## from monday 24 march to friday 28 march 2014

The workshop wil take place in the UCPA center, located in Aber Wrac'h, Brittany

### Focus

Law of rare events and extreme value theory, from a dynamical and probabilist point of view.

### Participants

• Pierre Ailliot (Univ. Brest),
• Miguel Abadi (Univ. Sao Paulo),
• Davide Azevedo (Univ. Porto),
• Jean-René Chazottes (Polytechnique),
• Yves Coudène (Univ. Brest),
• Ana Cristina Freitas (Univ. Porto),
• Jorge Freitas (Univ. Porto),
• Loïc Hervé (INSA Rennes),
• Gerhard Keller (Univ. Erlangen),
• Xavier Leoncini (Univ. Marseille),
• Glenn Merlet (Aix-Marseille Université),
• Arnaldo Nogueira (Aix-Marseille Université),
• Françoise Pène (Univ. Brest),
• Nicolas Raillard (Ifremer Brest),
• Mathieu Ribatet (Université de Lyon 1),
• Jérôme Rousseau (Univ. Bahia),
• Anne Sabourin (Telecom ParisTech),
• Benoît Saussol (Univ. Brest),
• Richard Sharp (Univ. Warwick),
• Imre Péter Tóth (Univ. Budapest),
• Pascal Yiou (CEA, Gif-sur-Yvette),
• Roland Zweimüller (Univ. Wien).

### Program

 Monday 24th Tuesday 25th Wednesday 26th Thursday 27th Friday 28th 9:00 Yiou 9:00 Freitas J. 9:00 Azevedo 8:45 Nogueira 10:00 Coffee 10:00 Coffee 10:00 Coffee 9:45 Coffee 10:30 Freitas A. 10:30 Sabourin 10:30 Pène 10:15 Toth 11:30 Rousseau 11:30 Leoncini 11:30 Abadi 11:15 Saussol 12:45 Lunch 12:45 Lunch 12:45 Lunch 12:45 Lunch 14:30 Coffee F R E E 15:00 Pène 16:00 Coffee 16:00 Coffee 16:00 Raillard 16:30 Ribatet 16:30 Sharp 17:00 Chazottes 17:30 Keller 17:30 Zweimüller 20:00 Dinner 20:00 Dinner 20:00 Dinner 20:00 Conference dinner

### Abstracts (and slides of the talks)

• Introduction, Françoise Pène

• Fluctuations and maxima of the matching function of rare words, Miguel Abadi

Saussol et al. [02] and afraimovixh et al. [03] proved That the first return of an n-cylinder over n converges to one almost surely, under suitable conditions. We derive from it the matching function and present its fluctuations law. We also show the asympthotic relationship between its maxima and entropy.

• Statistical properties for systems with weak invariant manifolds, Davide Azevedo

We consider a discrete dynamical system $f : M \to M$, where $M$is a Riemannian manifold and $f$ is a diffeomorphism. We assume that the dynamical system has a Gibbs- Markov-Young structure, which consists of a reference set $\Lambda$ with a hyperbolic product structure that satisfies certain properties. The properties assumed here are the existence of a Markov partition $\Lambda_1,\ldots,\Lambda_d$ of $\Lambda$, polynomial contraction on stable leaves, polynomial backwards contraction on unstable leaves, a bounded distortion property and a certain regularity of the stable foliation. Our main goals are to prove results establishing a control on the decay of correlations and large deviations, as well as presenting an example of a dynamical system satisfying the Gibbs-Markov-Young structure described above. (Joint work with José F. Alves)

• On Poisson approximation for visits to balls, Jean-René Chazottes

I will present a result obtained with P. Collet about approximating the number of visits to balls, on the appropriate scale, by a Poisson law for a class of non uniformly hyperbolic dynamical systems.

• Periodicity and clustering of extreme events, Ana Cristina Freitas

The extremal index appears as a parameter in Extreme Value Laws for stochastic processes, characterising the clustering of extreme events. We apply this idea in a dynamical systems context to analyse the possible Extreme Value Laws for the stochastic process generated by observations taken along dynamical orbits with respect to various measures. In this context, we prove that the extremal index is associated with periodic behaviour. More precisely, we characterise the extremal behaviour at periodic points by proving that for these type of points we have an Extremal Index less than 1.

