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Topological contact dynamics II: Topological automorphisms, contact homeomorphisms, and non-smooth contact dynamical systems


Authors: Stefan Müller and Peter Spaeth
Journal: Trans. Amer. Math. Soc. 366 (2014), 5009-5041
MSC (2010): Primary 53D10, 57R17, 37J55, 22F50, 57S05
DOI: https://doi.org/10.1090/S0002-9947-2013-06123-5
Published electronically: December 3, 2013
MathSciNet review: 3217708
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Abstract: This sequel to our previous paper continues the study of topological contact dynamics and applications to contact dynamics and topological dynamics. We provide further evidence that the topological automorphism groups of a contact structure and a contact form are the appropriate transformation groups of contact dynamical systems. This article includes an examination of the groups of time-one maps of topological contact and strictly contact isotopies, and the construction of a bi-invariant metric on the latter. Moreover, every topological contact or strictly contact dynamical system is arbitrarily close to a continuous contact or strictly contact dynamical system with the same end point. In particular, the above groups of time-one maps are independent of the choice of norm in the definition of the contact distance. On every contact manifold we construct topological contact dynamical systems with time-one maps that fail to be Lipschitz continuous, and smooth contact vector fields whose flows are topologically conjugate but not conjugated by a contact $ C^1$-diffeomorphism.


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Additional Information

Stefan Müller
Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801 – and – Korea Institute for Advanced Study, Seoul, South Korea
Email: stefanm@illinois.edu

Peter Spaeth
Affiliation: Department of Mathematics, Pennsylvania State University, Altoona, Pennsylvania 16601 – and – Korea Institute for Advanced Study, Seoul, South Korea
Email: spaeth@psu.edu

DOI: https://doi.org/10.1090/S0002-9947-2013-06123-5
Keywords: Contact metric, contact topology, topological or continuous contact or strictly contact isotopy, Hamiltonian function, conformal factor, $C^0$-rigidity of contact isotopies and diffeomorphisms and conformal factors, topological automorphism of contact structure or contact form, transformation law, topological group, left invariant metric, $L^{(1,\infty)}$-norm, $L^\infty$-norm, reparameterization of contact or strictly contact isotopy, energy-capacity inequality, coarse equals fine contact energy, bi-invariant metric on strictly contact homeomorphism group, non-smooth contact homeomorphism, topologically conjugate contact vector fields or Reeb orbits, not $C^1$-conjugate
Received by editor(s): April 2, 2012
Received by editor(s) in revised form: November 22, 2012, and February 21, 2013
Published electronically: December 3, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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