Holomorphic Dynamics in one and several variables

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Dynamical systems are mathematical models that allow researchers to study physical phenomenon that evolve with time: from motion of the planets to stock markets. The PI studies holomorphic dynamics, which refers to iterations of certain complex-valued maps that admit a rich geometry. The subject has numerous interactions with other areas of mathematics, in particular with the important subjects of Teichmuller Theory and Multidimensional Complex Analysis. The PI will continue her work concerning geometric aspects of holomorphic dynamics and in particular she will work to extend certain parts of the the theory from dynamics of one complex variable to several complex variables. The PI will continue to provide special topical lectures to undergraduate and graduate students in the hope of exciting the interest of students in the field of dynamical systems and, more broadly, in mathematics.The link between holomorphic dynamics and Teichmuller Theory can be to a large extent explained by holomorphic motions. The PI, together with collaborators, suggest a geometric approach to the proof of Slodkowski's lambda lemma. We relate holomorphic motions to the technique of filling totally real manifolds by holomorphic discs. We plan to use this approach to investigate the h-principle for holomorphic motions over Riemann surfaces. Thurston's theorem on postcritically finite rational maps is one of the central results in one-dimensional holomorphic dynamics. Building on Thurston's work, A. Epstein introduced deformation spaces, which give a unified approach to transversality results in holomorphic dynamics. The PI, together with collaborators, will investigate certain topological properties of these deformation spaces. In one-dimensional holomorphic dynamics to a large extent the orbits of critical points determine the dynamics of the map. For an automorphism of complex plane there are no critical points however there are various analogs of the critical points in different dynamically defined regions. The research will investigate these critical loci and their application to the construction of topological models. We also study the non-linearizable germs of analytic diffeomorphisms of complex plane.