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Symplectic rigidity for Anosov hypersurfaces

Published online by Cambridge University Press:  11 September 2006

D. BURNS  and
R. HIND
D. BURNS
Affiliation:
Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI 48109-1043, USA (e-mail: dburns@umich.edu)
R. HIND
Affiliation:
Department of Mathematics, Hurley Hall, University of Notre Dame, Notre Dame, IN 46556-4618, USA (e-mail: hind.1@nd.edu)
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Abstract

We prove that for a suitable (open) class of open, smoothly bounded domains in the cotangent bundle of a surface of genus $g \geq 2$ any exact symplectomorphism is homotopic to one which is smooth up to the boundary. In particular, such boundaries are not unseen in the sense of Eliashberg–Hofer. The contact boundaries of these domains have Anosov Reeb flows. The methods employed include the properties of such flows in three dimensions, and analysis of two-dimensional transverse flow maps along the lines of Thurston, and thus appear restricted to the case of three-dimensional contact boundaries.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 26 , Issue 5 , October 2006 , pp. 1399 - 1416
Copyright
2006 Cambridge University Press

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