Dynamics of Polynomials

This project analyzes the structure and dynamical properties of families of complex polynomials of degree three. Nonlinear mappings arise in mathematical models across a host of scientific and applied fields, and a key issue is to understand how the behavior of such mappings changes as the underlying parameters vary. Among the simplest nonlinear mappings are complex polynomials. The structure and dynamical properties of the space of complex quadratic polynomials has been intensively studied since the early 1980s, culminating in a detailed understanding of the celebrated Mandelbrot set. Analyzing the structure of spaces of complex cubic polynomials is at the heart of this project. The project also provides research opportunities for graduate students and contributes to the training and mentoring of undergraduate students. In addition, the principal investigator continues to serve as director of an outreach program aimed at Alabama high school students. The project develops the dynamical and structural theory of moduli spaces of complex polynomials of degree three from several perspectives. A first line of inquiry concerns the construction of locally connected models of the cubic connectedness locus. By analogy with classical combinatorial models for the Mandelbrot set, the project also studies a combinatorial model in the cubic case based upon critical portraits. This work relies on recent laminational results recently developed by the PI and collaborators. A further approach to be investigated involves analytic tools. Estimates for the moduli of annuli will be used to show that Julia sets of polynomials of degree three are generated by rational cuts and admit a description in terms of rational laminations. If successful, this line of inquiry will validate conjectural laminational models of such Julia sets as well as certain subsets of the cubic connectedness locus. 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.