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

Müller, Stefan; Spaeth, Peter

doi:10.1090/S0002-9947-2013-06123-5

Topological contact dynamics II: Topological automorphisms, contact homeomorphisms, and non-smooth contact dynamical systems
HTML articles powered by AMS MathViewer

by Stefan Müller and Peter Spaeth PDF

Trans. Amer. Math. Soc. 366 (2014), 5009-5041 Request permission

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.

References

Augustin Banyaga, The structure of classical diffeomorphism groups, Mathematics and its Applications, vol. 400, Kluwer Academic Publishers Group, Dordrecht, 1997. MR 1445290, DOI 10.1007/978-1-4757-6800-8
Augustin Banyaga, On the group of strong symplectic homeomorphisms, Cubo 12 (2010), no. 3, 49–69 (English, with English and Spanish summaries). MR 2779373, DOI 10.4067/s0719-06462010000300004
Augustin Banyaga and Paul Donato, Lengths of contact isotopies and extensions of the Hofer metric, Ann. Global Anal. Geom. 30 (2006), no. 3, 299–312. MR 2256527, DOI 10.1007/s10455-005-9011-7
Augustin Banyaga and Peter Spaeth, On the uniqueness of contact Hamiltonians of topological strictly contact isotopies, available at http://newton.kias.re.kr/~spaeth/, 2012.
Garrett Birkhoff, A note on topological groups, Compositio Math. 3 (1936), 427–430. MR 1556955
Lev Buhovsky and Sobhan Seyfaddini, Uniqueness of generating Hamiltonians for topological Hamiltonian flows, J. Symplectic Geom. 11 (2013), no. 1, 37–52. MR 3022920
Manfred Denker, Christian Grillenberger, and Karl Sigmund, Ergodic theory on compact spaces, Lecture Notes in Mathematics, Vol. 527, Springer-Verlag, Berlin-New York, 1976. MR 0457675
Ya. M. Eliashberg, A theorem on the structure of wave fronts and its application in symplectic topology, Funktsional. Anal. i Prilozhen. 21 (1987), no. 3, 65–72, 96 (Russian). MR 911776
A. Fathi, Structure of the group of homeomorphisms preserving a good measure on a compact manifold, Ann. Sci. École Norm. Sup. (4) 13 (1980), no. 1, 45–93. MR 584082
Hansjörg Geiges, An introduction to contact topology, Cambridge Studies in Advanced Mathematics, vol. 109, Cambridge University Press, Cambridge, 2008. MR 2397738, DOI 10.1017/CBO9780511611438
Mikhael Gromov, Partial differential relations, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 9, Springer-Verlag, Berlin, 1986. MR 864505, DOI 10.1007/978-3-662-02267-2
M. Gromov, Soft and hard symplectic geometry, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) Amer. Math. Soc., Providence, RI, 1987, pp. 81–98. MR 934217
Morris W. Hirsch, Differential topology, Graduate Texts in Mathematics, vol. 33, Springer-Verlag, New York, 1994. Corrected reprint of the 1976 original. MR 1336822
H. Hofer, On the topological properties of symplectic maps, Proc. Roy. Soc. Edinburgh Sect. A 115 (1990), no. 1-2, 25–38. MR 1059642, DOI 10.1017/S0308210500024549
Shizuo Kakutani, Über die Metrisation der topologischen Gruppen, Proc. Imp. Acad. Tokyo 12 (1936), no. 4, 82–84 (German). MR 1568424
V. L. Klee Jr., Invariant metrics in groups (solution of a problem of Banach), Proc. Amer. Math. Soc. 3 (1952), 484–487. MR 47250, DOI 10.1090/S0002-9939-1952-0047250-4
François Lalonde and Dusa McDuff, The geometry of symplectic energy, Ann. of Math. (2) 141 (1995), no. 2, 349–371. MR 1324138, DOI 10.2307/2118524
Stefan Muller, The group of Hamiltonian homeomorphisms and topological symplectic topology, ProQuest LLC, Ann Arbor, MI, 2008. Thesis (Ph.D.)–The University of Wisconsin - Madison. MR 2711770
Stefan Müller, The group of Hamiltonian homeomorphisms in the $L^\infty$-norm, J. Korean Math. Soc. 45 (2008), no. 6, 1769–1784. MR 2449929, DOI 10.4134/JKMS.2008.45.6.1769
Yong-Geun Oh and Stefan Müller, The group of Hamiltonian homeomorphisms and $C^0$-symplectic topology, J. Symplectic Geom. 5 (2007), no. 2, 167–219. MR 2377251
Stefan Müller and Peter Spaeth, Helicity of vector fields preserving a regular contact form and topologically conjugate smooth dynamical systems, Ergod. Th. & Dynam. Sys. 33 (2013).
—, Topological contact dynamics I: symplectization and applications of the energy-capacity inequality, arXiv:1110.6705 [math.SG], 2012.
—, Topological contact dynamics III: uniqueness of the topological Hamiltonian, arXiv reference available.
Yong-Geun Oh, The group of Hamiltonian homeomorphisms and continuous Hamiltonian flows, Symplectic topology and measure preserving dynamical systems, Contemp. Math., vol. 512, Amer. Math. Soc., Providence, RI, 2010, pp. 149–177. MR 2605316, DOI 10.1090/conm/512/10062
Claude Viterbo, On the uniqueness of generating Hamiltonian for continuous limits of Hamiltonians flows, Int. Math. Res. Not. , posted on (2006), Art. ID 34028, 9. MR 2233715, DOI 10.1155/IMRN/2006/34028