Skip to main content
    • Aa
    • Aa

A new proof of the Brouwer plane translation theorem

  • John Franks (a1)

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).

Hide All
[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.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 11 *
Loading metrics...

Abstract views

Total abstract views: 185 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 18th October 2017. This data will be updated every 24 hours.