Project 4. Entropy in Complex Dynamics

Complex dynamics is the study of holomorphic maps under iteration (holomorphic = complex; iteration = dynamics). One simple yet famous family of homomorphic maps are the quadratic polynomials f_{c}(z) = z^{2} + c, for any parameter c in the complex plane. If you have not seen this example before*, I encourage you to google the Mandelbrot set and explore the various “Julia sets” associated to various parameters c.

It turns out that the global dynamics of these maps is determined by the behavior of the maps’ critical points. Among other things, this allows us to build simple models which coarsely approximate the dynamics in question. One important invariant of these models is that of entropy, which measures something like the complexity of the Julia set. For example, the study of the Mandelbrot set polynomials fc(z) gives rise to the Figure 4a, and try it yourself to see if you can recover the “shape” of the Mandelbrot set from the core entropy (hint, compare with Figure 4b).

In this project we aim to use core entropy plots to understand the shape of generalizations of the Mandelbrot set, such as nice families of quadratic rational maps or higher degree polynomials. Coding experience (python, matlab, mathematica, sage math, etc.) is a plus, however, students willing to learn as they go historically get by just fine.