Joksimović, Dušan \(C^0\)-rigidity of Poisson diffeomorphisms. (English) Zbl 1526.53079 Lett. Math. Phys. 113, No. 3, Paper No. 69, 10 p. (2023). Let \((M, \{\cdot, \cdot\})\) be a Poisson manifold, and let \(\mathrm{Poiss}(M,\{\cdot, \cdot\})\) denote the group of Poisson diffeomorphisms of \(M.\) In this work, the author considers \(\mathrm{Diff}(M),\) the group of diffeomorphisms of \(M,\) equipped with the \(C^0\)-topology, i.e., the compact-open topology. The main result is that \(\mathrm{Poiss}(M,\{\cdot, \cdot\})\) is a closed subset of \(\mathrm{Diff}(M),\) which is the Poisson version of the Eliashberg-Gromov \(C^0\)-rigidity theorem for symplectic diffeomorphisms. The proof makes use of a Poisson version of the energy-capacity inequality. Reviewer: Luen-Chau Li (University Park) MSC: 53D17 Poisson manifolds; Poisson groupoids and algebroids 57S05 Topological properties of groups of homeomorphisms or diffeomorphisms Keywords:Poisson diffeomorphisms; \(C^0\)-rigidity; energy-capacity inequality × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] Arnaud, M-C, Rigidity in topology \(C^0\) of the Poisson bracket for Tonelli Hamiltonians, Nonlinearity, 28, 8, 2731-2742 (2015) · Zbl 1342.37060 · doi:10.1088/0951-7715/28/8/2731 [2] Burago, D.; Ivanov, S.; Polterovich, L., Conjugation-invariant norms on groups of geometric origin, Groups diffeomorphisms, 52, 221-250 (2008) · Zbl 1222.20031 [3] Buhovsky, L.; Opshtein, E., Some quantitative results in \({C}^0\) symplectic geometry, Invent. Math., 205, 1, 1-56 (2016) · Zbl 1348.53074 · doi:10.1007/s00222-015-0626-4 [4] Buhovsky, L., The \(2/3\)-convergence rate for the Poisson bracket, Geom. Funct. 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