• DOI:10.1090/S0002-9939-1982-0643757-6
• Corpus ID: 121058940

A symplectic fixed point theorem on open manifolds

@inproceedings{Colvin1982ASF,
  title={A symplectic fixed point theorem on open manifolds},
  author={Michael R. Colvin and Kent E. Morrison},
  year={1982}
}

• M. Colvin, Kent E. Morrison
• Published 1 April 1982
• Mathematics

In 1968 Bourgin proved that every measure-preserving, orientationpreserving 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'-fine topology has a fixed point. This result follows from a more general result on open manifolds and symplectic diffeomorphisms. Introduction. Fixed point theorems for area-preserving mappings have a history which dates back to Poincare's "last geometric theorem", i.e., any area-preserving mapping of an annulus which twists the boundary curves in opposite directions has at least two fixed points. More recently it has been proved that any area-preserving, orientation-preserving mapping of the two-dimensional sphere into itself possesses at least two distinct fixed points (see [N, Si]). In the setting of noncompact manifolds, Bourgin [B] showed that any measure-preserving, orientation-preserving homeomorphism of the open two-cell B2 has a fixed point. For Bourgin's theorem one assumes that the measure is finite on B2 and that the measure of a nonempty open set is positive. Bourgin also gave a counterexample to the generalization of the theorem for the open ball in RI35 and asked the question whether his theorem remains valid for the open balls in low dimensions. In [As] Asimov constructed counterexamples for all dimensions greater than two and actually got a flow of measure-preserving, orientation-preserving diffeomorphisms with no periodic points. To formulate our results and place the comments above into our framework, we need some concepts from symplectic geometry. A smooth manifold is called symplectic if there exists a nondegenerate, closed, differentiable 2-form w defined on M. A differentiable mapping f of M into itself is called symplectic if f preserves the form w. We refer to the texts by Abraham and Marsden [A & M] and Arnold [A] for the general background in symplectic geometry. We reformulate Bourgin's question to ask: does every symplectic mapping of a 2n-dimensional cell, equipped with a symplectic structure, have a fixed point? Using a generalization of a theorem of Weinstein [W2], we answer this question affirmatively for mappings sufficiently close to the identity. Received by the editors January 22, 1981 and, in revised form, July 2, 1981; presented to the Society at the annual meeting in San Francisco, January 8, 1981. 1980 Mathematics Subject Classification. Primary 58D05; Secondary 53C15, 55M20, 70H15.