Rubber Bands to Rational Maps

Indiana University, Bloomington IN

Investigators

Abstract

The dynamics of complex polynomials and rational maps has been intriguing and inspiring mathematicians for many years. Their Julia sets provide prototypical examples of fractals and exhibit fascinating dynamical behavior starting from very simple rules. In this project, the PI will work to better understand the geometry of fractals, particularly those generated by "rational maps", generalizing the polynomials used in the Mandelbrot set. The PI has developed a simple description of such maps using graphs, thought of as networks of rubber bands. This promises to be more accessible than previous approaches, lowering the barrier to participation by undergraduates and beginning graduate students, and letting them experience and understand first-hand the structure behind the beautiful pictures. In particular, the project will allow for a classification of the simplest rational maps, and a new test for when a loose (topological) description of a rational map can be tightened up into the rigid picture that generates the fractal. In more technical terms, the project will study certain maps from the sphere to itself (topological branched self-covers) that are related to rational maps. To determine the behavior of the rational map, the key points to look at are the critical points, where the map branches. A map can be described by giving an appropriate spine for the complement of the post-critical set on the sphere, along with its inverse image. One key question is to characterize when a branched self-cover can be made "rigid" into a rational map. William Thurston gave one answer in 1982, in terms of certain obstructions, collections of annuli that could not exist inside actual rational maps. This answer is complicated to state and difficult to apply, but it nevertheless provides one of the few tools for constructing rational maps. In this project, the PI will study a complementary criterion, which loosely says that the spine (with elastic weights on the edges) get looser under repeated backwards iteration. The new criterion promises to be both simple and powerful, and the PI will work with other researchers, graduate students, and undergraduate students to explore it. They will use the new characterization to, for instance, construct a new census of rational maps. They will also investigate extensions of the main theorem and study the new structure it reveals.