Extreme Value Laws in Dynamical Systems for Non-smooth Observations

Abstract

Abstract. We prove the equivalence between the existence of a non-trivial hitting time statistics law and Extreme Value Laws in the case of dynamical systems with measures which are not absolutely continuous with respect to Lebesgue. This is a counterpart to the result of the authors in the absolutely continuous case. Moreover, we prove an equivalent result for returns to dynamically defined cylinders. This allows us to show that we have Extreme Value Laws for various dynamical systems with equilibrium states with good mixing properties. In order to achieve these goals we tailor our observables to the form of the measure at hand.

Extremes and Recurrence in Dynamical Systems

“…It first appeared in the pioneering paper of Collet, [72], which has been an inspiration for plenty of the research on this issue. Then the subject has been further addressed and developed in many subsequent contributions including [134,73,135,74,136,137,138,139,49,140,81,46,76,77,78,141,142,79,44].…”

“…Alternatively, for observables φ : X → R tailored to the measure µ, so that µ{φ(x) > u n } is regularly varying in u, convergence rates can again be achieved. Such observables are considered in [136].…”

Decay of correlations and laws of rare events for transitive random maps

“…It turns out that these are just two views on the same phenomena: there is a link between these two approaches. This link was already perceivable in the pioneering work of Collet, [9], however it has been formally proved in [13,14], where Freitas et al showed that under general conditions on the observable functions, the existence of HTS/RTS is equivalent to the existence of EVLs. These observable functions achieve a maximum (possibly ∞) at some chosen point ζ in the phase space so that the rare event of an exceedance of a high level occurring corresponds to an entrance in a small ball around ζ.…”

“…In [14], the authors carried the connection further to include more general measures, which, in particular, allowed the coauthors to obtain the connection in the random setting in [6]. For that, it was sufficient to use the skew product map to look at the random setting as a deterministic system and to take the observable ϕ • π : M × Ω → R ∪ {+∞} defined as in (2.6) with ϕ : M → R ∪ {+∞} as in [14, equation (4.1)].…”

“…where ζ is a chosen point in the phase space M and the function g : [0, +∞) → R ∪ {+∞} is such that 0 is a global maximum (+∞ is allowed), g is a strictly decreasing bijection in a neighbourhood of 0 and has one of the three types coming from the Classical Extreme Value Theory; see [14]. Then [14, Theorems 1 and 2] guarantee that if we have an EVL, in the sense that (2.10) holds for some d.f.…”