Complex and Real Low Dimensional Dynamics

We propose a research program on several intertwined geometric themes of complex and real low-dimensional dynamics, primarily on the polynomial dynamics on the Riemann sphere and the dynamics in the dissipative real and complex Henon family. Most of these themes are unified by the idea of renormalization as a powerful tool of penetrating into small-scale structure of dynamical objects, aimed towards their complete classification. We particularly emphasize the following themes: the problem of local connectivity of the Mandelbrot set, Feigenbaum Julia sets of positive area and Siegel Renormalization Theory, the Palis Conjecture on finiteness of attractors for strongly dissipative real Henon maps, and stability and bifurcations for moderately dissipative complex Henon maps. The proposed activity will result in deeper insights into small scale structure of dynamical systems, in training of highly qualified graduate students and postdocs who will apply their skills in academia and industry, in broader interactions between experts in various branches of real and complex dynamics, in promotion of communication between the field of dynamics and related areas of physics and applied mathematics.