Dynamical Data: from Algebra to Topology

Dynamical systems are ubiquitous: they govern the motion of the planets, the weather, the stock market, and the ecosystems. These systems depend on different parameters, and as these parameters change, the corresponding system changes. Sometimes there are special values of the parameters for which the corresponding dynamical system is relatively simple. In complex dynamics, these are known as postcritically finite parameters. The postcritically finite parameters form a thin but sprawling collection that helps the understanding of the global structure of the parameter space. This project aims to extensively study the postcritically finite parameters and record several essential data that uniquely encode them. Broader impacts of the project are through mentoring at all levels, including a Math Corps summer camp for middle school campers and high school mentors from Ypsilanti, Michigan. In complex dynamics, one typically studies rational maps on the Riemann sphere from the point of view of iteration. A main principle in the subject asserts that in order to understand the dynamical behavior of a rational map, it is necessary to study the orbits of the critical points of the map. Those rational maps for which all critical points have finite forward orbits are quite special; they are known as postcritically finite rational maps. These maps possess only repelling and superattracting cycles, and their Julia sets are locally connected. Furthermore, (excluding well-understood exceptional cases), postcritically finite maps are inherently algebraic: every map can be conjugated by an automorphism of the Riemann sphere so that the resulting map has algebraic coefficients. In the moduli space of rational maps of a given degree, the postcritically finite locus can be used to study the geometry of this space. This project aims to investigate and explore dynamical data associated to postcritically finite maps. The projects in the proposal incorporate complex analysis, Teichmueller theory, topology, and geometric group theory to better understand these data. 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.