Sandro Vaienti

A Probabilistic Approach To Intermittency

We present an original approach which allows to investigate the statistical properties of a non-uniform hyperbolic map of the interval. Based on a stochastic approximation of the deterministic map, this method gives essentially the optimal polynomial bound for the decay of correlations, the degree depending on the order of the tangency at the neutral xed point.

Recurrence, Dimensions, and Lyapunov Exponents

We show that the Poincaré return time of a typical cylinder is at least its length. For one dimensional maps we express the Lyapunov exponent and dimension via return times.

Hitting and return times in ergodic dynamical systems

Given an ergodic dynamical system (X, T,μ), and U C X measurable with μ(U) > 0, let μ(U)τ U (x) denote the normalized hitting time of x ∈ X to U. We prove that given a sequence (U n ) with μ(U n ) → 0, the distribution function of the normalized hitting times to U n converges weakly to some subprobability distribution F if and only if the distribution function of the normalized return time converges weakly to some distribution function F, and that in the converging case, (=) F(t)= ∫ 0 t (1-F(s))ds, t ≥ 0. This in particular characterizes asymptotics for hitting times, and shows that the asymptotics for return times is exponential if and only if the one for hitting times is also.

Return time statistics via inducing

We prove that the return time statistics of a dynamical system do not change if one passes to an induced (i.e. first return) map. We apply this to show exponential return time statistics in (i) smooth interval maps with nowhere-dense critical orbits and (ii) certain interval maps with neutral fixed points. The method also applies to (iii) certain quadratic maps of the complex plane.

Numerical Convergence of the Block-Maxima Approach to the Generalized Extreme Value Distribution

In this paper we perform an analytical and numerical study of Extreme Value distributions in discrete dynamical systems. In this setting, recent works have shown how to get a statistics of extremes in agreement with the classical Extreme Value Theory. We pursue these investigations by giving analytical expressions of Extreme Value distribution parameters for maps that have an absolutely continuous invariant measure. We compare these analytical results with numerical experiments in which we study the convergence to limiting distributions using the so called block-maxima approach, pointing out in which cases we obtain robust estimation of parameters. In regular maps for which mixing properties do not hold, we show that the fitting procedure to the classical Extreme Value Distribution fails, as expected. However, we obtain an empirical distribution that can be explained starting from a different observable function for which Nicolis et al. (Phys. Rev. Lett. 97(21): 210602, 2006) have found analytical results.

Conformal measure and decay of correlation for covering weighted systems

We show that for a large class of piecewise monotonic transformations on a totally ordered, compact set one can construct conformal measures and obtain the exponential mixing rate for the associated equilibrium state. The method is based on the study of the Perron–Frobenius operator. The conformal measure, the density of the invariant measure and the rate of mixing are deduced by using an appropriate Hilbert metric, without any compactness arguments, even in the case of a countable to one transformation.

Laws of Rare Events for Deterministic and Random Dynamical Systems

The object of this paper is twofold. From one side we study the dichotomy, in terms of the Extremal Index of the possible Extreme Value Laws, when the rare events are centred around periodic or non periodic points. Then we build a general theory of Extreme Value Laws for randomly perturbed dynamical systems. We also address, in both situations, the convergence of Rare Events Point Processes. Decay of correlations against L 1 observables will play a central role in our investigations.

Extreme value theory for singular measures.

In this paper, we perform an analytical and numerical study of the extreme values of specific observables of dynamical systems possessing an invariant singular measure. Such observables are expressed as functions of the distance of the orbit of initial conditions with respect to a given point of the attractor. Using the block maxima approach, we show that the extremes are distributed according to the generalised extreme value distribution, where the parameters can be written as functions of the information dimension of the attractor. The numerical analysis is performed on a few low dimensional maps. For the Cantor ternary set and the Sierpinskij triangle, which can be constructed as iterated function systems, the inferred parameters show a very good agreement with the theoretical values. For strange attractors like those corresponding to the Lozi and Hènon maps, a slower convergence to the generalised extreme value distribution is observed. Nevertheless, the results are in good statistical agreement with the theoretical estimates. It is apparent that the analysis of extremes allows for capturing fundamental information of the geometrical structure of the attractor of the underlying dynamical system, the basic reason being that the chosen observables act as magnifying glass in the neighborhood of the point from which the distance is computed.

Extremes and Recurrence in Dynamical Systems

This book provides a comprehensive introduction for the study of extreme events in the context of dynamical systems. The introduction provides a broad overview of the interdisciplinary research area of extreme events, underlining its relevance for mathematics, natural sciences, engineering, and social sciences. After exploring the basics of the classical theory of extreme events, the book presents a careful examination of how a dynamical system can serve as a generator of stochastic processes, and explores in detail the relationship between the hitting and return time statistics of a dynamical system and the possibility of constructing extreme value laws for given observables. Explicit derivation of extreme value laws are then provided for selected dynamical systems. The book then discusses how extreme events can be used as probes for inferring fundamental dynamical and geometrical properties of a dynamical system and for providing a novel point of view in problems of physical and geophysical relevance. A final summary of the main results is then presented along with a discussion of open research questions. Finally, an appendix with software in Matlab programming language allows the readers to develop further understanding of the presented concepts.

Extreme Value Statistics for Dynamical Systems with Noise

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.