A Short Proof of the Cartwright-Littlewood Fixed Point Theorem

O. H. Hamilton

doi:10.4153/CJM-1954-056-8

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The purpose of this paper is to give a short proof of the Cartwright-Littlewood fixed point theorem (2, p. 3, Theorem A).

Theorem A. If T is a (1-1) continuous and orientation preserving transformation of the Euclidean plane E onto itself which leaves a bounded continuum M invariant and if M does not separate E, then some point of M is left fixed by T.

References

1. Brouwer, L. E. J., Beweis des ebenen Translationssatzes, Math. Ann., 72 (1912), 37–54.Google Scholar

2. Cartwright, M. L. and Littlewood, J. E., Some fixed point theorems, Ann. Math., 54 (1951), 1–37.Google Scholar

3. Kerékjartó, B. V., Topologie (Berlin, 1923).Google Scholar

4. Newman, M. H. A., Topology of plane sets of points (Cambridge, 1951).Google Scholar

Crossref Citations

This article has been cited by the following publications. This list is generated based on data provided by Crossref.

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Mason, W. K. 1975. Fixed points of pointwise almost periodic homeomorphisms on the two-sphere. Transactions of the American Mathematical Society, Vol. 202, Issue. 0, p. 243.

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Firmo, S. Ribón, J. and Velasco, J. 2015. Fixed points for nilpotent actions on the plane and the Cartwright–Littlewood theorem. Mathematische Zeitschrift, Vol. 279, Issue. 3-4, p. 849.

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BOROŃSKI, J. P. 2017. On a generalization of the Cartwright–Littlewood fixed point theorem for planar homeomorphisms. Ergodic Theory and Dynamical Systems, Vol. 37, Issue. 6, p. 1815.

Kucharski, Przemysław 2022. On Cartwright–Littlewood fixed point theorem. Indagationes Mathematicae, Vol. 33, Issue. 4, p. 861.

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A Short Proof of the Cartwright-Littlewood Fixed Point Theorem

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