Complex Dynamics: Renormalization, Geometry, and Algebra

The mathematical area of dynamical systems studies the long-term evolution of physical or mathematical systems. The evolution often has chaotic nature (i.e., sensitive dependence on initial conditions), and fractal sets often come up when studying systems with chaotic behavior. An important fractal set is the Mandelbrot set, which has played a major role in dynamical systems and is one of the most recognizable fractal sets in mathematics.

Among central themes of this project is understanding certain self-similarity aspects of the Mandelbrot set. The Mandelbrot set encodes how a quadratic polynomial depends on a parameter. It is a test object for several fundamental problems in dynamics, in particular on how exactly chaos arises, even in simple systems.

The proposed activity also covers topics involving higher degree rational maps (i.e., ratios of polynomials). The projects will involve multiple collaborations with early-career researchers, and the training of graduate students. One of the goals of the principal investigator is to promote broader interactions between experts in different branches of mathematics.

The principal investigator will continue developing the near-neutral renormalization theory responsible for the self-similarity features of the Mandelbrot set near its main cardioid. Renormalization (interplay between different scales) appears in many branches of mathematics and physics; the exact dictionary is yet to be formalized. In the case of quadratic polynomials, central conjectures are essentially equivalent to establishing rigorous theories of the associated renormalizations (for example, proving hyperbolicity of the renormalization operators).

The principal investigator will also study the geometric and topological properties of parameter spaces of higher degree rational maps. A concrete theme is understanding the boundaries of hyperbolic components using near-neutral renormalization. The PI will continue designing the algebraic and algorithmic theories of self-branched coverings of a two-sphere, such as maps obtained from post-critically finite rational maps by forgetting the complex structure. The last topic is closely related to the theories of mapping class groups and self-similar groups.

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.