Means and Extremes in Dynamical Systems

The main goal of the study of dynamical systems is to analyze the long-term behaviour of evolving systems and their orbits. We are particularly interested in chaotic systems whose unpredictable evolution seems to be practically random. In many situations, erratic phenomena can be modeled by chaotic dynamical systems such as the famous Lorenz equations or Hénon maps usually connected to meteorology, which brings special interest to the subject for its potential of application in many practical situations. Although we try to avoid the technical details for now, we can already say that the general goal of this project is to contribute for the understanding of evolutionary systems, especially the so-called chaotic systems, from a probabilistic point of view, with three major guiding vectors: i) existence and properties of physical measures or, more generally, equilibrium states; ii) analysis of the asymptotic behaviour described by limit theorems which characterize the mean and extremal properties of the system, in particular, associated with the study of rare events and recurrence; iii) contribute to the understanding of the properties of dynamical systems in a broader sense, whether through random perturbations or actions of free semigroups. An important issue that has motivated several results in the area is the so-called Palis conjecture [P05], which establishes that dynamical systems in finite-dimensional manifolds typically have a finite number of SRB measures. A probability measure is called Sinai-Ruelle-Bowen (SRB), if there is a set of initial states with positive volume such that time averages of any continuous potential evaluated along the orbits converge to the respective spatial average computed with respect to that measure. More generally, the thermodynamic formalism aims at determining the existence of invariant probability measures which maximize the topological pressure (a concept that generalizes entropy by the use of weights, defined by a potential function, along the orbits) and to describe their statistical properties. Such measures are called equilibrium states, and include as specific examples measures of maximal (associated to the null potential) entropy and SRB measures (associated to the logarithm of the Jacobian). The success of the probabilistic approachto study such properties of chaotic systems lies fundamentally in the fact that these systems loose memory very fast, which restores asymptotic independence. This property of the systems is usually referred to as mixing. The speed of mixing is particularly captured by the rates of decay of correlations, which became one of the most studied issues in Ergodic Theory. Sufficiently fast decay of correlations is an important tool to prove limit theorems that describe the average behaviour, through central limit theorems, laws of large numbers, functional limit theorems, but also the extremal behaviour related to the occurrence of rare events and recurrence to critical regions of the phase space, which are characterized by extreme value laws, hitting times statistics or weak convergence of point processes, for example. Quite often, modeling natural phenomena gives rise to non- autonomous systems for which the mathematical formulation of the iterative process requires different choices of the law that describes the dynamics at each step in the process. These choices can be made from a finite sample of possible options or a wide range of small perturbations of an initial system. We are naturally led to free semigroup actions, in the first case, or to random perturbations, in the second one. Sequential and random dynamical systems generalize the classical dynamical concepts, and thus enclose many difficulties related to either the non-stationarity or the possible combination of several degrees of non-hyperbolic behaviour.