The minimal Denjoy index of a symmetric perfect set (Q5906948)

Let \(P \subset \mathbb R\) be a perfect set and \(x \in P\). Let \((a,b)\) and \((c,d)\) be any complementary intervals such that \( a < b \leq x \leq c < d\). If \((c,d) \cap (2x-b,2x-a) \neq \emptyset \), then the intersection is denoted by \((x + s, x + t)\). The Denjoy index \(\alpha (x)\) of \(x\) is given by \(\alpha (x) = \lim _{h \to 0+} \sup \{t/s;\;t < h\}\), where the supremum is taken over all quotients \(t/s\) with \(t < h\) obtained in the way described above (the supremum of empty set is defined as \(1\)). The authors investigate the values of the Denjoy index \(\alpha \) at points of uniform symmetric perfect subsets of the real line. Among other results they prove that for every symmetric perfect set \(P\) there is a point of \(P\) where \(\alpha \) attains its minimum on \(P\).