Extreme values for Benedicks-Carleson quadratic maps

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).Comment: 18 page

Extreme Value Laws for non stationary processes generated by sequential and random dynamical systems

We develop and generalize the theory of extreme value for non-stationary stochastic processes, mostly by weakening the uniform mixing condition that was previously used in this setting. We apply our results to non-autonomous dynamical systems, in particular to {\em sequential dynamical systems}, given by uniformly expanding maps, and to a few classes of random dynamical systems. Some examples are presented and worked out in detail

Extreme Value Laws for sequences of intermittent maps

We study non-stationary stochastic processes arising from sequential dynamical systems built on maps with a neutral fixed points and prove the existence of Extreme Value Laws for such processes. We use an approach developed in \cite{FFV16}, where we generalised the theory of extreme values for non-stationary stochastic processes, mostly by weakening the uniform mixing condition that was previously used in this setting. The present work is an extension of our previous results for concatenations of uniformly expanding maps obtained in \cite{FFV16}.Comment: To appear in Proceedings of the American Mathematical Society. arXiv admin note: substantial text overlap with arXiv:1510.0435

Extreme Value Laws in Dynamical Systems for Non-smooth Observations

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

Speed of convergence for laws of rare events and escape rates

We obtain error terms on the rate of convergence to Extreme Value Laws for a general class of weakly dependent stochastic processes. The dependence of the error terms on the `time' and `length' scales is very explicit. Specialising to data derived from a class of dynamical systems we find even more detailed error terms, one application of which is to consider escape rates through small holes in these systems

Complete convergence and records for dynamically generated stochastic processes

We consider empirical multi-dimensional Rare Events Point Processes that keep track both of the time occurrence of extremal observations and of their severity, for stochastic processes arising from a dynamical system, by evaluating a given potential along its orbits. This is done both in the absence and presence of clustering. A new formula for the piling of points on the vertical direction of bi-dimensional limiting point processes, in the presence of clustering, is given, which is then generalised for higher dimensions. The limiting multi-dimensional processes are computed for systems with sufficiently fast decay of correlations. The complete convergence results are used to study the effect of clustering on the convergence of extremal processes, record time and record values point processes. An example where the clustering prevents the convergence of the record times point process is given

Rare Events for the Manneville-Pomeau map

We prove a dichotomy for Manneville-Pomeau maps $f:[0,1]\to [0, 1]$: given any point $\zeta\in [0,1]$, either the Rare Events Point Processes (REPP), counting the number of exceedances, which correspond to entrances in balls around $\zeta$, converge in distribution to a Poisson process; or the point $\zeta$ is periodic and the REPP converge in distribution to a compound Poisson process. Our method is to use inducing techniques for all points except 0 and its preimages, extending a recent result by Haydn, Winterberg and Zweim\"uller, and then to deal with the remaining points separately. The preimages of 0 are dealt with applying recent results by Ayta\c{c}, Freitas and Vaienti. The point $\zeta=0$ is studied separately because the tangency with the identity map at this point creates too much dependence, which causes severe clustering of exceedances. The Extremal Index, which measures the intensity of clustering, is equal to 0 at $\zeta=0$, which ultimately leads to a degenerate limit distribution for the partial maxima of stochastic processes arising from the dynamics and for the usual normalising sequences. We prove that using adapted normalising sequences we can still obtain non-degenerate limit distributions at $\zeta=0$

Clustering of extreme events created by multiple correlated maxima

We consider stochastic processes arising from dynamical systems by evaluating an observable function along the orbits of the system. The novelty is that we will consider observables achieving a global maximum value (possible infinite) at multiple points with special emphasis for the case where these maximal points are correlated or bound by belonging to the same orbit of a certain chosen point. These multiple correlated maxima can be seen as a new mechanism creating clustering. We recall that clustering was intimately connected with periodicity when the maximum was achieved at a single point. We will study this mechanism for creating clustering and will address the existence of limiting Extreme Value Laws, the repercussions on the value of the Extremal Index, the impact on the limit of Rare Events Points Processes, the influence on clustering patterns and the competition of domains of attraction. We also consider briefly and for comparison purposes multiple uncorrelated maxima. The systems considered include expanding maps of the interval such as Rychlik maps but also maps with an indifferent fixed point such as Manneville-Pommeau maps

Extreme Value Laws for dynamical systems with countable extremal sets

We consider stationary stochastic processes arising from dynamical systems by evaluating a given observable along the orbits of the system. We focus on the extremal behaviour of the process, which is related to the entrance in certain regions of the phase space, which correspond to neighbourhoods of the maximal set $\mathcal M$, i.e. the set of points where the observable is maximised. The main novelty here is the fact that we consider that the set $\mathcal M$ may have a countable number of points, which are associated by belonging to the orbit of a certain point, and may have accumulation points. In order to prove the existence of distributional limits and study the intensity of clustering, given by the Extremal Index, we generalise the conditions previously introduced in \cite{FFT12,FFT15}.Comment: arXiv admin note: text overlap with arXiv:1505.0155

Extremal dichotomy for uniformly hyperbolic systems

We consider the extreme value theory of a hyperbolic toral automorphism $T: \mathbb{T}^2 \to \mathbb{T}^2$ showing that if a H\"older observation $\phi$ which is a function of a Euclidean-type distance to a non-periodic point $\zeta$ is strictly maximized at $\zeta$ then the corresponding time series $\{\phi\circ T^i\}$ exhibits extreme value statistics corresponding to an iid sequence of random variables with the same distribution function as $\phi$ and with extremal index one. If however $\phi$ is strictly maximized at a periodic point $q$ then the corresponding time-series exhibits extreme value statistics corresponding to an iid sequence of random variables with the same distribution function as $\phi$ but with extremal index not equal to one. We give a formula for the extremal index (which depends upon the metric used and the period of $q$). These results imply that return times are Poisson to small balls centered at non-periodic points and compound Poisson for small balls centered at periodic points.Comment: 21 pages, 4 figure