Project Details
Description
ABSTRACT:
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).
Status | Finished |
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Effective start/end date | 7/1/00 → 6/30/06 |
Funding
- National Science Foundation: $495,110.00