Geometry and probability in dynamical systems: fundamentals and applications

Abstract

The study of statistical properties of Dynamical Systems is currently present in almost all fields of science, from fundamental Mathematics to applied modelling. One, for example, is interested in knowing how robust the conclusions drawn from a model is. This task is particularly intricate when the dynamics evolve under random bounded perturbations. In this case, the study of statistical properties of the dynamics is usually carried out in terms of Markov chains and/or perturbations on the space of maps. The main challenge is that this requires strong assumptions on the properties of the perturbations themselves, on the existence of nearby orbits, on the space where it takes place, and on the classes of systems. Contrasting with stochastic processes under unbounded perturbations where tools from statistical analysis are applied, in the case of bounded perturbations only some punctual progress on this direction has been made. Furthermore, there is no general approach to address some of the most fundamental questions in this field. This project aims at studying statistical properties of some of the most important classes of dynamical systems and its geometric properties. In particular, it is intended to make substantial contributions connecting structural properties with statistical ones. The issues that should be addressed include a formulation of a general criterion for stochastic stability in light of the possibility of representation of random maps, thoroughly established in terms of shadowing property, and a formulation of a general principle to study statistical stability in terms of Optimal Transport Theory and the solution of the Monge problem. Furthermore, it is also intended to make substantial contribution towards applications. (AU)