Proof of the main conjecture on g-areas

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2015, 22: 92-102. doi: 10.3934/era.2015.22.92

Proof of the main conjecture on $g$-areas

François Lalonde ^1, and Egor Shelukhin ^2,

1.	Département de Mathématiques et de Statistique, Université de Montréal, C.P. 6128, Succ. Centre-ville, Montréal H3C 3J7, Québec
2.	Centre de Recherches Mathématiques, Université de Montréal, C.P. 6128, Succ. Centre-ville, Montréal H3C 3J7, Québec, Canada

Received June 2015 Revised August 2015 Published November 2015

Abstract
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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.

Keywords: Hofer's geometry, Poisson bracket., g-area, commutator length.

Mathematics Subject Classification: 53C15, 53D22, 53D40, 53D45, 57R58, 57S05, 58B2.

Citation: François Lalonde, Egor Shelukhin. Proof of the main conjecture on $g$-areas. Electronic Research Announcements, 2015, 22: 92-102. doi: 10.3934/era.2015.22.92

References:

[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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References:

[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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