Complex dynamics via tropical moduli spaces

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Dynamics is the study of systems evolving with time. One striking feature is that a system evolving under a very simple rule can exhibit extremely complicated long-term behaviour. This often manifests through the ubiquitousness of fractals --- infinitely complicated shapes. Complex dynamics is the study of the behavior of holomorphic maps --- for example (z-->z^2-1) --- under repeated application. When studying a dynamical system, the main question is: how does long term behaviour depend on initial condition? The "Julia set" of a dynamical system is the boundary demarcating initial conditions with different long-term behaviour. In the above example, as well as more generally, the Julia set is a fractal. It is natural to investigate not just the dynamical behavior of one map in isolation, but also the variation of dynamical behavior within families of maps. For example, instead of looking just at (z-->z^2-1), one could consider all dynamical systems of the form (z-->z^2+constant). The field of complex dynamics underwent a transformation with Douady and Hubbard's exploration of the Mandelbrot set, which lives inside the space of dynamical systems (z-->z^2+constant). The boundary of the Mandelbrot set is a fractal that demarcates dynamical systems with qualitatively different long-term behaviour. The dynamical behaviour of a map is reflected by the long-term behaviour of critical points --- points where the derivative is zero. For example, the Julia set of (z-->z^2+constant) is connected if and only only if the critical point "0" has bounded orbit. Post-critically finite (PCF) maps are maps for which every critical point eventually lands in a periodic cycle. Their dynamical behaviour can be encoded combinatorially, and understanding PCF maps is crucial for understanding complex dynamics more generally. PCF maps have a very special distribution in families of rational maps, for example they are dense in the boundary of the Mandelbrot set. PCF maps also provide a fascinating link between dynamics in one variable and in many variables. Thurston proved a consequential rigidity result for PCF maps by constructing dynamical systems called "Thurston pullback maps", whose fixed points are PCF maps in one variable. Thurston's pullback maps act on high-dimensional Teichmuller spaces: understanding their dynamical behavior "near infinity" is of crucial importance for understanding PCF maps, as well as for understanding degenerations of rational maps. By work of Koch, Thurston's pullback maps have algebro-geometric "shadows" called Hurwitz correspondences. This provides a new opportunity to use tools from combinatorial algebraic geometry of moduli spaces in order to address questions in complex dynamics. Tropical geometry is the study of degenerations in algebraic geometry: it is a very well-suited framework to use to study the dynamics of Hurwitz correspondences and Thurston's pullback map. However, it has not yet been applied to this setting. In this research, we will use the dynamics of tropical Hurwitz correspondences in order to unify objects that are active topics of research, but in different fields. We will link dynamical degrees (algebraic dynamics in many variables) with Thurston obstructions (Teichmuller-theoric objects). This will link the global algebraic dynamics of Hurwitz correspondences to the local dynamics near infinity of Thurston's pullback map. We will link Hubbard trees (rational dynamics in one variable) with admissible covers (combinatorial algebraic geometry) and tropical curves.