In one month I'll be giving a talk to motivated high schools students on a topic of my choice from dynamical systems and/or ergodic theory.
I'm having trouble coming up with a topic compelling enough to keep their interest, yet elementary enough to be comprehensible. In particular I need to keep clear of concepts of measure and multivariate calculus, unless I can somehow compartmentalize the necessary measure concepts in some neat way (e.g. Lebesgue measure on a manifold is probably okay).
I have some inclination towards giving a talk on dynamical billiards on polygons (very easy in terms of prerequisites), but I thought I would ask anyway to hear some different ideas. For instance, can anybody think of a high-school friendly topic in celestial mechanics?