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Introduction to the Modern Theory of Dynamical Systems

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An Introduction To The Theory Of Point Processes

The Bootstrap and Edgeworth Expansion

An Introduction to Probability Theory and its Applications, Volume I

We review work on extreme events, their causes and consequences, by a group of European and American researchers involved in a three-year project on these topics The review covers theoretical aspects of time series analysis and of extreme value theory, as well as of the deterministic modeling of extreme events, via continuous and discrete dynamic models The applications include climatic, seismic and socio-economic events, along with their prediction Two important results refer to (i) the complementarity of spectral analysis of a time series in terms of the continuous and the discrete part of its power spectrum; and (ii) the need for coupled modeling of natural and socio-economic systems Both these results have implications for the study and prediction of natural hazards and their human impacts

https://npg.copernicus.org/articles/18/295/2011/npg-18-295-2011.pdf

Extreme events: dynamics, statistics and prediction

Effect of processing and storage on the volatile profile of sugarcane honey: A four-year study.

We prove that the distributional limit of the normalised number of returns to small neighbourhoods of periodic points of non-uniformly hyperbolic dynamical systems is compound Poisson. The returns to small balls around a fixed point in the phase space correspond to the occurrence of rare events, or exceedances of high thresholds, so that there is a connection between the laws of Return Times Statistics and Extreme Value Laws. The fact that the fixed point in the phase space is a repelling periodic point implies that there is a tendency for the exceedances to appear in clusters whose average sizes is given by the Extremal Index, which depends on the expansion of the system at the periodic point. 
We recall that for generic points, the exceedances, in the limit, are singular and occur at Poisson times. However, around periodic points, the picture is different: the respective point processes of exceedances converge to a compound Poisson process, so instead of single exceedances, we have entire clusters of exceedances occurring at Poisson times with a geometric distribution ruling its multiplicity. 
The systems to which our results apply include: general piecewise expanding maps of the interval (Rychlik maps), maps with indifferent fixed points (Manneville-Pomeau maps) and Benedicks-Carleson quadratic maps.

The compound Poisson limit ruling periodic extreme behaviour of non-uniformly hyperbolic dynamics

We consider the quadratic family of maps given by $f_{a}(x)=1-a x^2$ with $x\in [-1,1]$, where $a$ is a Benedicks-Carleson parameter. For each of these chaotic dynamical systems we study the extreme value distribution of the stationary stochastic processes $X_0,X_1,...$, given by $X_{n}=f_a^n$, for every integer $n\geq0$, where each random variable $X_n$ is distributed according to the unique absolutely continuous, invariant probability of $f_a$. Using techniques developed by Benedicks and Carleson, we show that the limiting distribution of $M_n=\max\{X_0,...,X_{n-1}\}$ is the same as that which would apply if the sequence $X_0,X_1,...$ was independent and identically distributed. This result allows us to conclude that the asymptotic distribution of $M_n$ is of Type III (Weibull).

/pdf/extreme-values-for-benedicks-carleson-quadratic-maps-553qepnwsx.pdf

Extreme values for Benedicks-Carleson quadratic maps

We study the asymptotic distribution of the partial maximum of observable random variables evaluated along the orbits of some particular dynamical systems. Moreover, we show the link between Extreme Value Theory and Hitting Time Statistics for discrete time non-uniformly hyperbolic dynamical systems. This relation allows to study Hitting Time Statistics with tools from Extreme Value Theory, and vice versa.

Statistical Properties of the Maximum for Non-Uniformly Hyperbolic Dynamics

We prove an abstract result establishing that one can obtain the convergence of Rare Events Point Processes counting the number of orbital visits to a sequence of shrinking target sets from the convergence of corresponding point processes for some induced system and matching shadowing shrinking sets inside the base of the inducing scheme. We apply this result to prove a dichotomy for two classes of non-uniformly hyperbolic interval maps: Misiurewicz quadratic maps and doubly intermittent maps. The dichotomy holds in the sense that the shrinking target sets may accumulate in any individual point $\zeta$ chosen in the phase space and then one either obtains a limiting homogeneous Poisson process at every non-periodic point $\zeta$ or a limiting compound Poisson process with geometric multiplicity distribution at every periodic point. We also highlight the reconstruction performed in order to recover the multiplicity distribution for a periodic orbit sitting outside the base of the induced map.

Jorge Milhazes Freitas

Papers

Effect of processing and storage on the volatile profile of sugarcane honey: A four-year study.

The compound Poisson limit ruling periodic extreme behaviour of non-uniformly hyperbolic dynamics

Extreme values for Benedicks-Carleson quadratic maps

Statistical Properties of the Maximum for Non-Uniformly Hyperbolic Dynamics

Inducing techniques for quantitative recurrence and applications to Misiurewicz maps and doubly intermittent maps