• Extremal dichotomy and rates of convergence for laws of rare events, Jorge Freitas

It is well known that the Extremal Index (EI) measures the intensity of clustering of extreme events in stationary processes. We sill see that for some certain uniformly expanding systems there exists a dichotomy based on whether the rare events correspond to the entrance in small balls around a periodic point or a non-periodic point. In fact, either there exists EI in (0,1) around (repelling) periodic points or the EI is equal to 1 at every non-periodic point. The main assumption is that the systems have sufficient decay of correlations of observables in some Banach space against all $L^1$-observables. Under the same assumption, we obtain convergence rates for the asymptotic extreme value limit distribution. The dependence of the error terms on the time' and length' scales is made very explicit.

• Rare events and extremal indices via spectral perturbation, Gerhard Keller

I will discuss how an eigenvalue perturbation formula for transfer operators of dynamical systems is related to exponential hitting time distributions and extreme value theory for processes generated by chaotic dynamical systems. The talk will be based on the papers
1) G. Keller, C. Liverani: Rare Events, Escape Rates and Quasistationarity: Some Exact Formulae, Journal of Stat. Phys. 135 (2009) 519-534.
2) G. Keller: Rare events, exponential hitting times and extremal indices via spectral perturbation . Dynamical Systems 27 (2012) 11-27.

• When rare events dominate transport: stickiness in Hamiltonian systems, Xavier Leoncini

In this talk I will discuss the problems related to anomalous transport and stickiness in low dimensional Hamiltonian systems. We firts introduce an exponent to characterize transport based on convergence of finite time observable average distributions. Using tools inspired from this approach we shall study the stickiness of regular island, and exhibit a potential problem with Kac Lemma. This problem actually imply that the ergodic measure is singular, leaving most of physically relevant initial conditions associated to the invariant lebesgue measure irrelevant: rare becomes important.

• Asymptotical behavior of piecewise contractions of the interval, Arnaldo Nogueira

An injective map $f:[0,1)\to [0,1)$ is a piecewise contraction of $n$ intervals, if there exist a partition of $[0,1)$ into intervals $I_1, \ldots, I_n$ and $0<\kappa <1$ such that the restrictions $f\vert_{I_i}$ are $\kappa$-Lipschitz. Poincaré first return maps induced by some Cherry flows on transverse intervals are, up to topological conjugacy, piecewise contractions. These maps also appear in dynamical systems describing the time evolution of manufacturing process adopting some decision-making policies. We have proved that any piecewise contraction of $n$ intervals has at most $n$ periodic orbits. In the talk we will show that a typical piecewise contraction is asymptotically periodic.
The talk is based in a joint work with B. Pires and R. Rosales (see in Arxiv).

• Asymptotic Poissonity for the number of visits to a small ball, Françoise Pène

We establish a general result of convergence in distribution to a Poisson random variable for the number of visits to a shrinking ball. Our result applies to any invertible dynamical system (with invariant probability measure) that can be modeled by a Gibbs-Markov-Young tower with polynomial tail distribution and for which the measure of a very small corona is neglectable with respect to the measure of the small ball containing it. We study namely the solenoid with intermittency and the billiard in a stadium. (This is joint work with Benoît Saussol)

• Modeling extreme values of processes observed at irregular time steps. Application to significant wave height, Nicolas Raillard

This work is motivated by the analysis of the extremal behavior of buoy and satellite data describing wave conditions in the North Atlantic Ocean. The available datasets consist of time series of significant wave height (Hs) with irregular time sampling. In such a situation, the usual statistical methods for analyzing extreme values cannot be used directly. The method proposed here is an extension of the peaks over threshold (POT) method, where the distribution of a process above a high threshold is approximated by a max-stable process whose parameters are estimated by maximizing a composite likelihood function. The efficiency of the proposed method is assessed on an extensive set of simulated data. It is shown, in particular, that the method is able to describe the extremal behavior of several common time series models with regular or irregular time sampling. The method is then used to analyze Hs data in the North Atlantic Ocean. The results indicate that it is possible to derive realistic estimates of the extremal properties of Hs from satellite data, despite its complex space-time sampling.

• Conditional simulation of max-stable processes, Mathieu Ribatet

Since many environmental processes are spatial in extent, a single extreme event may affect several locations, and their spatial dependence has to be appropriately taken into account. This paper proposes a framework for conditional simulation of max-stable processes and gives closed forms for the regular conditional distributions of some well known max-stable processes. We test the method on simulated data and give an application to extreme rainfall around Zurich and extreme temperatures. The proposed framework provides accurate conditional simulations and can handle real-sized problems.

