Complex Dynamics in Higher Dimensions

University Of Notre Dame, Notre Dame IN

Investigators

Abstract

DMS-0140408 Complex Dynamics in Higher Dimension Proposal Abstract The principal investigator proposes to study the dynamics of meromorphic self-maps of compact complex manifolds. If such a map expands some cohomology class of the manifold, then the main idea is that there ought to be a distinguished invariant current representing this class. Intersections between invariant currents should in turn give rise to invariant measures that describe the distribution of orbits where the action of the map is most complicated. Hence the linear action of a meromorphic map on cohomology should go a long way toward determining a probabilistic picture of the pointwise dynamics of the map. To justify these ideas, the investigator proposes to use a combination of techniques from complex geometry, smooth ergodic theory, and pluripotential analysis. Various problems in physics and mathematics reduce to questions about the dynamics of particular meromorphic maps. The investigator proposes to use some of these as starting points and test cases for his research. The study of dynamical systems has grown increasingly important in the past several decades. Put briefly, a dynamical system is anything--e.g. the weather, the stock-market, the solar system, etc--that evolves in time according (at least theoretically) to definite and codifiable rules. The general goal in studying a dynamical system is to make predictions about its future state given its present one. If the future is very far off then this, as any honest forecaster will admit, is typically quite difficult. Small changes in the present state of the system can imply huge differences in eventual outcome. The research proposed here is devoted to better understanding instability in dynamical systems--when it occurs, why it occurs, and how to recognize it. The research is also devoted to attaining partial control over instability by making statistical predictions about the eventual behavior of an unstable system that do not depend on perfect understanding of its present state. The dynamical systems that the investigator proposes to study are mathematical in nature, but they are also archtypes for the systems that one analyzes in connection with real-world phenomena. The idea is that weather and the stockmarket and the solar system ought not be treated as a separate and totally unrelated dynamical systems but as particular examples of the same general phenomena and subject to similar kinds of analyses. **********************************