Actual problems in fractals.

Question

I am preparing a report for student conference on the mathematics. The main objective of a report is to overview history of fractals, both from theoretical and applied viewpoint. The only problem is that my current version, as I see it, doesn't contain enough examples of actual problems related to fractals.

So, the question is: what are examples of such problems which are available to understand for average math student? (studied Calculus I and II, Linear Algebra, Differential Equations and some functional analysis)

UPD. Because of preference of the conference to functional analysis, it is desirable to look at fractals as fixed points of some contraction mappings.

Answer 1

You may start with Cantor set, the simple H-fractal, the Binary tree,the Sierpinski's triangle, the Koch fractal,the Koch island, the Minkowski island, the Dragon curve,..., and then go to the nature and find fractals such as shells, amonite, etc.

Answer 2

Differential equations! If you look at some 3+ dimensional differential equations, you can see that the set of limit points for all possible initial values (called the attractor) often has fractal geometry. You may want to try out search terms such as "fractal strange attractor" for some easy introductions and nice pictures. One particularly famous attractor is the Lorenz attractor, generated from a simplified form of the Navier-Stokes fluid flow equations (whose analysis is one of the million dollar problems), and has a recognizable "butterfly" shape; it has a Hausdorff dimension of approximately $2.0627$.

An important thing to note, which you will see from strange attractors, is that fractals are not necessarily self-similar. You should also look into the box counting algorithm for calculating fractal dimension.