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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Action minimizing properties and distances on the group of Hamiltonian diffeomorphisms

Alfonso Sorrentino and Claude Viterbo

Geometry & Topology 14 (2010) 2383–2403
Abstract

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 β 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
Mathematical Subject Classification 2010
Primary: 37J05, 37J50
Secondary: 53D35
References
Publication
Received: 20 February 2010
Accepted: 21 September 2010
Published: 27 November 2010
Proposed: Leonid Polterovich
Seconded: Yasha Eliashberg, Danny Calegari
Authors
Alfonso Sorrentino
Department of Pure Mathematics and Mathematical Statistics
University of Cambridge
Wilberforce Road
Cambridge
CB3 0WB
UK
http://www.dpmms.cam.ac.uk/~as998
Claude Viterbo
Centre de Mathématiques Laurent Schwartz
UMR 7640 du CNRS
École Polytechnique
91128 Palaiseau
France