Uniqueness of Denjoy minimal sets for twist maps with zero entropy (Q6557854)

This paper investigates Denjoy minimal sets for monotone twist maps with zero topological entropy on the cylinder \(S^ 1 \times \mathbb{R}\). The main result is that the set of recurrent points with irrational rotation number \(\alpha\) can be characterized by one orientation-preserving circle homeomorphism. Consequently, there is either an invariant circle or a unique Denjoy minimal set with rotation number \(\alpha\).\N\NPrior work by \textit{J. N. Mather} in [Comment. Math. Helv. 60, 508--557 (1985; Zbl 0597.58015)] had shown that for area-preserving monotone twist diffeomorphisms, there are uncountably many Denjoy minimal sets with rotation number \(\omega\) if the system has no invariant circle with rotation number \(\omega\), and in such a case the system has positive topological entropy.\N\NThe authors examine what happens for monotone twist maps with zero topological entropy. These are not necessarily area-preserving.\N\NThe main result is that if \(f\) is an orientation-preserving monotone twist homeomorphism on the cylinder with zero topological entropy with lift \(\tilde{f}: \mathbb{R}^2 \rightarrow \mathbb{R}^2\), then for all irrational \(\alpha\) in the rotation set of \(\tilde{f}\) the nonwandering set of \(\tilde{u_0}\), (the lift of \(u_0 \in S^1 \times \mathbb{R}\)) is Birkhoff, and the recurrent set is an ordered set.