Hofer geometry of a subset of a symplectic manifold

Abstract

To every closed subset X of a symplectic manifold (M, ω) we associate a natural group of Hamiltonian diffeomorphisms Ham(X, ω). We equip this group with a semi-norm · X,ω, generalizing the Hofer norm.We discuss Ham(X, ω) and · X,ω if X is a symplectic or isotropic submanifold. The main result involves the relative Hofer diameter of X in M. Its first part states that for the unit sphere in R2n this diameter is bounded below by π2 , if n ≥ 2. Its second part states that for n ≥ 2 and d ≥ n there exists a compact subset X of the closed unit ball in R2n, such that X has Hausdorff dimension at most d + 1 and relative Hofer diameter bounded below by π/ k(n, d), where k(n, d) is an explicitly defined integer.