The theory of Dynamical systems (with discrete time) studies the long-term behavior of trajectories described by a certain iteration procedure, and the way this phase portrait depends on the parameters of the system. Very interesting fractal objects (like Julia sets and the Mandelbrot set) may appear as phase and parameter diagrams for such systems. In the project, the PI will focus on complex and real low-dimensional dynamical systems described by simple quadratic equations. Despite simplicity of the description, these systems are known to display complicated chaotic behavior serving as a good model for various phenomena that appear in celestial mechanics, fluid dynamics, biology, and other branches of natural science. The proposed activity will result in deeper insights into small scale structure of dynamical systems, in training of highly qualified postdocs and graduate students who will apply their skills in academia and industry, in broader interactions between experts in various branches of real and complex dynamics, in publishing a book that would help a broad student and research community to acquire background in the area, in promotion of communication in the field by organizing conferences and scientific programs, giving mini-courses, and maintaining a dynamics web site (http//www.math.stonybrook/dynamics).
The PI will conduct a broad research program on several intertwined geometric themes of complex and real low-dimensional dynamics, making a gradual transition from the one-dimensional to the two-dimensional world. The PI will work on the Dynamics of dissipative complex Henon maps and attractors for typical dissipative real Henon maps. Specific themes include exploring the problem of existence of wandering domains, building up puzzle techniques, and the study of local dynamics near semi-Cremer fixed points. The PI will keep pursuing several one-dimensional projects unified by the idea of renormalization, a powerful tool of penetrating into small-scale structure of dynamical objects aimed towards completing their classification. They include the Siegel Renormalization Theory, scaling of Mandelbrot limbs, and a priori bounds for primitively infinitely renormalizable quadratic polynomials. The PI will finish the first volume of a book "Conformal Geometry and Dynamics of Quadratic Polynomials".
