New examples of Cantor sets in S 1 that are not C 1 -minimal.

For a homeomorphism f, a set K is minimal if it is compact, invariant, and nonempty, and minimal (with respect to inclusion) with respect to compactness and invariance. Fixed points, periodic orbits, alphaand omega-limit sets are examples of minimal sets. The author focuses on homeomorphisms of the circle f : S 1 → S 1 . It is known that every Cantor subset of S ′ is the minimal set of some homeomorphism of the circle. However, not every such set is minimal for a C 1 -diffeomorphism of S 1 . In this paper the author constructs new examples of Cantor sets in S 1 that are not minimal for any diffeomorphism of S 1 . The author defines a “p-separation condition” and proves that if a Cantor set in S 1 satisfies such a condition, then it is not minimal for any diffeomorphism. Then he constructs a Cantor set in S 1 with a p = 1 separation condition. D. McDuff [Ann. Inst. Fourier 31, No. 1, 177–193 (1981; Zbl 0439.58020)] had previously shown essentially a p = 0 example.