Holomorphic families of complex dynamical systems

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ABSTRACT. The Principal Investigator (PI) aims to study complex dynamical systems in holomorphic families. The questions are centered on the notions of stability within families, bifurcations, and degenerations. More specifically, the PI will pursue (1) questions about the compactification of the moduli space of rational maps, where it remains open to construct a natural boundary which captures the information of dynamical degeneration, and to describe precisely which stable families are bounded in the moduli space; (2) characterizations of stability for families of higher-dimensional dynamical systems, where the techniques from one-dimensional dynamics do not generalize, but there has been recent progress using pluripotential theory; (3) an investigation of families of polynomials in one variable and the associated space of trees with dynamics, which provides a continuous combinatorial model for polynomials in all degrees, and should model the structure of the escape locus in the moduli space of polynomials; and (4) properties of the transfinite diameter in connection with families of polynomials in all dimensions, where the discussion involves a combination of analytic and arithmetic methods.A more general overview:There are many open questions around the long-term effects of perturbations of a dynamical system. The PI studies the iteration of rational functions of one variable, one of the simplest examples of a non-invertible system with non-trivial dynamics. In this project, she aims to answer questions of a global nature: what is the structure of the stable regime in a complex-analytic family of rational functions? Or, what type of degenerations can take place and what do they tell us about the bifurcation locus? The motivation for studying these particular families comes from interesting connections with hyperbolic geometry, algebraic and arithmetic geometry, and potential theory.