HOLOMORPHIC DYNAMICS AND RELATED THEMES

The theory of dynamical systems studies the long-term behavior of trajectories described by iteration procedures, and how such behavior depends on the parameters of the system. Intricate fractal objects (like Julia sets and the Mandelbrot set) may appear as phase and parameter diagrams for such systems. This project focuses on complex and real low-dimensional dynamical systems described by simple quadratic equations. Despite the simplicity of the model, such systems are known to display complicated chaotic behavior indicative of various phenomena appearing in celestial mechanics, fluid dynamics, statistical mechanics, biology, and other branches of natural science. The proposed activity will result in deeper insights into the small scale structure of dynamical systems, in the training of highly qualified postdoctoral fellows and graduate students, in broader interactions between senior and junior experts in various branches of real and complex dynamics, and in the preparation of a book to assist the research community in acquiring background in the area. In addition, the Principal Investigator will facilitate communication within the field through the organization of international conferences and scientific programs and by maintaining a dynamics-related web site. The project addresses several geometric themes within complex low-dimensional dynamics, making a gradual transition from the one-dimensional to the two-dimensional world. In connection with dynamics in one dimension, renormalization will be investigated as a unifying and powerful tool for elucidating the small-scale structure of dynamical objects. Specific topics under consideration include a semi-local theory of neutral maps and a priori bounds for infinitely renormalizable quadratic polynomials with applications to the problem of local connectivity of the Mandelbrot set. Other topics of study include the dynamics generated by Schwarz reflections in quadrature domains and the dynamics of dissipative complex Henon maps. In connection with the latter, specific themes include the construction of two-dimensional examples of real and complex wild attractors and the development of a general theory of unimodal Henon maps. The project will also explore applications of higher dimensional holomorphic dynamics to the spectral theory of self-similar groups. Finally, the principal investigator will continue to work on a multi-volume book on the conformal geometry and dynamics of quadratic polynomials. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.