Topics in Dynamical Systems and Ergodic Theory

  • Katok, Anatoly (PI)

Project: Research project

Project Details



The project primarily deals with rigidity phenomena in dynamical systems.

One aspect of rigidity concerns situations when a weaker structure

(i.e. measurable) determines a stronger one (i.e. differentiable)

within certain classes of systems. Another aspect deals with

preservation of the differentiable orbit structure or some of its

important elements under small perturbations (local differentiable

rigidity) or within certain classes of systems (global differentiable

rigidity). Yet another type of rigidity appears when a certain

property (such as a relation between invariants, or a regularity of an

invariant structure) forces the system to belong to a specialized

narrow class. A part of the research program deals with the

identification of various rigidity phenomena in classical dynamical

systems. An essential characteristic of proposed work is a synthetic

approach which looks simultaneously into all three principal classes of

behavior which appear in dynamics: elliptic, parabolic and hyperbolic

exploring both similarities and contrasts among these three

paradigms. Among the major goals is the identification of new

situations where measurable orbit structure determines differentiable

orbit structure. Another part of the program builds upon PI's earler

successes in identifying and classifying rigidity phenomena for actions

of higher--rank abelian groups, i.e. dynamical systems with

multidimensional ``time'' which displays behavior essentially different

from the classical case. Among other goals of the program is the

development of new techniques for construction of real--analytic

dynamical systems with uniform ergodic behavior including solution of

the long--standing problems of existence of such systems near elements

of periodic flows on some simple low--dimensional manifolds based on

recent advances in that direction .

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

most basic manifestations can be seen at the level just above

high school algebra: 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) than 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. Various aspects of rigidity appear at

the junction of several major mathematical disciplines, including

differential geometry, the theory of Lie groups and the theory of

dynamical systems. The research program under the present grant aims at

identifying various rigidity phenomena both in classical dynamics

when time is one--dimensional and for dynamical systems with

multidimensional time where such phenomena are more pronounced and

prevalent. Another central theme of the proposed research is a general

classification of dynamical phenomena into hyperbolic and partially

hyperbolic (roughly 'chaotic' in lay parlance) elliptic (stable

behavior) and parabolic (intermediate complexity accompanied by

peculiar special features).

Effective start/end date7/1/006/30/06


  • National Science Foundation: $495,110.00


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