Rigidity Phenomena for Higher Rank Abelian Actions

Proposal Number: DMS-0140513

PI: Boris Kalinin

ABSTRACT

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.