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Proof of the main conjecture on $g$-areas

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  • In this paper, we prove the main conjecture on $g$-areas arising from [7]. That conjecture was announced by the first author in 2004. It~states that the $g$-area of any Hamiltonian diffeomorphism $\phi$ is equal to the positive Hofer pseudo-distance between $\phi$ and the subspace of Hamiltonian diffeomorphisms that can be expressed as a product of at most $g$ commutators.
    Mathematics Subject Classification: 53C15, 53D22, 53D40, 53D45, 57R58, 57S05, 58B20.

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  • [1]

    A. Banyaga, Sur la structure du groupe des difféomorphismes qui péservent une forme symplectique, Comment. Math. Helv., 53 (1978), 174-227.doi: 10.1007/BF02566074.

    [2]

    L. Buhovsky and Y. Ostrover, On the uniqueness of Hofer's geometry, Geom. Funct. Anal., 21 (2011), 1296-1330.doi: 10.1007/s00039-011-0143-6.

    [3]

    M. Entov, Commutator length of symplectomorphisms, Comment. Math. Helv., 79 (2004), 58-104.doi: 10.1007/s00014-001-0799-0.

    [4]

    F. Lalonde, A field theory for symplectic fibrations over surfaces, Geom. Topol., 8 (2004), 1189-1226.doi: 10.2140/gt.2004.8.1189.

    [5]

    F. Lalonde and D. McDuff, Hofer's $L^{\infty}$-geometry: Energy and stability of Hamiltonian flows. Part II, Invent. Math., 122 (1995), 35-69.doi: 10.1007/BF01231437.

    [6]

    F. Lalonde and D. McDuff, Symplectic stuctures on fiber bundles, Topology, 42 (2003), 309-347; Erratum, Topology, 44 (2005), 1301-1303.doi: 10.1016/S0040-9383(01)00020-9.

    [7]

    F. Lalonde and A. Teleman, The $g$-areas and commutator length, Internat. J. Math., 24 (2013), 1350057, 13 pp.doi: 10.1142/S0129167X13500572.

    [8]

    D. McDuff, Geometric variants of the Hofer norm, J. Symplectic Geom., 1 (2002), 197-252.

    [9]

    L. Polterovich, The Geometry of the Group of Symplectic Diffeomorphisms, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2001.doi: 10.1007/978-3-0348-8299-6.

    [10]

    G. M. Tuynman, The Hamiltonian?, in Geometric Methods in Physics. XXXII Workshop, Białowie.za, Poland, June 30-July 6, 2013, Springer International Publishing, Switzerland, 2014, 287-290.doi: 10.1007/978-3-319-06248-8_25.

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