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

Abstract

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

Extremes and extremal indices for level set observables on hyperbolic systems *

“…Corollary 2.2 of the recent preprint [42] implies in this case that the extremal index is one. We include an alternate proof for completeness.…”

“…If L is part of a local unstable manifold and T n L has no self-intersections with L then the extremal index is one. The proofs we give in the case of the hyperbolic toral automorphism for this scenario break down but the techniques of the recent preprint [42] probably extend to this case. If L contains a periodic point ζ of period q then the extremal index would be roughly θ ∼ 1 − 1 |DT u (ζ)| q where DT u (ζ) is the expansion in the unstable direction at ζ with a correctional factor due to the conditional measure on the unstable manifold which contains L. If L does not contain a periodic point but its continuation in the unstable manifold does contain a periodic point of period q then as in case (5) of theorem 2.1, if T q L ∩ L = ∅ then θ = 1, otherwise we expect θ to lie roughly in the range 1 −…”