Topological contact dynamics I: symplectization and applications of the energy-capacity inequality

Details Topological contact dynamics I: symplectization and applications of the energy-capacity inequality Topological contact dynamics I: symplectization and applications of the energy-capacity inequality

2011-10-31

arxiv.org/abs/1110.6705v2

Mathematics

Symplectic Geometry

44 pages; v2: minor corrections, moved original section 12, improved presentation of parts of what was section 11 and is now s

Scientific paper

We introduce topological contact dynamics of a smooth manifold carrying a cooriented contact structure, and a group of contact homeomorphisms, generalizing previous work in the case of a symplectic structure [MO07] or a contact form [BS11a]. 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 nondegeneracy of a Hofer-like bi-invariant pseudo-metric on the group of strictly contact diffeomorphisms constructed in [BD06], with no restriction on the contact form. The topological automorphism group of the contact structure exhibits rigidity properties analogous to those of symplectic diffeomorphisms, including C^0-rigidity of contact and strictly contact diffeomorphisms. Other consequences are the extension of a non-vanishing contact invariant defined in [Ban00] in the smooth case, and proper essentiality of the topological automorphism group.

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