What topics could be covered in a course on fractals?

Question

I'd like to propose a class on fractals to my department in the next few years.

One issue is that there seems to be no consensus on what a fractal is (see the wikipedia talk page on fractals, for instance). The following are all things that I would consider fractals or fractal-like:

1. Subsets of $\mathbb{R}^n$ whose Hausdorff dimension does not agree with the topological dimension.
2. Space-filling curves
3. Infinite self-similar tilings (like the Penrose tiling and other tilings from subdivision rules)

Perhaps the first class of fractals would be the most important, while the other two would be short appendices at the end. Within the first class, one could cover

1. Julia sets
2. The Mandelbrot set
3. The Cantor set, Sierpinski carpet, and Menger sponge

And so on.

The issue is, these topics are more or less all over the place, and require complex analysis and topology. I'd prefer to keep this to a senior-level course, so that means cutting some things out.

What subset of the above topics (or additional concepts about fractals) can be reasonably covered in a one-semester course aimed at senior undergraduates who have taken real analysis and abstract algebra? Also, should complex analysis be a prerequisite?

Answer 1

Some years ago, I attended a course where Julia and Mandelbrot sets were part of. It was called "Chaotic Dynamical Systems", and was held once a week for 90 minutes for a total of 14 lectures. It was made for master students in math, although advanced bachelor students could also have participated. This doesn't completely fit the topics of your course, but it might be a solid base. Without understanding of complex numbers, this course wouldn't have worked, although complex analysis wasn't strictly required.

The contents of the course were:

1. Introduction of discrete dynamical systems; the definition mentions topological spaces, but the script says "Usually, we have more than that". Orbit, invariant sets, conjugate systems, examples are usually on $\mathbb{R},\mathbb{R}^+$ and intervals. Main toy example is the logistic system/family.
2. Fixed points, periodic points, stability of orbits, attractive points, repellent points, characterization of these points for functions which are continouosly differentiable on $\mathbb{R}$ or $\mathbb{C}$, Bifurcation for a family of dynamical systems, Sharkovsky's theorem (no proof), theorem of Li-Yorke (with proof)
3. Definition of a chaotic system (Devaney, see references), symbolic dynamics
4. Complex dynamic systems: Newton's method, Julia sets, Fatou set, Mandelbrot set. proofs partly omitted, partly shortened to a sketch

References for the course:

• R. L. Devaney: An introduction to chaotic dynamical systems

• D. Gulick: Encounters with chaos

The desired topics were only covered in the last few weeks of the course, but I think that if you go lighter on the chaotic part, you might be able to give more emphasis to the sets you are interested in.

I think that a course on discrete dynamical systems is a good context for these kind of sets - unfortunately I can't quite tell how to fit in Cantor/Sierpinski/Merger - usually I'd include these in a course on measure theory.

Answer 2

Background: In my senior year of undergrad, I was a TA for our school's "Fractal Geometry" course, having worked with the professor before on a research project in fractals. We followed some of the materials provided (publicly!) in Yale's Fractal Geometry course.

There was a particular focus on Iterated Function Systems, which are great ways to generate fractals of "type 1", per OP's list. (Indeed, the Yale course was created by Michael Frame, who was a collaborator on our research into IFS.) There are some great visuals and applets provided to explore the main ideas and appeal to some more visual/hands-on learners. I highly recommend at least looking through the course's contents, whether or not you end up using all of it.

If you are looking to address a student audience of senior math majors, you might want to home in on some of the following sections. Note: This list is certainly not exhaustive; this is just what I found scanning through now. But these will at least require the students to use some of their math knowledge and maturity.

• Algebra of Dimensions: Does knowing the dimension of $A$ and $B$ tell us anything about the dimensions of $A\cap B$, $A\cup B$, $A\times B$, $A$ projected onto $B$?
• Combinatorics of the Mandelbrot set
• Newton's Method for complex numbers, followed by a use of such a method on the Mandelbrot set
• Various topics in the chapter on Chaos

In addition, there some appendix resources:

• Lesson plans and a list of required mathematical topics, sorted by fractal topic
• "Lab" activities for exploring certain topics (could serve as in-class activities or take-home projects)

Regarding your question about whether to require complex analysis: You don't have to. This will depend on how much material on the Mandelbrot/Julia sets you want to cover, and to what depth you do so. I would recommend tending towards the side of not requiring much familiarity with complex analysis, and only a working knowledge of the algebra of complex numbers; hopefully, this will make more students want to take this course!

Answer 3

You certainly can have even a one-year course on fracals without complex analysis.

For example, even homotheticaly self-similar fractals are rich, with easy-to-state open questions; e.g. the precise behavior of the Favard length (average measure of $1$-dimensional projections) of the four-corner Cantor set in the plane (product of two middle-half Cantor sets on the line) is still unknown.

You can also investigate the notion of rectifiability, or study snowflaked metrics, i.e. metrics as $d(x,y)^\alpha$ where $d$ is the Euclidean metric on $\mathbb{R}^n$ and $\alpha\in(0,1)$. It is known that such metric spaces can be bilipschitz embedded into $\mathbb{R}^N$ for some $N=N(n,\alpha)$ (a theorem of Assouad) but apart when $n=1$, we do not know how big $N$ must be (of course, it must be at least $n/\alpha$, the Hausdorff dimension of $d^\alpha$).

More basically, already exploring the notion of Hausdorff dimension can be interesting (e.g. constructing two sets of dimension $0$ whose product has dimension $1$). Mattila has a great book to learn these things.

I mentioned open problems because they can be motivating to students, but of course there are uncountably many proved results to teach in these directions. I mentioned more than can be covered in a one-semester course, but all this is highly flexible.