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Area-preserving homeomorphisms of the open disk without fixed points

Author: Steve Alpern
Journal: Proc. Amer. Math. Soc. 103 (1988), 624-626
MSC: Primary 55M20; Secondary 28A99, 54H25, 57N05, 58C30
MathSciNet review: 943094
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Abstract: D. G. Bourgin has proved that every measure-preserving orientation-preserving homeomorphism of the open two-dimensional disk $ D$ has a fixed point. He suggested that the "result is perhaps valid even if the condition of orientability preservation be dropped." We show that on the contrary there exist fixed point free homeomorphisms of $ D$ which preserve any given finite nonatomic locally positive Borel measure. Examples are also constructed in all higher dimensions.

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  • [An] S. Andrea, On homeomorphisms of the plane which have no fixed points, Abh. Math. Sem. Univ. Hamburg 30 (1967), 61-74. MR 0208588 (34:8397)
  • [As] D. Asimov, On volume-preserving homeomorphisms of the open $ n$-disk, Houston J. Math. 2 (1976), 1-3. MR 0400312 (53:4147)
  • [Bo] D. G. Bourgin, Homeomorphisms of the open disk, Studia Math. 31 (1968), 433-438. MR 0235548 (38:3852)
  • [Br] L. E. J. Brouwer, Beweis des ebenen Translationssatzes, Math. Ann. 72 (1912), 37-54. MR 1511684
  • [CM] M. Colvin and K. Morrison, A symplectic fixed point theorem on open manifolds, Proc. Amer. Math. Soc. 84 (1982), 601-604. MR 643757 (83h:58021)
  • [Mn] D. Montgomery, Measure preserving transformations at fixed points, Bull. Amer. Math. Soc. 51 (1945), 949-953. MR 0013905 (7:216b)
  • [M] K. Morrison, Symplectic flows on the open ball, J. Differential Equations 46 (1982), 59-62. MR 677583 (84a:58024)
  • [Mo] J. Moser, On the volume elements of a manifold, Trans. Amer. Math. Soc. 120 (1965), 286-294. MR 0182927 (32:409)
  • [OU] J. Oxtoby and S. Ulam, Measure preserving homeomorphisms and metrical transitivity, Ann. of Math. (2) 42 (1941), 874-920. MR 0005803 (3:211b)

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Article copyright: © Copyright 1988 American Mathematical Society

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