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A symplectic fixed point theorem on open manifolds

Authors: Michael Colvin and Kent Morrison
Journal: Proc. Amer. Math. Soc. 84 (1982), 601-604
MSC: Primary 58C30; Secondary 55M20, 57S99, 58D05, 58F10
MathSciNet review: 643757
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Abstract: In 1968 Bourgin proved that every measure-preserving, orientation-preserving homeomorphism of the open disk has a fixed point, and he asked whether such a result held in higher dimensions. Asimov, in 1976, constructed counterexamples in all higher dimensions. In this paper we answer a weakened form of Bourgin's question dealing with symplectic diffeomorphisms: every symplectic diffeomorphism of an even-dimensional cell sufficiently close to the identity in the $ {C^1}$-fine topology has a fixed point. This result follows from a more general result on open manifolds and symplectic diffeomorphisms.

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Additional Information

Keywords: Symplectic manifold, fixed points, open manifold, symplectic diffeomorphism
Article copyright: © Copyright 1982 American Mathematical Society