Topics in Dynamical Systems and Ergodic Theory

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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 withpreservation of the differentiable orbit structure or some of itsimportant elements under small perturbations (local differentiablerigidity) or within certain classes of systems (global differentiablerigidity). Yet another type of rigidity appears when a certainproperty (such as a relation between invariants, or a regularity of aninvariant structure) forces the system to belong to a specializednarrow class. A part of the research program deals with theidentification of various rigidity phenomena in classical dynamicalsystems. An essential characteristic of proposed work is a syntheticapproach which looks simultaneously into all three principal classes ofbehavior which appear in dynamics: elliptic, parabolic and hyperbolicexploring both similarities and contrasts among these threeparadigms. Among the major goals is the identification of newsituations where measurable orbit structure determines differentiableorbit structure. Another part of the program builds upon PI's earlersuccesses in identifying and classifying rigidity phenomena for actionsof higher--rank abelian groups, i.e. dynamical systems withmultidimensional ``time'' which displays behavior essentially differentfrom the classical case. Among other goals of the program is thedevelopment of new techniques for construction of real--analyticdynamical systems with uniform ergodic behavior including solution ofthe long--standing problems of existence of such systems near elementsof periodic flows on some simple low--dimensional manifolds based onrecent advances in that direction . Mathematical concept of "rigidity" has many facets. Its simplest andmost basic manifestations can be seen at the level just abovehigh school algebra: a small number of equations or inequalities of aspecial type may imply much larger number of equation. For example, ifthe 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 conceptuallysimilar albeit technically much more sophisticated: a compactsurface of negative curvature, i.e. a bounded geometric shape where anygeodesic triangle has the sum of its angles less than 180 degrees, forwhich two numbers characterizing global and statistical volume growth(topological and metric entropy) coincide has constant negativecurvature, i.e. the sum of the angles of a geodesic triangle isuniquely determined by the area. Various aspects of rigidity appear atthe junction of several major mathematical disciplines, includingdifferential geometry, the theory of Lie groups and the theory ofdynamical systems. The research program under the present grant aims atidentifying various rigidity phenomena both in classical dynamicswhen time is one--dimensional and for dynamical systems withmultidimensional time where such phenomena are more pronounced andprevalent. Another central theme of the proposed research is a generalclassification of dynamical phenomena into hyperbolic and partiallyhyperbolic (roughly "chaotic" in lay parlance) elliptic (stablebehavior) and parabolic (intermediate complexity accompanied bypeculiar special features).