• Hitting time statistics for observations of dynamical systems , Jérôme Rousseau

We study the distribution of hitting and return times for observations of dynamical systems. We apply this results to get an exponential law for the distribution of hitting and return time for rapidly mixing random dynamical systems. In particular, it allows us to obtain an exponential law for random toral automorphisms, random circle maps expanding in average and randomly perturbed dynamical systems.

• Non parametric modeling of multivariate extremes with Dirichlet mixtures , Anne Sabourin

The dependence structure of multivariate extremes, defined as random vectors which jointly exceed large thresholds, can be characterized, up to marginal standardization, by an angular measure on the simplex, only subject to first moment constraints. Estimating the angular measure is thus, by nature, a non parametric problem. Finite Dirichlet mixtures can be used to approach weakly such angular measures but, in practice, the moment constraints make Bayesian inference very challenging in dimension greater than three. We present a re-parametrization of the Dirichlet mixture model, in which the moment constraints are automatically satisfied. This allow for a natural prior specification as well as a simple implementation of a reversible-jump MCMC. Posterior consistency and ergodicity are verified. We illustrate the methods with a four-variate streamflow dataset, including historical information (the earliest flood has been recorded in 1604), which results in censored and missing data. Advantage is taken of the conditioning and marginalization properties of Dirichlet distributions to resolve censored likelihood issues within a data augmentation framework.

• Quenched hitting time distribution in random subshifts, Benoît Saussol

We study the distribution of hitting times for a class of random dynamical systems. We prove that for invariant measures with super-polynomial decay of correlations hitting times to dynamically defined cylinders satisfy exponential distribution. Similar results are obtained for random expanding maps. We emphasize that what we establish is a quenched exponential law for hitting times. (This is joint work with Jérôme Rousseau and Paulo Varandas).

• Fluctuation theorems with shrinking intervals, Richard Sharp

Fluctuation theorems arise in statistical mechanics and describe systems away from equilibrium. They have been rigorously proved for hyperbolic dynamical systems and are related to large deviations of ergodic averages. We will discuss a version where averages lie in intervals that shrink (at an appropriately slow rate) as the systems evolves. (This is joint work with Mark Pollicott.)

• Convergence of the energies to a Markov jump process in a slow-fast dynamical system , Imre Péter Toth

We consider a Hamiltonian flow with two particles moving almost independently. One particle's motion is uniformly hyperbolic, the other one's motion is fully integrable. In addition, they can collide with each other and exchange energy. We study the "rare interaction limit", when such energy exchange collisions happen very seldom. If the initial state is random and we rescale time appropriately, it is natural to expect that in the limit the energies evolve according to a Markov pure jump process. We attempt to give a rigorous proof of this convergence, for a wide class of initial distributions on the Hamiltonian phase space. This is motivated by the Gaspard-Gilbert heat conduction model with seldom colliding billiard disks. Our model is a simplification which is easier to treat rigorously, but still features a pure jump process in the limit. This is the most important difference from another modification which has been treated rigorously by Dolgopyat and Liverani. The methods used are similar: the dynamical part of the proof is based on the standard pair method and coupling techniques developed by Chernov and Dolgopyat. At the heart of the argument one needs a strong result about fast correlation decay of the hyperbolic flow. This is work in progress, joint with Péter Bálint, Domokos Szász and Tamás Tasnádi.

• Analogues de la circulation atmosphérique: un outil pour la détection d'événements climatiques rares, Pascal Yiou

les méthodes d'analogues de circulation sont connues depuis les années 1970, grâce notamment à E.N. Lorenz. Ces méthodes statistiques ont servi principalement à effectuer des prévisions du temps, avant l'utilisation de supercalculateurs pour la prévision numérique. Il a été montré, de manière expérimentale, que cette technique peut également servir à reconstruire des champs atmosphériques tridimensionnels à partir de données de surface. Dans cet exposé, je parcourrai plusieurs applications en climatologie des méthodes d'analogues, de la détection/attribution d'événements extrêmes aux générateurs de temps stochastiques.

• The general asymptotic return-time process, Roland Zweimüller

I will discuss limit processes for consecutive return-times of asymptotically rare events in general ergodic probability preserving systems. Natural questions regard the relation between asymptotic hitting- and return- time processes, the class of all possible limit processes, and their occurrence in every (non-trivial) system.

### Organizers

Françoise Pène, Benoît Saussol, Jean-René Chazottes, Sandro Vaienti

### Partners

 Laboratoire de Mathématiques de Bretagne Atlantique UMR 6205 du CNRS ANR PERTURBATIONS CNRS