Extreme value theory for synchronization of coupled map lattices,

Correlation dimension and phase space contraction via extreme value theory

“…The technique just described was firstly proposed by Keller [23] as an alternative way to get EVT for systems with exponential mixing and it is based on a perturbative result by Keller and Liverani [24]. We defer to [23] and to our paper [17] for a detailed presentation of that theory. It can be applied to conformal mixing repellers and it provides the preceding estimates, namely the asymptotic scaling for the maximal eigenvalue.…”

“…We would like to point out that with our choice for the observable ψ, the perturbative approach just sketched gives the Gumbel's law in a very direct and natural manner. The computation of q 0 proceeds now as in [17] with a substantial difference: the nature of the conformal measure does not imply necessarily that the ratio ν β (B(T x,r)) ν β (B(x,r)) is constant, which happened when the conformal measure was Lebesgue. This difficulty could be partially overcame by supposing that the potential is constant, otherwise we could bound q 0 from above and below with (close) approximations of the potential.…”

“…It is worth mentioning that whenever the conformal measure is Lebesgue (β = 1), the above computation can be made rigorous as in Proposition (5.3) in [17] and it gives…”

“…Moreover, an exponent of the limit law, the extremal index, is related, for hyperbolic attractors, to the positive Lyapunov exponent in dimension two and to the metric entropy in higher dimensions. The idea of the relationship between EVT and CD comes from a previous work [17] where we used extreme value theory to detect and quantify the onset of synchronization in coupled map lattices. The relationship between the extremal index and the Lyapunov exponent and the entropy is new and is particularly striking for maps with piecewise constant jacobian.…”

“…We want to point out that our DEI is a well defined quantity that can be used as a new indicator for the sensitivity associated to local hyperbolicity. We will present the theoretical results in the next section: some of those results can be obtained by generalizing the techniques introduced in [17]; we will also address the need to develop a more appropriate theory of EVT for diffeomorphisms in higher dimensions. We will then provide several examples of classical conceptual low-dimensional dynamical systems.…”