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A new proof of the Brouwer plane translation theorem

  • John Franks (a1)
  • DOI:
  • Published online: 01 September 2008

Let f be an orientation-preserving homeomorphism of ℝ2 which is fixed point free. The Brouwer ‘plane translation theorem’ asserts that every x0 ∈ ℝ2 is contained in a domain of translation for f i.e. an open connected subset of ℝ2 whose boundary is Lf(L) where L is the image of a proper embedding of ℝ in ℝ2, such that L separates f(L) and f−1(L). In addition to a short new proof of this result we show that there exists a smooth Morse function g: ℝ2 → ℝ such that g(f(x)) < g(x) for all x and the level set of g containing x0 is connected and non-compact (and hence the image of a properly embedded line).

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[A]S. Andrea . Abh. Math. Sem. Univ. Hamburg30 (1967), 61–61.

[B]L. E. J. Brouwer . Beweis des ebenen Translationssatzes. Math. Ann. 72 (1912), 3754.

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

[S]Edward E. Slaminka . A Brouwer Translation Theorem for Free Homeomorphisms. Trans. Amer. Math. Soc. 306 (1988), 277291.

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Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
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