Multivariable complex dynamics

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This proposal concerns problems in the intersection between several complex variables, complex algebraic geometry and dynamical systems. The problems stem from a very general program for constructing and analyzing measures of maximal entropy for rational self-maps of projective space. The work will begin with and be guided by the particular case of rational maps preserving a meromorphic two form. Specific issues to be investigated include `algebraic stability' for degree growth of maps, laminarity of invariant currents in codimension greater than one, and how to intersect dynamically natural closed currents to produce invariant measures. A dynamical system, at its most general, is any 'thing'-e.g. the weather, a bacteria colony, an economy, or a solar system-that evolves in time according to definite mathematical rules. Given the present state of the system, one often wants to know something about its future state. Will the earth warm significantly in the next century? Will the solar system fly apart? What are the long term financial effects of a given tax cut or government intervention? In a broad mathematical sense, all of these questions reduce to understanding whether some aspect of a dynamical system is 'stable' or 'unstable'. That is, does it vary slowly and predictably as the system evolves, and or is it prone to change rapidly and chaotically with a small variation in the system. The research proposed here aims at better understanding mathematics of dynamical systems, and in particular at determining and describing those parts of a system which are most unstable. Funding for this proposal will also support the two PhD students the PI is currently advising. It's broader impact will be felt through the PI's involvement with a Notre Dame summer program for high school math teachers and with the Riverbend Math Center, which is a local independent non-profit organization dedicated to promoting math education at all levels.