Advanced High School Course: Fractals and Dynamics

What was the course about?

This course had as objective to discuss the Non wandering Domain theorem of Dennis Sullivan. Not the proof of it, of course, but the result itself and see it with examples.

We covered the following topics with varying degrees of depth:

  • Complex numbers and the Fundamental Theorem of Algebra
  • Newton’s Method for Polynomials.
  • The concept of a Dynamical System.
  • Newton’s Dynamical System and the theorem of Cayley (for quadratic polynomials).
  • Examples of Newton Fractals.
  • Equivalent Systems and Mobius Transformations
  • Examples of fractals created by iteration: Sierpinski tetrahedron, Sierpinski Carpet, Menger Sponge, Koch Curve, Cantor Set.
  • Fatou and Julia Sets
  • Connected Component and Wandering Domain
  • Sullivan’s Non-wandering Domain Theorem!

What motivated me?

I have had a fascination with Fractals and Dynamics for a very long time. Some months before the course started, Dennis Sullivan was awarded the Abel Prize. This motivated me to create this course for the students, trying to explain to them what this famous theorem of someone was who just got recognized with one of the highest awards in mathematics.

A moment I remember 🙂

A student was so sad the course was over that he almost cries. It was very touching.

I later worked with this student in a mentorship program.

Exercises we worked on:

Note: The exercise section is not always complete. Many extra things were added on the spot as the course moved forward, depending on the questions of the public, and sometimes I do not have record of all of it.

Fractals_and_Dynamics__Problem_Set_1Download

Other resources:

Note: The list of resources we used might be incomplete. Many extra things were added on the spot as the course moved forward, depending on the questions of the public, and sometimes I do not have record of all of it.

A wonderful book to learn about the history of dynamics is A History of Complex Dynamics by Daniel S. Alexander.