Dynamically defined Cantor sets under the conditions of McDuff ’s conjecture.

It is well known that if f : S 1 → S 1 is a C 1 -diffeomorphism of the circle S 1 without periodic points, then there exists a unique set Ω(f) ⊂ S 1 minimal for f. The set Ω(f) is referred to as C 1 -minimal and is either a Cantor set or S 1 . Examples of C 1 -minimal Cantor sets are due to Denjoy. However, as shown by D. McDuff [ Ann. Inst. Fourier 31, No. 1, 177–193 (1981; Zbl 0439.58020)] and A. N. Kercheval [Ergodic Theory Dyn. Syst. 22, No. 6, 1803–1812 (2002; Zbl 1018.37023)], the usual middle thirds and affine Cantor sets are not C 1 -minimal. Let K be a Cantor subset of S 1 and let Kc = ∪ Ij , where Ij is a connected component of Kc . The spectrum of K is the ordered set {λj}, λj+1 < λj , where λj is the length of Ij for some j. McDuff’s conjecture is as follows: if λn/λn+1 does not tend to 1 as n → +∞, then the Cantor set K is not C 1 -minimal. The paper presents some results towards proving this conjecture. Namely, it is shown that if the Cantor set K, dynamically defined by a function S ∈ C 1+α, satisfies the assumptions of McDuff’s conjecture, then it cannot be C 1 -minima