The Combinatorial Mandelbrot Set


January 19


Alexander Blokh and Lex Oversteegen, UAB


Quadratic complex polynomials P_c(z)=z^2+c have critical point 0 and critical value c=P_c(0). We say that c is non-escaping if the sequence 0, P_c(0)=c, P^2(0), ... is bounded. The set of all non-escaping c's is called the filled Mandelbrot set and its boundary is called the Mandelbrot set. This famous fractal set has been extensively studied since the 1980-s. Adrien Douady, John Hamal Hubbard and William Thurston constructed a combinatorial model for the Mandelbrot set, called by Douady the pinched disk model.

In this talk we will describe this model and several simplifications of it. We are motivated by the fact that such simplifications reveal the structure of the Mandelbrot set and may allow generalizations to polynomials of higher degree.