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

Abstract

Consider an ergodic measure preserving dynamical system (T, X, μ), and an observable ϕ : X → R . For the time series X n (x) = ϕ(T n (x)), we establish limit laws for the maximum process M n = max k⩽n X 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.

Escape rate and conditional escape rate from a probabilistic point of view

“…]. The other cases use similar arguments and correspond to case (2) to (6) of[3, Theorem 2.1].Below we verify the assumptions of Theorem 5.2. Put δ n = e −un .…”

“…αℓ (K, U n ) is the conditional probability of having at least (ℓ − 1) returns to U n before time K. We shall assume that the limit αℓ (K) = lim n→∞ αℓ (K, U n ) exists for for K ∈ N large enough and every ℓ ≥ 1. By monotonicity the limits (3) αℓ = lim K→∞ αℓ (K) exist as αℓ (K) ≤ αℓ (K ′ ) ≤ 1 for all K ≤ K ′ . Since we put τ 0 U = 0, it follows that α1 = 1.…”

“…Proof. We follow the proof of Theorem 1 in [15] and observe that by the previous lemma, for k ∈ [1,3] that is an integer multiple of q −1 , we have…”

“…Also denote by λ > 1 the eigenvalue of T . Following [3] we take Λ to be a line segment with finite length l(Λ). We will lift Λ to Λ ⊂ R 2 and parametrize Λ by p 1 + tv for some p 1 ∈ R 2 and t ∈ [0, l(Λ)].…”

Number of visits in arbitrary sets for $ϕ$-mixing dynamics

“…The statistics of the number of visits was proven in Haydn & Vaienti (2020), with a technique different from the spectral approach: we recall it in the next subsection and then, by considering a very simple example in Section 5.2 we will exhibit clearly how different can be this approach from the symbolic approach developed in Section 4.4. We remind that an alternate probabilistic approach in a coupled maps setting is proposed in Carney et al (2021). We conclude with Section 5.3, considering the case of continuous state Markov chains, a third situation, in which yet another behaviour is displayed.…”