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A symplectic manifold is a space with a global structure on which Hamiltonian equations are defined. A classical result by Darboux says that every symplectic manifold locally looks standard, so it has been interesting to study global properties of symplectic manifolds. Since Gromov invented his famous theory of J-holomorphic curves in 1985, symplectic rigidity phenomena have been found in many different ways. In this talk, we explore it in terms of the symplectic mapping class groups and entropies.