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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On the sphere conjecture of Birkhoff


Author: Richard Jerrard
Journal: Proc. Amer. Math. Soc. 102 (1988), 193-201
MSC: Primary 54H20,; Secondary 55M99,58F08
DOI: https://doi.org/10.1090/S0002-9939-1988-0915743-5
MathSciNet review: 915743
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Abstract: Birkhoff's sphere conjecture, now known to be false, says that if $ f$ is a measure preserving homeomorphism of $ {S^2}$ with the poles $ N$ and $ S$ fixed, and with no other periodic points, then $ f$ is topologically conjugate to an irrational rotation of $ {S^2}$. In this setting we say that $ D \subset {S^2}$ is maximal for $ f$ if $ f\left( D \right) \cap D = \emptyset $ and $ D$ is maximal with respect to that property. Also, $ f$ is $ 2$-small if for any circular ball $ B$ such that $ f\left( B \right) \cap B = \emptyset ,{f^{ - 1}}\left( B \right) \cap f\left( B \right) = \emptyset $ also.

Theorem. Any $ f$ as above and also $ 2$-small has a maximal set $ D$ which is an open ball; its boundary contains $ N$ and $ S$ and is locally connected, and the area of $ D$ is an irrational fraction of the area of $ {S^2}$. This theorem gives another way of looking at the maps involved in the Birkhoff conjecture.


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DOI: https://doi.org/10.1090/S0002-9939-1988-0915743-5
Keywords: Sphere homeomorphism, area preserving
Article copyright: © Copyright 1988 American Mathematical Society