Abstract
We study the distribution of maxima (extreme value statistics) for sequences of observables computed along orbits generated by random transformations. The underlying, deterministic, dynamical system can be regular or chaotic. In the former case, we show that, by perturbing rational or irrational rotations with additive noise, an extreme value law appears, regardless of the intensity of the noise, while unperturbed rotations do not admit such limiting distributions. In the case of deterministic chaotic dynamics, we will consider observables specially designed to study the recurrence properties in the neighbourhood of periodic points. Hence, the exponential limiting law for the distribution of maxima is modified by the presence of the extremal index, a positive parameter not larger than one, whose inverse gives the average size of the clusters of extreme events. The theory predicts that such a parameter is unitary when the system is perturbed randomly. We perform sophisticated numerical tests to assess how strong the impact of noise level is when finite time series are considered. We find agreement with the asymptotic theoretical results but also non-trivial behaviour in the finite range. In particular, our results suggest that, in many applications where finite datasets can be produced or analysed, one must be careful in assuming that the smoothing nature of noise prevails over the underlying deterministic dynamics.
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Footnotes
- 8
This choice is dictated by the fact that the stationary measure will be equivalent to Lebesgue in all the examples considered below.
- 9
Of course we could make other choices, but these two spaces will play a major role in the subsequent theory.
- 10
The result is even more general and applies to multidimensional maps too, but for our concerns, especially for rotations, the 1D case is enough. Later, we will discuss generalization to piecewise continuous maps.
- 11
With respect to the previous notations, we changed Ωε into [−1, 1], ω = εξ, with ξ ∈ [−1, 1] and finally
ε becomes dξ over [−1, 1]. - 12
- 13
We recall that θ = θ(z) = 1 − |det D(f−p(z)|, where z is a periodic point of prime period p, see [20, theorem 3].
- 14
See the discussion and the examples about the so-called Standard map in [14].
- 15
We recall that Benedicks and Carleson [5] proved that there exists a set of positive Lebesgue measure S in the parameter space such that the Hénon map has a strange attractor whenever a, b ∈ S. The value of b is very small and the attractor lives in a small neighbourhood of the x-axis. For those values of a and b, one can prove the existence of the physical measure and of a stationary measure under additive noise, which is supported in the basin of attraction and that converges to the physical measure in the zero noise limit [6]. It is still unknown whether such results could be extended to the 'historical' values that we consider here.