Volume 14, issue 4 (2010)

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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 $\beta$ 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].

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