Action minimizing properties and distances on the group of Hamiltonian diffeomorphisms

Alfonso Sorrentino (1, 2) , Claude Viterbo (3)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 DPMMS - Department of Pure Mathematics and Mathematical Statistics
3 CMLS - Centre de Mathématiques Laurent Schwartz

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}.

Domaines: Géométrie symplectique [math.SG]; Systèmes dynamiques [math.DS]

Alfonso Sorrentino, Claude Viterbo. Action minimizing properties and distances on the group of Hamiltonian diffeomorphisms. Geometry and Topology, 2010, pp.2383-2403. ⟨hal-00458323⟩