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Recurrence and fixed points of surface homeomorphisms

  • John Franks (a1)
  • DOI:
  • Published online: 10 December 2009

We prove that if f is a homeomorphism of the annulus which is homotopic to the identity and has a compact invariant chain transitive set L, then either f has a fixed point or every point of L moves uniformly in one direction: clockwise or counterclockwise. If f is area-preserving, then the annulus itself is a chain transitive set, so, in the presence of a boundary twist condition, one obtains a fixed point. The same techniques apply to homeomorphisms of the torus T2. In this setting we show that if f is homotopic to the identity, preserves Lebesgue measure and has mean translation 0, then it has at least one fixed point.

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[3]C. Conley . Isolated Invariant Sets and the Morse Index. CBMS Regional Conf. Series in Math. 38. AMS, Providence, RI (1978).

[4]C. Conley & E. Zehnder . The Birkhoff-Lewis fixed point theorem and a conjecture of Arnold. Invent. Math. 73 (1983), 3349.

[6]J. Oxtoby . Diameters of arcs and the gerrymandering problem. Amer. Math. Monthly 84 (1977), 155162.

[7]J. Oxtoby & S. Ulam . Measure preserving homeomorphisms and metrical transitivity. Ann. Math. 42 (1941), 874920.

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Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
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