New examples of Cantor sets in \(S^{1}\) that are not \(C^{1}\)-minimal (Q2480848)

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, alpha- and omega-limit sets are examples of minimal sets. The author focuses on homeomorphisms of the circle \(f: S^1\to S^1\). 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. \textit{D. McDuff} [Ann. Inst. Fourier 31, No. 1, 177--193 (1981; Zbl 0439.58020)] had previously shown essentially a \(p= 0\) example.