seen in a Poincaré section analysis. These predictions are verified in an exactly solvable model.