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Action minimizing
properties and distances on the group of Hamiltonian
diffeomorphisms
Alfonso Sorrentino and Claude Viterbo |
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Geometry & Topology 14 (2010) 2383–2403
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Abstract |
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In this article we prove that for a smooth fiberwise convex Hamiltonian, the asymptotic Hofer distance from the identity gives a strict upper bound to the value at 0 of Mather’s function, thus providing a negative answer to a question asked by Siburg [Duke Math. J. 92 (1998) 295-319]. However, we show that equality holds if one considers the asymptotic distance defined in Viterbo [Math. Ann. 292 (1992) 685-710]. |
Keywords
Aubry–Mather theory, Mather theory, Hofer distance, Viterbo
distance, Mather's minimal average action, Mather's beta
function, symplectic homogenization, action-minimizing
measure
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Mathematical Subject Classification 2010
Primary: 37J05, 37J50
Secondary: 53D35
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References |
Publication
Received: 20 February 2010
Accepted: 21 September 2010
Published: 27 November 2010
Proposed: Leonid Polterovich
Seconded: Yasha Eliashberg, Danny Calegari
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Authors |
| Alfonso Sorrentino | |
| Department of Pure Mathematics and
Mathematical Statistics University of Cambridge Wilberforce Road Cambridge CB3 0WB UK |
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| http://www.dpmms.cam.ac.uk/~as998 | |
| Claude Viterbo | |
| Centre de Mathématiques Laurent
Schwartz UMR 7640 du CNRS École Polytechnique 91128 Palaiseau France |
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