Uniqueness of Denjoy minimal sets for twist maps with zero entropy. (English) Zbl 07867472
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 \(\alpha\).
Prior work by 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.
The authors examine what happens for monotone twist maps with zero topological entropy. These are not necessarily area-preserving.
The 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.
Prior work by 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.
The authors examine what happens for monotone twist maps with zero topological entropy. These are not necessarily area-preserving.
The 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.
Reviewer: William J. Satzer Jr. (St. Paul)
MSC:
| 37E40 | Dynamical aspects of twist maps |
| 37E45 | Rotation numbers and vectors |
| 37B40 | Topological entropy |
| 37B20 | Notions of recurrence and recurrent behavior in topological dynamical systems |
| 37E10 | Dynamical systems involving maps of the circle |
Citations:
Zbl 0597.58015References:
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