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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Realizing rotation vectors for torus homeomorphisms


Author: John Franks
Journal: Trans. Amer. Math. Soc. 311 (1989), 107-115
MSC: Primary 58F22
DOI: https://doi.org/10.1090/S0002-9947-1989-0958891-1
MathSciNet review: 958891
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Abstract: We consider the rotation set $ \rho (F)$ for a lift $ F$ of a homeomorphism $ f:{T^2} \to {T^2}$, which is homotopic to the identity. Our main result is that if a vector $ v$ lies in the interior of $ \rho (F)$ and has both coordinates rational, then there is a periodic point $ x \in {T^2}$ with the property that

$\displaystyle \frac{{{F^q}({x_0}) - {x_0}}}{q} = v$

where $ {x_0} \in {R^2}$ is any lift of $ x$ and $ q$ is the least period of $ x$.

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DOI: https://doi.org/10.1090/S0002-9947-1989-0958891-1
Article copyright: © Copyright 1989 American Mathematical Society