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 \({\Vert\cdot\Vert^{X, \omega}}\), generalizing the Hofer norm. We discuss Ham (X, ω) 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 \({\mathbb{R}^{2n}}\) this diameter is bounded below by \({\frac{\pi}{2}}\) , 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 \({\mathbb{R}^{2n}}\), 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.
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Swoboda, J., Ziltener, F. Hofer geometry of a subset of a symplectic manifold. Geom Dedicata 163, 165–192 (2013). https://doi.org/10.1007/s10711-012-9743-z
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DOI: https://doi.org/10.1007/s10711-012-9743-z