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We prove that there exists an open subset of the set of real-analyticHamiltonian diffeomorphisms of a closed surface in which diffeomorphismsexhibiting fast growth of the number of periodic points are dense. We alsoprove that there exists an open subset of the set of smooth area-preservingdiffeomorphisms of a closed surface in which typical diffeomorphisms exhibitfast growth of the number of periodic points.

Abundance of fast growth of the number of periodic points in 2-dimensional area-preserving dynamics