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Transactions of the American Mathematical Society

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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$.

References [Enhancements On Off] (What's this?)

  • [B] R. F. Brown, The Lefschetz fixed point theorem, Scott Foresman and Co., Glenview, Ill., 1978. MR 0283793 (44:1023)
  • [Br] M. Brown, A new proof of Brouwer's lemma on translation arcs, Houston J. Math. 10 (1984), 35-41. MR 736573 (85h:54080)
  • [C] C. Conley, Isolated invariant sets and the Morse index, CBMS Regional Conf. Ser. in Math., no. 38, Amer. Math. Soc., Providence, R. I., 1978. MR 511133 (80c:58009)
  • [Fa] A. Fathi, An orbit closing proof of Brouwer's lemma on translation arcs, Enseign. Math. 33 (1987), 315-322. MR 925994 (89d:55004)
  • [F] J. Franks, Recurrence and fixed points of surface homeomorphisms, Ergodic Theory Dynamical Systems 8* (1988), 99-107. MR 967632 (90d:58124)
  • [F2] -, A variation on the Poincaré-Birkhoff theorem, Hamiltonian Dynamics, Contemp. Math., Amer. Math. Soc., Providence, R. I. (to appear). MR 986260 (90e:58095)
  • [HDK] H. Hadwiger, H. Debrunner, and V. Klee, Combinatorial geometry in the plane, Holt Rinehart and Winston, New York, 1964. MR 0164279 (29:1577)
  • [ML] R. MacKay and J. Llibre, Rotation vectors and entropy for homeomorphisms homotopic to the identity, preprint.
  • [MZ] M. Misiurewicz and K. Ziemian, Rotation sets of toral maps (to appear).
  • [Ox] J. Oxtoby, Diameters of arcs and the gerrymandering problem, Amer. Math. Monthly 84 (1977), 155-162. MR 0433333 (55:6310)

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Additional Information

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

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