Complex Dynamics in Higher Dimension

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We investigate three aspects of complex dynamics in higher dimension. 1. We consider birational mappings in dimension 2 and higher. We use the tools of complex analysis and algebraic geometry to analyze fundamental properties of these mappings. In particular, we investigate how to determine the dynamical degrees in all dimensions. 2. We investigate the existence of automorphisms in complex surfaces and 3-folds. 3.We work with the complex Henon family, which has served as an important model family to exhibit complicated dynamical behaviors, and has been important because many "observed" phenomena can be proved mathematically in the complex case. We will focus on semi-parabolic implosion, which is an important bifurcation phenomenon.The broader impact of this work will come from its interaction with other areas of mathematics and physics. In particular, for physics, the research will have an impact on the area of lattice statistical mechanics. In mathematics, the research should result in a positive cross-fertilization with algebraic geometry. Results of this research should give new insights, too, into the tools of complex analysis.