Mapping Class Semigroups and the Classification of Conformal Dynamical Systems

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Dynamical systems are mathematical objects that describe the evolution of systems over time. They model many types of behavior, from animal neurons to epidemics to weather modeling. In some cases, we have a classification of the simplest types, and know how they are arranged. For example, we have now a good understanding in the real quadratic case. This achievement was the result of work by many researchers from different fields. It remains a central challenge to give a classification of examples beyond the real quadratic setting. This project takes up this challenge using tools from several different areas of mathematics. It focuses on systems that involve complex numbers, and on more general types called conformal. The conformal dynamical systems studied in this project include newly discovered, more exotic examples. Though our understanding of those more general systems is poorer, we have many new tools, from algebra to analysis, with which to study them. The project applies techniques from the well-developed theory of mapping class groups to the classification problem. Mapping class groups are symmetries of two-dimensional objects and are studied by both mathematicians and physicists. This project generalizes the notion of a mapping class group in a way that includes these two-dimensional complex dynamical systems. It also applies new tools from the recently developed theory of self-similar groups. This research area has an abundance of accessible problems. Students who engage with these topics will come to appreciate the essential unity of mathematics and the excitement of research. This will contribute to the development of a pool of mathematical talent that is broadly trained. Special software designed for this study allows for rich experimentation and the development of technical skills. The combinatorial foundations of complex dynamical systems were laid by A. Douady, J. Hubbard, and W. Thurston. The lack of a sufficiently natural algebraic framework delayed progress on fundamental problems until new techniques were introduced by L. Bartholdi and V. Nekrashevych in 2006. These selfsimilar group techniques are now standard. The recent developments in this new field are paralleling those in the theory of mapping class groups. The natural objects–branched self-covers of the sphere whose forward orbits of branch points form a finite set—may be fruitfully regarded as representing branched mapping classes in a countable semigroup. That this semigroup is in addition a biset over the pure mapping class group makes the combinatorial structure immensely rich. Just as with mapping class groups, on the semigroup side, there are similarly very deep connections to Teichmueller theory, dynamics, geometry, and algorithmic questions. The fuller exploration of these connections is the focus of this project. Basic dynamical finiteness results, and the development of an appropriate notion of relative hyperbolicity, are currently lacking. Intriguingly, the notions of relative hyperbolicity seem to lead naturally to connections with both coarse geometry and arithmetic dynamics on Berkovich spaces.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.