Complex and real topological dynamics

University Of Alabama At Birmingham, Birmingham AL

Investigators

Abstract

We plan to extend Thurston's model of the Mandelbrot set onto polynomials of higher degrees (e.g., describe a model of the boundary of the cubic connectedness locus) and generalize onto higher degrees Thurston's results about unlinked quadratic minors. We want to extend the one-dimensional spectraldecomposition for topological polynomials. In addition, we study expanding polymodials, including c-tent plane maps (which combine properties of tent maps with properties of complex quadratic maps and should play the same role for quadratic complex maps as tent maps play on the interval) and seek to replace holomorphic tools by tools based upon the expansiveness of maps. Finally, we plan to describe all points such that small perturbations around them change the dynamics of the interval map (this is a pointwise version of stability) and to further develop rotation theory for interval maps, complex quadratic maps, and billiards. The results on laminations may have a strong impact on other areas as they should lead to a model of a higher dimensional self-similar set analogous to the Mandelbrot set; moreover, this set appears in a natural fashion. The analog of spectral decomposition for topological polynomials may provide examples of limit behavior for complicated non-invertible systems from applications (e.g., from population dynamics). Structural results on expanding polymodials would give another class of maps for which analogs of well-known complex dynamical results hold. Finally, topics in one-dimensional dynamics should yield a better understanding of stability of one-dimensional maps and the way different types of limit behavior of a one-dimensional map coexist. The project can have an educational impact as well as impact upon human resources development as some of the topics (such as new results/tools on cubic laminations, on spectral decomposition for topological polynomials and on dynamics of expanding polymodials) could serve as the basis of new courses for graduate students and could be developed by students in their dissertations (in fact, a local seminar is devoted to laminations).