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.
Research Description: The research funded by this grant is about holomorphic dynamical systems in higher dimensions. Such a system consists of a complex manifold X and a holomorphic (or sometimes rational) mapping f from X to X. For any point x in X one studies how the sequence of iterates (x, f(x), f(f(x)), ...., f^n(x),...) behaves as n increases. Such systems serve as idealized models of the types of higher-dimensional dynamical behavior that can occur in real world systems. Examples including the Henon map and Newton's method have driven the major conjectures and breakthroughs in the early days of the subject. The purpose of this project was to generate several new examples of holomorphic (and rational) maps f: X->X which are used to make new conjectures and discover new general results. Primary results: 1) We showed that the stable manifold for an invariant circle the Migdal Kadanoff renormalization operator for the Ising model on the diamond hierarchical lattice is not real analytic. This has interesting consequences about the distribution of Lee-Yang and Lee-Yang-Fisher zeros, hence interesting consequences about Statistical Physics. This very specific result led to a general result about analyticity of stable manifolds of circles when the rate of transversal superattraction is greater than or equal to the degree of the mapping within the line. (Joint with S. Kaschner.) 2) Proof of algebraic stability for a family of postcritically finite rational maps with indeterminate points that arises from Thurston's topological classification of rational maps. This was followed by a computation of the dynamical degrees for these mappings. (Joint with S. Koch.) 3) Use of intersection theory to develop an algebraic approach to prove criteria for a composition of rational maps to act nicely (functorially) on cohomology. 4) Found examples of rational maps of the 2-dimensional projective space with equal first and second dynamical degrees and no invariant foliation. This answered a question of V. Guedj. (Joint with S. Kaschner and R. Perez.) 5) Lower an upper bounds for the number of solutions to p(z) = overline{z} and r(z) = overline{z}, where p and r are holomorphic polynomials and rational functions were proved by Petters, Khavinson-Swiatec, and Khavinson- Neumann. We showed that between these two bounds any number of solutions allowed by the argument principle occurs and nothing else occurs, off of a proper real- algebraic set of parameters. This solves a question from astrophysics about how many images a star can have under a certain type of gravitational lens. (Joint with P. Bleher, Yo. Homma, and L. Ji.) 6) Consideration of systems of n Gaussian random polynomials in n variables. We showed that for a natural (SO(n+1) invariant) ensemble of these polynomials, the correlations between zeros behaves like C t^{2-n}, where t is the distance between points. This illustrates the same behavior as proved for the complex case in a seminal paper by Bleher-Shiffman- Zelditch. (Joint with P. Bleher and Yu. Homma.) These results have considerable intellectual merit as they have resulted in new techniques, led to new fundamental questions, and established new connections between various areas of mathematics and physics. All of these results have been disseminated widely through research presentations and papers (published, submitted, and in preparation). Educational Projects and Broader Impact: 1) One-on-one supervision of research students. While supported by this grant, I supervised Scott Kaschner's Ph.D. thesis and taught him how to give research talks, write papers, etc. He is now a teaching postdoc at the University of Arizona. During the first year of support from this grant, Pavel Bleher and I led two high school students Youkow Homma and Lyndon Ji on a research project relating complex variables to astrophysics. They presented their results at the Intel International Science fair. The results were then published as a joint paper. Both students now major in mathematics at Yale University. During the second and third years supported from this grant, Pavel Bleher and I led another high school student, Yushi Homma, on a research project about the zeros of random polynomials. He submitted his work to the Intel Science Talent Search, making it to the top 40 and visiting Washington DC for the finals. We are now finishing a joint paper for publication. Yushi is now a math major at Stanford University. 2) During the period supported by this grant, I helped to organize 2 special sessions on complex dynamics and one conference on dynamical systems. Many of our speakers were young mathematicians (graduate students and postdocs). 3) During the period supported by this grant, I have co-organized (with Jeff Watt) the IUPUI high school math contest, which reaches between 50 and 100 talented high school students from throughout Indiana. All students who participate and their teachers are invited to an award ceremony where a public math lecture is presented.
