Dans ce travail, nous étudions les propriétés statistiques de certains systèmes dynamiques déterministes et stochastiques. Nous nous intéressons particulièrement aux valeurs extrêmes et à la récurrence. Nous montrons l’existence de Lois pour les Valeurs Extrêmes(LVE) et pour les Statistiques des Temps d’Entrée (STE) et des Temps de Retour (STR) pour des systèmes avec décroissance des corrélations rapide. Nous étudions aussi la convergence du Processus Ponctuel d’Evènements Rares (PPER).Dans la première partie, nous nous intéressons aux systèmes dynamiques déterministes, et nous caractérisons complètement les propriétés précédentes dans le cas des systèmes dilatants. Nous montrons l’existence d’un Indice Extrême (IE) strictement plus petit que 1 autour des points périodiques, et qui vaut 1 dans le cas non-périodique, mettant ainsi en évidence une dichotomie dans la dynamique caractérisée par l’indice extrême. Dans un contexte plus général, nous montrons que le PPER converge soit vers une distribution de Poisson pour des points non-périodiques, soit vers une distribution de Poisson mélangée avec une distribution multiple de type géométrique pour des points périodiques. De plus, nous déterminons explicitement la limite des PPER autour des points de discontinuité et nous obtenons des distributions de Poisson mélangées avec des distributions multiples différentes de la distribution géométrique habituelle. Dans la deuxième partie, nous considérons des systèmes dynamiques stochastiques obtenus en perturbant de manière aléatoire un système déterministe donné. Nous élaborons deux méthodes nous permettant d’obtenir des lois pour les Valeurs Extrêmes et les statistiques de la récurrence en présence de bruits aléatoires. La première approche est de nature probabiliste tandis que la seconde nécessite des outils d’analyse spectrale. Indépendamment du point choisi, nous montrons que l’IE est constamment égal à 1 et que le PPER converge vers la distribution de Poisson standard. / In this work, we study the statistical properties of deterministic and stochastic dynamical systems. We are particularly interested in extreme values and recurrence. We prove the existence of Extreme Value Laws (EVLs) and Hitting Time Statistics (HTS)/ ReturnTime Statistics (RTS) for systems with decay of correlations against L1 observables. We also carry out the study of the convergence of Rare Event Point Processes (REPP). In the first part, we investigate the problem for deterministic dynamics and completely characterise the extremal behaviour of expanding systems by giving a dichotomy relying on the existence of an Extremal Index (EI). Namely, we show that the EI is strictly less than 1 for periodic centres and is equal to 1 for non-periodic ones. In a more general setting, we prove that the REPP converges to a standard Poisson if the centre is non-periodic, and to a compound Poisson with a geometric multiplicity distribution for the periodic case. Moreover, we perform an analysis of the convergence of the REPP at discontinuity points which gives the convergence to a compound Poisson with a multiplicity distribution different than the usual geometric one.In the second part, we consider stochastic dynamics by randomly perturbing a deterministic system with additive noise. We present two complementary methods which allow us to obtain EVLs and statistics of recurrence in the presence of noise. The first approach is more probabilistically oriented while the second one uses spectral theory. We conclude that, regardless of the centre chosen, the EI is always equal to 1 and the REPP converges to the standard Poisson. / Neste trabalho, estudamos as propriedades estatısticas de sistemas dinâmicos deterministicos e estocasticos. Estamos particularmente interessados em valores extremos e recorrência. Provamos a existência de Leis de Valores Extremos (LVE) e Estatısticas doTempo de Entrada (ETE) / Estatısticas de Tempo de Retorno (ETR) para sistemas comdecaimento de correlaçoes contra observaveis em L1. Também realizamos o estudo daconvergência dos Processos Pontuais de Acontecimentos Raros (PPAR). Na primeira parte, investigamos o problema para dinâmica determinıstica e caracterizamos completamente o comportamento extremal de sistemas expansores. Mostramos que ha uma dicotomia quanto 00E0 existência de um Indice de Extrema (IE). Nomeadamente, provamos que o IE é estritamente menor do que 1 em torno de pontos periodicos e é igual a 1 para pontos aperiodicos. Num contexto mais geral, mostramos que os PPAR convergem para um processo de Poisson simples ou um processo de Poisson composto, em que a distribuiçao de multiplicidade é geométrica, dependendo se o centro é um ponto aperiodico ou periodico, respectivamente. Além disso, realizamos uma analise da convergência dos PPAR em pontos de descontinuidade, o que conduziu à descoberta de convergência para um processo de Poisson composto com uma distribuiçao de multiplicidade diferente da usual distribuiçao geométrica. Na segunda parte, consideramos dinâmica estocastica obtida por perturbaçao aleatoria de um sistema determinıstico por inclusao de um ruıdo aditivo. Apresentamos duas técnicas complementares que nos permitem obter LVE e as ETE na presen¸ca deste tipo de ruıdo. A primeira abordagem é mais probabilıstica enquanto que a outra usa sobretudo teoria espectral. Conclui-se que, independentemente do centro escolhido, o IE é sempre igual a 1 e os PPAR convergem para o processo de Poisson simples.

In this work, we study the statistical properties of deterministic and stochastic dynamical systems. We are particularly interested in extreme values and recurrence. We prove the existence of Extreme Value Laws (EVLs) and Hitting Time Statistics (HTS)/ ReturnTime Statistics (RTS) for systems with decay of correlations against L1 observables. We also carry out the study of the convergence of Rare Event Point Processes (REPP). In the first part, we investigate the problem for deterministic dynamics and completely characterise the extremal behaviour of expanding systems by giving a dichotomy relying on the existence of an Extremal Index (EI). Namely, we show that the EI is strictly less than 1 for periodic centres and is equal to 1 for non-periodic ones. In a more general setting, we prove that the REPP converges to a standard Poisson if the centre is non-periodic, and to a compound Poisson with a geometric multiplicity distribution for the periodic case. Moreover, we perform an analysis of the convergence of the REPP at discontinuity points which gives the convergence to a compound Poisson with a multiplicity distribution different than the usual geometric one.In the second part, we consider stochastic dynamics by randomly perturbing a deterministic system with additive noise. We present two complementary methods which allow us to obtain EVLs and statistics of recurrence in the presence of noise. The first approach is more probabilistically oriented while the second one uses spectral theory. We conclude that, regardless of the centre chosen, the EI is always equal to 1 and the REPP converges to the standard Poisson.