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Action minimizing properties and distances on the group of Hamiltonian diffeomorphisms

Viterbo, Claude; Sorrentino, Alfonso (2010), Action minimizing properties and distances on the group of Hamiltonian diffeomorphisms, Geometry and Topology, 14, 4, p. 2383–2403. http://dx.doi.org/10.2140/gt.2010.14.2383

Type
Article accepté pour publication ou publié
External document link
http://hal.archives-ouvertes.fr/hal-00458323/fr/
Date
2010
Journal name
Geometry and Topology
Volume
14
Number
4
Publisher
Mathematical Sciences Publishers
Pages
2383–2403
Publication identifier
Metadata
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Author(s)
Viterbo, Claude cc
Sorrentino, Alfonso
Abstract (EN)
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 K. Siburg in \cite{Siburg1998}. However, we show that equality holds if one considers the asymptotic distance defined in \cite{Viterbo1992}.
Subjects / Keywords
Asymptotic Hofer distance; action-minimizing measure; symplectic homogenization; Mather's beta function; Mather's minimal average action; Viterbo distance; Mather theory; Aubry–Mather theory