Abundance of fast growth of the number of periodic points in   2-dimensional area-preserving dynamics
  We prove that there exists an open subset of the set of real-analytic Hamiltonian diffeomorphisms of a closed surface in which diffeomorphisms exhibiting fast growth of the number of periodic points are dense. We also prove that there exists an open subset of the set of smooth area-preserving diffeomorphisms of a closed surface in which typical diffeomorphisms exhibit fast growth of the number of periodic points. 
Masayuki Asaoka
math.DS