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

TL;DR

This paper extends the concept of contact dynamics to a topological setting using symplectization, establishing key inequalities and rigidity properties that mirror symplectic cases, thus broadening the understanding of contact structures in topology.

Contribution

It introduces topological contact dynamics via symplectization, proving an energy-capacity inequality and demonstrating rigidity properties similar to symplectic diffeomorphisms.

Findings

Proved an energy-capacity inequality for contact diffeomorphisms.

Established non-degeneracy of a Hofer-like pseudo-metric.

Showed C^0-rigidity of contact and strictly contact diffeomorphisms.

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 C^0-rigidity of contact and strictly contact diffeomorphisms.