Measure Theory and Geometric Topology in Dynamics

The goal of Dynamical Systems theory is to describe the long run behavior of a body or of a collection of particles subject to laws of motion. It is often the case that full knowledge of the particles' position and velocity are not exact. As a particle moves after a unit of time (say a day), the new position is not known exactly. In this project such systems are studied. The goal is to show that the long run behavior of the collection of particles is governed by some probability distribution in the phase space. The distribution should have nice geometric properties. While the "chaotic" behavior of these system is by definition impossible to be exactly predicted there are systems which are described by algebraic equations where researchers can probabilistically study the long run behavior. The random iteration of diffeomorphisms is an important model for different types of dynamical systems. The PI plans to study these random iterations and gives a precise (as precise as possible) description of the stationary measure associated with it, particularly its geometric properties. Once geometric properties are established, some statistical properties can be deduced as corollaries, for example, the equidistribution of almost every orbit with respect to volume. This project will develop in two additional directions. One concerns dynamical systems with multidimensional time (i.e. actions of higher rank groups) and the other concerns classical dynamical systems (i.e. one dimensional time) displaying some uniform (though possibly partial) hyperbolicity. The unifying theme in the study is the techniques used which involve the interactions between measure theoretical and topological properties of a system. In all cases a description of relevant invariant measures is desired, how this impacts the topological dynamics and ultimately how this affects the topology of the ambient manifold. The PI also seeks intermediate interactions, for instance how the topology of the manifold imposes restrictions on the dynamics.