Geometry and Ergodic Theory of Rational Maps

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The principal goal of this proposal is to understand the dynamics of rational self-maps of complex manifolds in terms of the induced linear actions on cohomology groups. The idea is to realize expanding eigenvectors geometrically as invariant positive closed currents carried by the manifold and, by intersecting currents, to produce and understand measures of maximal entropy for the maps. It is also proposed to push beyond ergodic theory for some families of rational maps with extra geometric structure and give more detailed, pointwise descriptions of dynamics.Mathematical laws for an evolving system typically prescribe the way the system changes. If the present state of the system is known, then such laws tell one what happens in the following instant. However, the distant future of the system, while in principle predictable from the same laws, is in practice often impossible to know: computations involved in making predictions can be overwhelming, or the computations can amplify small uncertainties in data about the present, so that predictions become hopelessly vague. Miraculously in such situations, one can often employ less direct means to obtain a rough probabilistic picture of the future. For instance, detailed knowledge of today's weather and the laws of physics will allow one to know the particulars of tomorrow's weather with some confidence, but they will avail one little in predicting the weather a year from tomorrow. Nevertheless, knowing only that today's date is Jan 1st, one can reasonably bet that it will be chill! y a year from today in Minneapolis. Understanding trends of this sort from a mathematical standpoint is called ergodic theory, and that is the subject of the research described in this proposal. It is devoted particularly to ergodic theory of systems known as `rational maps'. Such systems are cleaner and more abstract than weather forecasting, but they are nevertheless very rich and applicable in practical situations where computers are used to solve complicated equations or to optimize the outcome of a particular process.