Periodic approximation of non-uniformly hyperbolic systems

In this paper, we give a review of the periodic approximation properties of non-uniformly hyperbolic systems, including the approximation of state points by periodic points, the approximation of invariant measures by periodic measures (atomic measures supported on periodic orbits), the approximation of the Lyapunov exponents of invariant measures by the Lyapunov exponents of periodic measures, the approximation of the independent numbers and style numbers of invariant measures by the independent numbers and style numbers of periodic measures, and the approximation of the invariant bundles of the Oseledec decomposition of invariant measures by the Oseledec invariant bundles of periodic measures.