The proposed research lies at the area of smooth dynamical systems and ergodic theory. The main goal of the project is to investigate "higher rank" dynamical systems. In particular, the investigator will study actions of discrete and continuous higher rank abelian groups, which are natural generalizations of diffeomorphisms and flows on smooth manifolds. Higher rank dynamical systems appear naturally in the study of various geometric and algebraic objects. The prime examples of these systems include hyperbolic and partially hyperbolic actions by automorphisms and translations on compact cosets of Lie groups. Using dynamical, analytic, and group theoretic methods the investigator will study rigidity properties of such systems. The examples of possible rigidity properties include description of invariant measures, regularity of measurable isomorphisms, and existence of smooth isomorphisms to the algebraic models.
Dynamical systems and ergodic theory is a relatively new field of mathematics which studies the evolution of physical and mathematical systems over time, for example planet systems, air and fluid flows. This field originated from the classical studies in differential equations and celestial mechanics. Dynamics and ergodic theory introduced new mathematical tools into these areas of physics and mechanics, such as the study of the qualitative behavior in the long run as well as various analytic and probabilistic methods. New ideas and concepts in dynamics, such as fractals and chaos, have not only affected the field itself dramatically, but also fundamentally changed our understanding of the world. The influence of the studies in dynamical systems nowadays goes as far as meteorology, biology, and computer science.