Action minimizing properties and distances on the group of Hamiltonian diffeomorphisms

Sorrentino, Alfonso¹ ; Viterbo, Claude²

¹ Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WB, UK
² Centre de Mathématiques Laurent Schwartz, UMR 7640 du CNRS, École Polytechnique, 91128 Palaiseau, France

Geometry & topology, Tome 14 (2010) no. 4, pp. 2383-2403.

Résumé

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 $\beta$ 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].

Détail

DOI : 10.2140/gt.2010.14.2383

Classification : 37J05, 37J50, 53D35
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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