On the dynamics of homology-preserving homeomorphisms of the annulus

Bonino, Marc

doi:10.1090/S0002-9947-08-04688-6

On the dynamics of homology-preserving homeomorphisms of the annulus
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Trans. Amer. Math. Soc. 361 (2009), 1903-1923 Request permission

Abstract:

We consider the homeomorphisms of the compact annulus $\mathbb {A} = \mathbb {S}^1 \times [-1,1]$ isotopic to the symmetry $S_{\mathbb {A}}$ which interchanges the two boundary components. We prove that if such a homeomorphism is, in some sense, conservative and twisted, then it possesses a periodic orbit of period exactly two. This can be regarded as a counterpart of the Poincaré-Birkhoff theorem in the isotopy class of $S_{\mathbb {A}}$.

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