Examples for complex dynamics in several variables

Indiana University, Bloomington IN

Investigators

Abstract

This research project focuses on the dynamics of rational maps in two or more complex variables. Such maps have been studied for about twenty years, with specific examples playing a crucial role in the development of a more general theory. Based largely on these examples, there is a clear conjecture describing what one should expect from the "most unstable" part of the dynamics for these maps, i.e. the support of the measure of maximal entropy. Significant progress has been made and verification of the conjecture is almost complete for rational maps in two variables. The dynamics away from the support of this measure of maximal entropy is also of significant interest. In certain cases, there is a physically relevant invariant real slice that is away from the support of this measure. In other cases, one may be interested in the topology of the Fatou set, on which the dynamics of the map is stable. For these types of questions, we need new conjectures, which the PI believes will be best developed by studying appropriate examples. The PI proposes three different projects, each providing examples that can motivate a deeper theory of complex dynamics in several variables. They are: (1) study of rational maps arising as renormalization operators for the Ising model (a model for magnetic materials) on hierarchical lattices, (2) study of Fatou components for globally holomorphic maps of the complex projective plane, and (3) investigation of the first properties for a family of postcritically finite rational maps that have indeterminate points. All three of the proposed projects will work out details for the dynamical behavior of specific rational maps in several variables. This will push the contemporary theory and also help to develop new conjectures. Moreover, the project related to the Ising model will help to build new connections between three fields (real-dynamics, complex dynamics, and statistical physics), while potentially leading to a deeper theoretical understanding of magnetic materials. There is significant potential for broader impact, particularly by fostering better communication between researchers in dynamical systems and mathematical physics. Work related to these projects can influence students of all levels, from very talented high school students (with whom the PI actively works) through Ph.D. candidates, by exposing them to contemporary research that crosses between disciplines.