Hofer Geometry of a Subset of a Symplectic Manifold

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 $\Vert\cdot\Vert^{X,ω}$, generalizing the Hofer norm. We discuss $Ham(X,ω)$ and $\Vert\cdot\Vert^{X,ω}$ 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 $R^{2n}$ 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 $π/k(n,d)$, where $k(n,d)$ is an explicitly defined integer.

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