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Locally most robust circles and boundary circles for area-preserving maps

What is the rotation number of the last rotational invariant circle to break in a family of area-preserving maps? And what are the rotation numbers of the outermost invariant circles around elliptic islands? Under the hypothesis that the breakup of invariant circles of arbitrary rotation number for a significant class of area-preserving maps is governed by a critical set of a renormalization operator with certain properties, the authors show that for maps in this universality class the rotation number of locally most robust circles is always noble, and that the rotation number of every boundary circle lies on the stable manifold of a certain Cantor set of numbers of constant type under the Gauss map.