2011/10/31 by Stefan C. Müller, Müller, Stefan, Peter Spaeth +1 · 1 citation
Mathematics · #28D05 #37J55 #53D10 #54H20 #57R17 #57S05 #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Symplectic Geometry (math.SG)
paper · pdf · doi:10.48550/arxiv.1110.6705
openalex publication_date 2011/10/31 · openalex created_date 2022/10/05 · openalex updated_date 2026/07/28
We introduce topological contact dynamics of a smooth manifold carrying a\ncooriented contact structure, generalizing previous work in the case of a\nsymplectic structure [MO07] or a contact form [BS12]. A topological contact\nisotopy is not generated by a vector field; nevertheless, the group identities,\nthe transformation law, and classical uniqueness results in the smooth case\nextend to topological contact isotopies and homeomorphisms, giving rise to an\nextension of smooth contact dynamics to topological dynamics. Our approach is\nvia symplectization of a contact manifold, and our main tools are an\nenergy-capacity inequality we prove for contact diffeomorphisms, combined with\ntechniques from measure theory on oriented manifolds. We establish\nnon-degeneracy of a Hofer-like bi-invariant pseudo-metric on the group of\nstrictly contact diffeomorphisms constructed in [BD06]. The topological\nautomorphism group of the contact structure exhibits rigidity properties\nanalogous to those of symplectic diffeomorphisms, including C0-rigidity of\ncontact and strictly contact diffeomorphisms.\n