Action minimizing properties and distances on the group of Hamiltonian diffeomorphisms

TL;DR

This paper investigates the relationship between asymptotic Hofer distance and Mather's beta function for convex Hamiltonians, revealing a strict inequality except under a different distance measure, thus addressing a question in symplectic geometry.

Contribution

It establishes a new inequality linking asymptotic Hofer distance and Mather's beta function, and clarifies conditions for equality, advancing understanding in Hamiltonian dynamics.

Findings

Asymptotic Hofer distance bounds Mather's beta function from above.

Equality holds when using Viterbo's asymptotic distance.

Provides a negative answer to Siburg's question.

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 K. Siburg in \cite{Siburg1998}. However, we show that equality holds if one considers the asymptotic distance defined in \cite{Viterbo1992}.