Geometric Aspects of Low Dimensional Dynamics

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This research program focuses on several interconnected geometric themes of complex and real low-dimensional dynamics. It includes polynomial dynamics on the Riemann sphere and the interval, dynamics in the real and complex Henon family, dynamics in some special families of rational endomorphisms of projective space coming from statistical physics, and the geometry of associated Riemann surfaces and hyperbolic laminations. Most of these themes are unified by the idea of renormalization as a powerful tool for penetrating into the small-scale structure of dynamical objects, aimed towards a complete classification of these objects.The project will result in deeper insights into small-scale structure of dynamical systems, a subject that furnishes a key to the understanding of many of the most basic processes in nature, the fundamental objects of study in science and engineering. It will include the training of highly qualified graduate students and postdocs, who will apply their skills in diverse ways and venues: in academia and industry, in the creation broader interactions between experts in various branches of real and complex dynamics and statistical physics, and in promoting efficient communication within the field of dynamical systems.