Limiting laws ruling dynamical systems

In general terms, the main goal of this project is to study statistical properties of dynamical systems, which are particularly captured by the description provided by limiting laws.

The complexity of the orbital structure of chaotic systems brought special attention to the existence of such limiting laws, since they borrow at least some probabilistic predictability to the erratic behaviour of such systems.

The first step in this direction is the construction of invariant physical measures, which provide an asymptotic spatial distribution of the orbits in the phase space. Ergodicity then gives strong laws of large numbers.

The mixing properties of the system restore asymptotic independence and, in this way, allow to mimic iid processes and prove limiting laws for the mean, such as: central limit theorems, large deviation principles, invariance principles, etc.

While the limiting laws mentioned so far pertain to the mean or average behaviour of the system, in the recent years, the study of the extremal behaviour, ie, the laws that rule the appearance of abnormal observations along the orbits of the system has suffered an unprecedented development, in which the team members took a prominent role. In this field of Extreme Value Theory applied to dynamical systems, one is interested in the limiting law for the partial maxima of stochastic processes arising from the system. It turns out that this study is deeply connected with the recurrence properties to certain regions of the phase space, which is one of the most emblematic achievements of the team.

Part of the reason for the development of this field stems from the interest that Physicists working with meteorological data saw in this area. One of the great advantages of understanding the extremal behaviour of dynamical systems is that the latter often provide toy models for very complex physical phenomena. This is the case of the Lorenz equations, which show how a very simplified model for the atmospheric convection can yet capture so accurately the complex and chaotic character of weather. Hence, the study of this subject has direct applications to risk assessment associated with occurrence of meteorological abnormal phenomena.

The main purpose of this project is to carry further the study of the limiting laws for the average behaviour and the extremal behaviour of the systems, as well as to understand the connection between both.

In particular, in the extremal case, we intend to provide more information about the impact of the occurrence of extreme events and the structure of record observations, for which we plan to perform a more sophisticated analysis using tools such as the theory of point processes. Moreover, in the case of heavy tailed distributions, which in this context are associated to observables with polynomial type of singularities, the average is tied to the extremal behaviour. We intend to exploit this link and understand its consequence in the dynamical setting.