Holomorphic Dynamics

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Dynamical systems play an important role in all the sciences. They occur as models for predator and prey populations, orbits of planets and weather conditions. Some of the dynamical systems that are easiest to state are obtained by the iteration of a single map, viewed as a function from the real line to the real line, or from the complex plane to the complex plane. More recently researchers in physical sciences have been interested in models given by higher dimensional mappings. In the complex setting one can make use of complex analytic and geometric tools that are not available in the real one.The aim of this project is to describe the dynamics of a complex analytic map from a space of several variables to itself in the presence of a fixed point that is parabolic. The local study of holomorphic maps close to a fixed point is well understood (although not complete) in one complex dynamics. In several dimensions however, the situation becomes much more complicated. The PI will also focus on global dynamics in this setting and investigate the possible behaviors of orbits and their relation with multipliers at fixed points. Finally, the PI will explore applications of summability theory in the context of parabolic maps in several dimensions.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.