Topics in Dynamical Systems and Ergodic Theory

Pennsylvania State Univ University Park, University Park PA

Investigators

Abstract

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).