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Article
Topological Contact Dynamics I: Symplectization and Applications of the Energy- Capacity Inequality
Advances in Geometry (2015)
  • Stefan Müller, University of Illinois at Urbana-Champaign
  • Peter Spaeth, Penn State University
Abstract
We introduce topological contact dynamics of a smooth manifold carrying a cooriented contact structure, generalizing previous work in the case of a symplectic structure [MO07] or a contact form [BS12]. A topological contact isotopy is not generated by a vector field; nevertheless, the group identities, the transformation law, and classical uniqueness results in the smooth case extend to topological contact isotopies and homeomorphisms, giving rise to an extension of smooth contact dynamics to topological dynamics. Our approach is via symplectization of a contact manifold, and our main tools are an energy-capacity inequality we prove for contact diffeomorphisms, combined with techniques from measure theory on oriented manifolds. We establish non-degeneracy of a Hofer-like bi-invariant pseudo-metric on the group of strictly contact diffeomorphisms constructed in [BD06]. The topological automorphism group of the contact structure exhibits rigidity properties analogous to those of symplectic diffeomorphisms, including C0 -rigidity of contact and strictly contact diffeomorphisms.
Keywords
  • Contact energy-capacity inequality,
  • Bi-Invariant metric on strictly contact diffeomorphism group,
  • Contact C0 -rigidity,
  • Symplectization,
  • Topological contact dynamics,
  • Uniqueness of topological contact isotopy
Disciplines
Publication Date
July, 2015
DOI
10.1515/advgeom-2015-0014
Publisher Statement
This version of the paper was obtained from arXIV.org. In order for the work to be deposited in arXIV.org, the authors must hold the rights or the work must be under Creative Commons Attribution licenseCreative Commons Attribution-Noncommercial-ShareAlike license, or Create Commons Public Domain Declaration. The publisher's final edited version of this article is available at Advances in Geometry.
Citation Information
Stefan Müller and Peter Spaeth. "Topological Contact Dynamics I: Symplectization and Applications of the Energy- Capacity Inequality" Advances in Geometry Vol. 15 Iss. 3 (2015) p. 349 - 380
source:http://arxiv.org/abs/1305.6951
doi:10.1515/advgeom-2015-0014 Available at: http://works.bepress.com/stefan-muller/1/