Area-preserving homeomorphisms of the open disk without fixed points

Alpern, Steve

doi:10.1090/S0002-9939-1988-0943094-1

Area-preserving homeomorphisms of the open disk without fixed points
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Proc. Amer. Math. Soc. 103 (1988), 624-626 Request permission

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.

References

Stephen A. Andrea, On homoeomorphisms of the plane which have no fixed points, Abh. Math. Sem. Univ. Hamburg 30 (1967), 61–74. MR 208588, DOI 10.1007/BF02993992
Daniel Asimov, On volume-preserving homeomorphisms of the open $n$-disk, Houston J. Math. 2 (1976), no. 1, 1–3. MR 400312
D. G. Bourgin, Homeomorphisms of the open disk, Studia Math. 31 (1968), 433–438. MR 235548, DOI 10.4064/sm-31-4-433-438
L. E. J. Brouwer, Beweis des ebenen Translationssatzes, Math. Ann. 72 (1912), no. 1, 37–54 (German). MR 1511684, DOI 10.1007/BF01456888
Michael Colvin and Kent Morrison, A symplectic fixed point theorem on open manifolds, Proc. Amer. Math. Soc. 84 (1982), no. 4, 601–604. MR 643757, DOI 10.1090/S0002-9939-1982-0643757-6
Deane Montgomery, Measure preserving homeomorphisms at fixed points, Bull. Amer. Math. Soc. 51 (1945), 949–953. MR 13905, DOI 10.1090/S0002-9904-1945-08477-8
Kent Morrison, Symplectic flows on the open ball, J. Differential Equations 46 (1982), no. 1, 59–62. MR 677583, DOI 10.1016/0022-0396(82)90109-7
Jürgen Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc. 120 (1965), 286–294. MR 182927, DOI 10.1090/S0002-9947-1965-0182927-5
J. C. Oxtoby and S. M. Ulam, Measure-preserving homeomorphisms and metrical transitivity, Ann. of Math. (2) 42 (1941), 874–920. MR 5803, DOI 10.2307/1968772