Breakup of Kolmogorov-Arnold-Moser tori of arbitrary frequency in a two-parameter system

In the extended standard map, the critical line for breakup of an invariant circle contains, for almost all rotation numbers, a Cantor set of cusps in parameter space. The Farey route to a given rotation number illustrates how the cusps appear and allows the characterization of their pattern. The fact that the cusp-splitting rules seem attainable by the Farey route rather than by the convergents must be incorporated in the renormalization theory.