The emergence of chaotic systems and the erratic behavior that they enclose triggered a new approach in their analysis, more concerned with their statistical properties. In order to learn about the long-term behavior of these systems through a probabilistic perspective, one can just consider dynamically defined stochastic processes arising from these systems by simply evaluating an observable function along the time evolution of the orbits of the system. These processes will be our starting point.
The next step is to find invariant ergodic probability measures with physical relevance. The time invariance of these measures implies that the dynamically defined stochastic processes become stationary with respect to these measures. The physical relevance derives from the fact that we require that the law of large numbers holds for a set of initial conditions with positive volume. These measures are usually called Sinal-Ruelle-Bowen (SRB) measures, authors who in their remarkable works established their existence for uniformly hyperbolic dynamical systems. Moreover, besides providing a description of time averages, SRB measures can be identified as equilibrium states for certain chosen potentials. One of our main problems consists in studying the existence of SRB measures for dynamical systems exhibiting weak forms of hyperbolicity.
While SRB measures immediately supply laws of large numbers for dynamically defined stochastic processes, for other finer statistical properties, ergodicity is not enough and some stronger feature like mixing is needed. Rates of mixing, which establish how fast the system looses memory, are usually written (proved) in terms of decay of correlations. Sufficiently fast decay of correlations can be used to obtain central limit theorems or rates for large deviations. In the last two decades there were many significant advances in the non-uniformly hyperbolic context, but some interesting questions in this field still remain open.
Most of the statistical properties considered so far pertain to the mean or average behavior of the orbits. However, in certain circumstances like in risk assessment, one is more interested in extreme observations (exceeding high thresholds) that correspond to the occurrence of rare events. Extreme value laws are related to the hitting times statistics or return times statistics. We aim at giving contributions on this topic for slowly mixing systems and systems with sigma-finite measures.
Another important issue concerns the stability of systems. Despite of remarkable successes in uniformly hyperbolic systems, structural stability proved to be too strong a requirement for many applications, in particular, among some classes of chaotic dynamical systems. More recently, increasing emphasis has been put on expressing stability in terms of persistence of statistical properties of the system. A natural formulation, the one that concerns us most in this work, corresponds to continuous variation of physical measures as a function of the dynamical system, where in the space of probability measures we consider the weak* topology or, when it makes sense, the L^1 norm in the space of densities with respect to Lebesgue measure.
A fruitful way of facing chaotic dynamical systems is through random perturbations and stationary measures. We say that the original map is stochastically stable if the stationary measures converge (in the weak* topology or L^1 norm, if possible) to the SRB measure of the original map when the support of the probabilistic law giving the possible choices of iterations shrinks to the original map. Stochastic stability tries to reflect that the introduction of small random noise affects just slightly the statistical description of the dynamical system and it has been proved in great generality for non-uniformly expanding maps. One of our goals is to obtain stochastic stability for other classes of maps and deduce some properties of the stationary measures as well.
Finally, another of our goals is to understand mechanisms leading to the creation of chaotic dynamical systems. Changes in the dynamics that lead to the creation of strange phenomena is always a key step to obtain new general results. In our specific case, we will be particularly interested in bifurcations via heterodimensional cycles and the dynamics arising from heterodimensional cycles for diffeomorphisms, which on their own motivate the study of iterated function systems.
Another way to obtain chaotic behavior may be by introducing an impulse in any region in the phase space. The theory of impulsive dynamical systems comes from the impulsive differential equations and, so far, the most significant part of its development has been focused on geometric/topological problems related to the existence and uniqueness of solutions and characterization of the limit sets. A first result on the statistical properties of impulsive dynamical systems established sufficient conditions for the existence of invariant probability measures. In the wake of this result, many interesting questions arise towards an ergodic theory of impulsive dynamical systems, such as the validity of a Variational Principle, the existence of maximum entropy measures, the existence of physical measures, etc.