African Diaspora Journal of Mathematics

Introduction to the Group of Symplectomorphisms

Augustin Banyaga

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In these Lecture Notes of a mini-course delivered in the " Séminaire Itinérant de Géometrie et Physique Mathématique, " Geometry and Physics V" at the University Cheikh Anta Diop, Dakar in May 2007, we introduce the group of symplectic diffeomorphisms, the main results on its algebraic structure and on some of its local and global properties. This survey culminates with the most recent results on Hofer geometry, the definitions of the groups of symplectic and hamiltonian homeomorphisms, and the introduction to the $C^0$ symplectic topology.

Article information

Afr. Diaspora J. Math. (N.S.), Volume 9, Number 2 (2009), 120-138.

First available in Project Euclid: 31 March 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55D05
Secondary: 53D35: Global theory of symplectic and contact manifolds [See also 57Rxx]

$C^0$ symplectic topology Hofer geometry Hofer topology hamiltonian topology symplectomorphism hamiltonian diffeomorphisms Arnold conjecture Floer homology symplectic rigidity symplectic capacity symplectic homeomorphisms Hamiltonian homeomorphism


Banyaga, Augustin. Introduction to the Group of Symplectomorphisms. Afr. Diaspora J. Math. (N.S.) 9 (2009), no. 2, 120--138.

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  • A. Banyaga Sur la structure du groupe des difféomorphismes qui préservent une forme symplectique, Comment. Math. Helv. 53(1978) pp.174--227.
  • A. Banyaga The structure of classical diffeomorphisms groups, Mathematics and its applications vol 400. Kluwer Academic Publisher's Group, Dordrecht, The Netherlands (1997).
  • A. Banyaga A Hofer-like metric on the group of symplectic diffeomorphisms, Proceedings of the 2007 AMS-SIAM Summer Research Conference " Symplectic Topology and Measure presrving Dynamical Systems", Snowbird, UT, Contemporary Math. to appear.
  • A. Banyaga, P. Donato Lengths of Contact Isotopies and Extensions of the Hofer Metric, Annals of Global Analysis and Geometry 30(2006) 299-312
  • A. Banyaga, On fixed points of symplectic maps, Inventiones Math. 56(1980) 215-229
  • A. Banyaga, On isomorphic classical diffeomorphism groups, J. Differ.Geom. 28(1988) 23-35
  • A. Banyaga, On the group of strong symplectic homeomorphisms, Cubo, A Mathematical Journal, To appear.
  • A.Banyaga, On the group of symplectic homeomorphisms, C. R. Acad. Sci. Paris Ser I 346(2008) 867-872
  • A. Banyaga, D. Hurtubise, Lectures on Morse Homology, Kluwer Texts in the Mathematical Sciences Vol 29 Kluwer Acad. Publisher, 2004.
  • E. Calabi, On the group of automorphisms of a symplectic manifold, Problems in Analysis, ( a symposium in honor of S. Bochner) pp 1 -26, Princeton University Press, 1970.
  • A. Cannas da Silva Lectures on Symplectic Geometry, Springer Lecture Notes No 1764 (2000).
  • J. Eells, C. Earle, A fiber-bundle description of Teichmuller theory, Jour. Diff. Geometry 3(1999) 19-49
  • R. P. Flilipkiewicz, Isomorphisms between diffeomorphism groups, Ergodic Theory and Dynamical Systems 2(1982) 159-171.
  • V. Guillemin and S.Sternberg, Geometric asymptotics, Survey 17, AMS, Providence, Rhode Island (1977).
  • M. Herman, Sur le groupe des diffeomorphismes du tore Ann. Inst. Fourrier 23- 2(1974) 75-86.
  • H. Hofer On the topological properties of symplectic maps, Proc. Royal Soc. Edimburgh 115A (1990), pp.25--38.
  • H. Hofer, E. Zehnder Symplectic invariants and hamiltonnian dynamics, Birkhauser Advanced Texts, Birkhauser Verlag (1994).
  • F. Lalonde, D. McDuff The geometry of symplectic energy, Ann. Math.141 (1995) 349 - 37.
  • F. Lalonde, D. McDuff, L. Polterovich, Topological rigidity of hamiltonian loops and quantum homology, Invent. Math. 135 (1999) 369- 385
  • P. Liberman, C.M. Marle, Symplectic Geometry and Analytical Mechanics, D. Reidell Publishing Company, 1987
  • J. Moser On the volume elements on a manifold, Trans. Amer. Math. Soc. 120(1965) 286-294
  • D. McDuff, D.A. Salamon, J-holomorphic Curves and Symplectic Topology, Amer. Math. Soc. Colloquium Publication 52, American Mathematical Society, Providence RI (2004).
  • Y-G. Oh, S. Muller The group of hamiltonian homeomorphisms and $C^0$ symplectic topology, Journal of Symplectic Geometry 5(2007). 167-220.
  • K. Ono Floer-Novikov cohomology and the flux conjecture, Geometric Funct. Analysis 16(2006) no 5, 981-1020
  • Y. Ostrover, A comparison of Hofer's metrics on hamiltoinan diffeomorphisms and lagrangian submanifolds, Commun. Contemp. Math. 5(2003) no 5, 803-811
  • L. Polterovich Symplectic displacement energy for Lagrangian submanifolds, Ergodic Theory and Dyamical Systems 13(1993), 357-367.
  • L. Polterovich The Geometry of the group of symplectic diffeomorphisms, Lect. in Math., ETH Zurich, Birkhauser (2001).
  • F. Sergeraert, Un théorème des fonctions implicites sur certains espaces de Fréchet et applications, Ann. Sc. Ec. Nor. Sup 4, 5(1972) 599-660.
  • S. Smale, Diffeomorphisms of the 2-sphere, Proceed. Amer. Math. Soc 10(1959) 599-62.
  • M. Schwarz, On the action spectrum for closed symplectically aspherical manifolds, Pacific Jour. Math. 193(2000) 419-461.
  • C. Viterbo Symplectic topology as the geometry of generating functions, Math. Annalen 292(1992), 685-710.
  • F. Warner Foundations of differentiable manifolds and Lie groups, Scott, Foresman and Company (1971).
  • A. Weinstein Lectures on symplectic manifolds, Regional Conference Series in Mathematics, 29 Amer. Math. Soc. Providence (1977).