Journal of Symplectic Geometry

Volume 14 (2016)

Number 1

Topological contact dynamics III: Uniqueness of the topological Hamiltonian and $C^0$-rigidity of the geodesic flow

Pages: 1 – 29

DOI: http://dx.doi.org/10.4310/JSG.2016.v14.n1.a1

Authors

Stefan Müller (Department of Mathematical Sciences, Georgia Southern University, Statesboro, Ga., U.S.A.; and Korea Institute for Advanced Study, Seoul, South Korea)

Peter Spaeth (GE Global Research, Niskayuna, New York, N.Y., U.S.A.; and Korea Institute for Advanced Study, Seoul, South Korea)

Abstract

We prove that a topological contact isotopy uniquely defines a topological contact Hamiltonian. Combined with previous results from “Topological contact dynamics I: symplectization and applications of the energy-capacity inequality” [Stefan Müller and Peter Spaeth, Adv. Geom., 15, 2015, no. 3, 349–380], this generalizes the classical one-to-one correspondence between smooth contact isotopies and their generating smooth contact Hamiltonians and conformal factors to the group of topological contact dynamical systems. Applications of this generalized correspondence include $C^0$-rigidity of smooth contact Hamiltonians, a transformation law for topological contact dynamical systems, and $C^0$-rigidity of the geodesic flows of Riemannian manifolds.

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Published 24 June 2016