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 | References | Similar Articles | Additional Information

Abstract: Birkhoff's sphere conjecture, now known to be false, says that if is a measure preserving homeomorphism of with the poles and fixed, and with no other periodic points, then is topologically conjugate to an irrational rotation of . In this setting we say that is *maximal for* if and is maximal with respect to that property. Also, is *-small* if for any circular ball such that also.

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

**[1]**G. D. Birkhoff,*Some unsolved problems of theoretical dynamics*, Science**94**(1941), 598-600. MR**0006260 (3:279j)****[2]**Michael Handel,*A pathological**diffeomorphism of the plane*, Proc. Amer. Math. Soc.**86**(1982), 163-168. MR**663889 (84f:58040)****[3]**B. V. Kerékjártó,*Vorlesungen uber Topologie*. I, Grundlehren Math. Wiss., Bd. VII, Springer-Verlag, Berlin and New York, 1923, p. 195.**[4]**L. Markus,*Three unresolved problems of dynamics*, Preprint, University of Warwick, 1985.**[5]**D. Montgomery,*Measure preserving homeomorphisms at fixed points*, Bull. Amer. Math. Soc.**51**(1945), 949-953. MR**0013905 (7:216b)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1988-0915743-5

Keywords:
Sphere homeomorphism,
area preserving

Article copyright:
© Copyright 1988
American Mathematical Society