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In this talk, we present rigidity properties of autonomous Hamiltonian systems which are Zoll (that is, they induce a free circle action) on a sequence of energies converging to a symplectic Morse-Bott minimum. We then specialize these techniques to magnetic systems on closed manifolds with symplectic magnetic form. In this setting, we show that the magnetic system defines an almost Kähler structure, whose holomorphic sectional curvature, corrected by a term measuring the non-integrability of the almost complex structure, is constant. In particular, we obtain a dynamical characterization of complex space forms among Kähler manifolds. This is joint work with Johanna Bimmermann and Samanyu Sanjay.

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