Homeomorphisms on minimal Cantor sets in the unimodal setting (Q2215624)

For a multimodal map \(f:[0,1] \rightarrow [0,1]\), \textit{K. M. Brucks} et al. [Ergodic Theory Dyn. Syst. 12, No. 3, 429--439 (1992; Zbl 0756.58044)] showed that if \(B \subseteq [0,1]\) is closed and infinite and \(f:B \rightarrow B\) is a homeomorphism, then some critical point \(c\) of \(f\) is in \(B\) and \(f|_{\omega(c)}\) is a minimal homeomorphism. While studying a related problem for a tent map \(T\), \textit{H. Bruin} [Contemp. Math. 246, 47--56 (1999; Zbl 0940.37013)] found sufficient conditions on the kneading map for \(T|_{\omega(c)}\) to be a homeomorphism. In the present paper the author characterizes unimodal maps \(f\) for which \(\omega(c)\) is a Cantor set and \(f:\omega(c) \rightarrow \omega(c)\) is a minimal homeomorphism in terms of the kneading sequence of \(f\).