Dynamical Systems, Rigidity and Related Topics

Abstract:

The central theme of the proposed program is the study of various

aspects of rigidity in dynamics.

New methods and insights have been recently introduced by the PI and

his collaborators which led to a significant progress in the problem

of rigidity of invariant measures and the differentiable rigidity of

orbit structure for actions of higher rank abelian groups. Advances

achieved based on these methods engendered fruitful applications to

Diophantine approximation problems in number theory and to the problem

in arithmetic quantum chaos. There are several major directions in the

proposed program:

1. Local differentiable rigidity for partially hyperbolic actions of

higher rank abelian groups with the emphasis on the combination of the

dynamical systems, harmonic analysis/group representation and

geometric methods.

2. Global rigidity of Anosov actions, using various approaches based

on invariant rigid geometric structures.

3. Rigidity of invariant measures using the innovative high entropy

and low entropy methods in the positive entropy case as well as new

approaches to the zero entropy case.

4. Problems of quantum unique ergodicity and existence of scars for

Finsler geodesic flows and billiards in polygons.

5. Precise asymptotic and multiplicative lower bounds for the growth

of the number of periodic orbits for broad classes of dynamical

systems with non-uniformly hyperbolic behavior.

6. The problem of smooth realization of measurable dynamical systems.

Mathematical concept of ``rigidity'' has many facets. Its simplest and

most basic manifestations can be seen from the following elementary

example: a small number of equations or inequalities of a

special type may imply much larger number of equation. For example, if

the arithmetic mean on n numbers coincides with the geometric mean

(one equation) then the numbers are all equal ( n-1 equations).

An example from the PI's earlier research is conceptually

similar albeit technically much more sophisticated: a compact

surface of negative curvature, i.e. a bounded geometric shape where any

geodesic triangle has the sum of its angles less than 180 degrees, for

which two numbers characterizing global and statistical volume growth

(topological and metric entropy) coincide has constant negative

curvature, i.e. the sum of the angles of a geodesic triangle is

uniquely determined by the area. The research under the present grant

involves both deeper

investigation of rigidity phenomena for dynamical systems with

multi-dimensional time,

and expansion and development of striking application to several areas

of mathematics and mathematical physics. Among the latter are: 1)

problems of simultaneous approximation of several irrational numbers by

rationals and 2) connection between the behavior of certain class of

quantum mechanical systems and their classical limits when Plank

constant goes to zero. The central idea is that certain properties of

classical limits

(such as hyperbolicity or ``chaos' on the one hand and presence of

certain types of periodic orbits on the other) is reflected in the

behavior of quantum systems such as

``unifrom distibution of quantum states ' and ``scars'.