Time:

2:30-5:10 PM, November 23.

3:10-6:00 PM, November 27, December 4, December 11, December18.

Abstract:

The notion of topological entropy, arising from information theory, is a fundamental tool to understand the complexity of a dynamical system. When the dynamical system varies in a family, the natural question arises of how the entropy changes with the parameter.

In the last decade, W. Thurston introduced these ideas in the context of complex dynamics by defining the "core entropy" of a quadratic polynomial as the entropy of a certain forward-invariant set of the Julia set, the Hubbard tree.

We will start the course by introducing the basic concepts on the dynamics of polynomials of one complex variable, defining and studying the basic properties of Julia sets and the Mandelbrot set.

Then, we will analyze the combinatorial structure of the Mandelbrot set, and define the core entropy. As we shall see, the core entropy is a purely topological / combinatorial quantity which nonetheless captures the richness of the fractal structure of the Mandelbrot set.

Finally, we will see how to compute the core entropy, and relate the variation of such a function to the geometry of the Mandelbrot set. We will also prove that the core entropy on the space of polynomials of a given degree varies continuously, answering a question of Thurston.