Rare event process and entry times distribution for arbitrary null sets on compact manifolds

We establish the general equivalence between rare event process for arbitrary continuous functions whose maximal values are achieved on non-trivial sets, and entry times distribution measure zero sets. then use it to show that differentiable maps a compact Riemannian manifold can be modeled by Young’s towers, limiting both converge compound Poisson distributions. A similar result is also obtained Gibbs–Markov systems, cylinders open give explicit expressions parameters of distribution, simple criterion Poisson. This applied large family observables achieve their maximum set with measure.