arXiv subjects
Publications and source records attributed to Dušan Joksimović.
We prove the Poisson version of the Gromov-Eliashberg's $C^0$-rigidity. More precisely, we prove that the group of Poisson diffeomorphisms is closed with respect to the $C^0$ topology inside the group of all diffeomorphisms. The proof relies on the Poisson version of the energy-capacity inequality.
We prove a Hölder-type inequality for Hamiltonian diffeomorphisms relating the $C^0$ norm, the $C^0$ norm of the derivative, and the Hofer/spectral norm. We obtain as a consequence that sufficiently fast convergence in Hofer/spectral metric forces $C^0$ convergence. The second theme of our paper is the study of pseudo-rotations that arise from the Anosov-Katok method. As an application of our Hölder-type inequality, we prove a $C^0$ rigidity result for such pseudo-rotations.
We remark that, as in the symplectic case, the Hofer norm on the Hamiltonian group of a Poisson manifold is non-degenerate. The proof is a straightforward application of tools from symplectic topology.
K. Cieliebak, H. Hofer, J. Latschev, and F. Schlenk (CHLS) posed the problem of finding a minimal generating set for the (symplectic) capacities on a given symplectic category. We show that if the category contains a certain one-parameter family of objects, then every countably Borel-generating set of (normalized) capacities has cardinality (strictly) bigger than the continuum. This appears to be the first result regarding the problem of CHLS, except for two results of D. McDuff about the category of ellipsoids in dimension 4. We also prove that every finitely differentiably generating set of capacities on a given symplectic category is uncountable, provided that the category contains a one-parameter family of symplectic manifolds that is ``strictly volume-increasing'' and ``embedding-capacity-wise constant''. It follows that the Ekeland-Hofer capacities and the volume capacity do not finitely differentiably generate all generalized capacities on the category of ellipsoids. This answers a variant of a question of CHLS. In addition, we prove that if a given symplectic category contains a certain one-parameter family of objects, then almost no normalized capacity is domain- or target-representable. This provides some solutions to two central problems of CHLS.
We prove that a closed manifold which admits a symplectic-Lipschitz structure has non-vanishing even-degree cohomology groups with real coefficients. In particular, spheres $S^{2n}, n \geq 2$ do not admit symplectic-Lipschitz structures.