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

Consider an ergodic measure preserving dynamical system (T, X, μ), and an observable $\phi :X\to \mathbb{R}$. For the time series X_n(x) = ϕ(Tⁿ(x)), we establish limit laws for the maximum process M_n = max_k⩽nX_k in the case where ϕ is an observable maximized on a line segment, and (T, X, μ) is a hyperbolic dynamical system. Such observables arise naturally in weather and climate applications. We consider the extreme value laws and extremal indices for these observables on hyperbolic toral automorphisms, Sinai dispersing billiards and coupled expanding maps. In particular we obtain clustering and nontrivial extremal indices due to self intersection of submanifolds under iteration by the dynamics, not arising from any periodicity.