It is well known that if \(f:S^1\to S^1\) is a \(C^1\)-diffeomorphism of the circle \(S^1\) without periodic points, then there exists a unique set \(\Omega(f)\subset S^1\) minimal for \(f\). The set \(\Omega(f)\) 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 \(K^c=\bigcup I_j\), where \(I_j\) is a connected component of \(K^c\). The spectrum of \(K\) is the ordered set \(\{\lambda_j\}\), \(\lambda_{j+1}<\lambda_j\), where \(\lambda_j\) is the length of \(I_j\) for some \(j\). McDuff’s conjecture is as follows: if \(\lambda_n/\lambda_{n+1}\) does not tend to 1 as \(n\to +\infty\), 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 \(S\in C^{1+\alpha}\), satisfies the assumptions of McDuff’s conjecture, then it cannot be \(C^1\)-minimal.