Hofer Geometry of a Subset of a Symplectic Manifold

Abstract

To every closed subset $X$ of a symplectic manifold $(M,\omega)$ we associate a natural group of Hamiltonian diffeomorphisms $Ham(X,\omega)$. We equip this group with a semi-norm $\Vert\cdot\Vert^{X,\omega}$, generalizing the Hofer norm. We discuss $Ham(X,\omega)$ and $\Vert\cdot\Vert^{X,\omega}$ 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 $R^{2n}$ this diameter is bounded below by $\frac\pi2$, if $n\geq2$. Its second part states that for $n\geq2$ and $d\geq n+1$ there exists a compact set in $R^{2n}$ of Hausdorff dimension at most $d$, with relative Hofer diameter bounded below by $\pi/k(n,d)$, where $k(n,d)$ is an explicitly defined integer.