A recurrent nonrotational homeomorphism on the annulus. (English) Zbl 0763.58025
In 1941, G. D. Birkhoff conjectured that if \(h:A\to A\) is an area- and orientation-preserving homeomorphism of an annulus \(A\) without periodic points then \(h\) is conjugate to an irrational rotation. The paper presents a construction of a \(C^ \infty\)-diffeomorphism of \(A\) which is a counterexample to Birkhoff’s conjecture. Other, more complicated counterexamples can be obtained by a modification of examples given by Handel and Herman.
Reviewer: R.Srzednicki (Kraków)
MSC:
| 37-XX | Dynamical systems and ergodic theory |
| 28D05 | Measure-preserving transformations |
| 54H20 | Topological dynamics (MSC2010) |
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