This project will explore several topics in holomorphic dynamical systems. One topic concerns the complex structure of domains arising naturally in holomorphic dynamical systems. It is conjectured that a stable manifold of a hyperbolic holomorphic automorphism is biholomorphically equivalent to complex Euclidean space. Instead of studying stable manifolds for one automorphism, one can translate the problem to the more general setting where an attracting basin of a sequence of automorphisms is studied. This topic is related to open problems in several complex variables and complex geometry. Another suggested topic is to study the polynomial automorphism groups in three or more complex variables. In two variables the structure of the polynomial automorphism group is known exactly, which is extremely helpful to those interested in iteration problems of polynomial automorphisms. A recent breakthrough by Shestakov and Umirbaev in dimension three has renewed the interest in this classical problem. In addition to discrete dynamical systems, the principal investigator will also study Riemann surface laminations in complex projective space. Examples of such laminations may occur by integrating holomorphic vector fields.
Dynamical systems are used throughout the sciences to model movement and change. Many dynamical systems suitable for rigorous analysis are generated by very special mappings, namely, by polynomials, rational functions, or other real analytic functions. It is natural to extend such functions to the complex plane so that the tools of complex analysis can be used to investigate these systems. Often the study of the corresponding complex analytic dynamical system gives valuable information regarding the original problem. This project considers dynamical systems in several complex variables. The problems put forth in this proposal are motivated by and related to problems in complex analysis, complex geometry, and affine algebraic geometry, areas that have a broad range of interest and application both within mathematics and within other disciplines.
