Measure Theory and Geometric Topology in Dynamics

Dynamical Systems theory describes the long run behavior of particles subject to a law. In order to achieve this, one needs to understand that, in most systems, the full knowledge of the particle's position and velocity, and even that of the law are not exact. This uncertainty can be modeled by the introduction of some randomness. As a particle moves after a unit of time, the new place is only known up to a random error. In this project this type of systems are studied and the aim is to show that the long run behavior of particles is governed by some distribution in the face space, which has nice geometric and even algebraic properties, thus introducing a purely mathematical and very powerful technology into the study of an otherwise intricate problem. Another important feature of this research is the classification problem. Finding invariants for a system helps its classification and a full set of invariants will give a full classification of the system. This set of invariants will depend a priori on the roughness one wants to classify the system. For chaotic systems it is not uncommon to expect that a set of invariants that is a priori very rough can lead to a finer classification automatically. This project will take advantage of this phenomenon and give some rigidity results. The geometry of invariant measures for systems displaying some hyperbolicity goes a long way when trying to describe its dynamics. In this project, the PI intends to deepen the knowledge of dynamical systems through the understanding of invariant measures with some nice geometric properties, how to make them appear and when to expect some uniqueness phenomena for them. This uniqueness comes attached with some uniform properties especially on information of equidistribution of orbits and nice statistical properties of the system. When this study gets in connection with the analysis of geometric objects or the actions by groups of larger rank, one often expects nice rigidity properties to show up. The principal investigator intends to develop this philosophy in several different frameworks, but in all cases some hyperbolicity will show up one way or another, sometimes from an intrinsic dynamical property, sometimes it will show up from the group acting, or from the randomness in the case of random dynamics. In all cases the principal investigator attempts to have a description of the relevant invariant measures and use this description to classify the dynamical systems or to get information of the geometry/topology of the support of the measure or to get strong statistical properties of the system. In some cases all these can be achieved. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.