Measure theory and geometric topology in dynamics

One of the goals of dynamical system theory is to understand the orbit structure of systems displaying some particular feature. In this project the principal investigator will focus on dynamics displaying some sort of hyperbolicity. The project will develop in two main directions, dynamical systems with multidimensional time (i.e. actions of higher rank groups) and positive entropy and classical dynamical systems (i.e. one dimensional time) displaying some uniform (though possibly partial) hyperbolicity. The aim is to develop techniques to understand the interactions between measure theoretical and topological properties of a system. This interaction happens to go both ways. For instance, in the direction of topology to measure theory it is observed that on 3-dimensional nilmanifolds partially hyperbolic systems are always K-systems. And in the direction of measure theory to topology it is known that a non-uniformly hyperbolic higher rank action on a 3-dimensional irreducible manifold can only exist on the torus.

Boltzmann's study of gas particles produced the nowadays accepted idea that several systems display some chaotic behavior. While the system may be of very diverse origin, e.g. from mechanical, biological, stock-market, etc., their abstract mathematical model can hopefully be classified and then studied with various mathematical technologies. The aim of this project is to classify systems displaying some hyperbolic behavior, the hallmark of chaotic dynamics, and prove that such systems should have some algebraic origin, thus introducing a purely mathematical and very powerful technology into the study of an otherwise intricate problem. While chaotic behavior is by definition impossible to be deterministically predicted, algebraic systems present symmetries that allows the researcher to study the long run behavior, and be able to describe the orbit structure of such systems. The PI plans to continue with the formation of graduate students, the writing of preparatory material on the subject (e.g. books, survey papers, lecture notes, etc.) and collaborate with the teaching of mini-courses, organization of workshop, seminars, etc.