In general terms the goal of this project is to study statistical properties of dynamical systems, both deterministic and stochastic (perturbed), with special emphasis on laws of rare events. The starting point of the analysis is a stochastic process. The dynamical system may appear in different ways. It can describe the time evolution simply by moving the process from one state to the succeeding one, acting, in this way, on the space of all realisations of the process. Or it can be a model for a natural phenomenon, which is described by a certain number of quantities (possibly infinite), such as position, velocity, acceleration, which characterise the state of the system in a certain moment. In these cases, the system gives rise to stochastic processes that correspond to the evaluation of a certain observable, like energy, for example, along the orbits of the system, i.e., the sequences of states that the system experiences, given an initial condition.

We are mostly interested in chaotic systems, which endorse some memory loss to the stochastic processes. Among the statistical properties of the systems, we are particularly interested in the study of rare (or extreme) events. These often correspond to severe deviations from the mean behaviour, which are most of the times related with adverse occurrences and hence of practical interest to be studied. The recurrence effect introduced by Poincaré is the starting point for a deeper analysis of the waiting times before the time evolution of the system leads the orbits to enter certain regions of the phase space where these unwanted, rare, consequences are doomed to occur. For these reasons, it became important to obtain the limit distribution of the elapsed time until the occurrence of a rare event, which is usually referred to as Hitting Time Statistics (HTS) and Return Time Statistics (RTS).

In the theory of Statistics, there exists a very well established branch called Extreme Value Theory (EVT) that is associated to the study of extreme events that correspond very high (or very low) observations, exceeding thresholds that make them unwanted. Despite its relevance in characterizing limit distributions of stochastic processes, EVT was used as a statistical tool in dynamical systems only very recently. The first remarkable contribution, due to Pierre Collet, dates back to the beginning of the century. Recent results by team members establish a connection between HTS/RTS and EVT for dynamical systems. This means that HTS/RTS and EVL are just two sides of the same coin. This brought new light to the subject and new techniques to develop the study of rare events. Furthermore, from the ergodic point of view, one major interest is to prove convergence theorems that can provide estimates for the Shannon entropy. The Ornstein-Weiss Theorem links HTS/RTS to the entropy and Wyner-Ziv have shown a duality between this return time and a matching function.

Recent results by team members show several links between the first possible return of a given sequence and the matching function as well as between this first possible return and the clustering phenomenon appearing in HTS/RTS and EVT. There is here a wide broad of topics that we intend to consider. The team in S. Paulo includes world's specialists on the subject and responsible for the early development of the theory of HTS/RTS. The team in Porto includes leading specialists on EVT for dynamical systems but also on other statistical properties such as decay of correlations and large deviations for dynamical systems. Hence, it is our goal to achieve a deep understanding of the theory of rare events in dynamical systems and its connection with other statistical properties of such systems to bring it to a whole new level of development.

**Coordinator:** Ana Cristina Moreira Freitas