I guess this is only for people who are familiar with dynamical systems and circle maps. Let f be a C^3 diffeomorphism of the circle. Prove that f is infinity renormalizable iff f has an irrational rotation number. Pf. Let's do the forward implication first. Assume that f is infinitely renormalizable. Then we can scale it down affinely as many times as we want. We want to derive that f cannot have any periodic points, which then implies that f has an irrational rotation number. WLOG, assume that f is a D minus function, ie where we cut the circle, c < v. So assume that f has a periodic point, and therefore f has a rational rotation number, let it be p/q. Therefore f as a point of period q. Let w be a point of period q. Now we want to derive a contradiction that either f is not infinitely renormalizable or f is not a diffeo or f is not a C^3 diffeo My guess is that since f has a periodic point, something gets messed up when we rescale affinely. But I can't get much further than that right now. For the converse, it's very similar reasoning. If f has an irrational rotation number, then it has no periodic points. So building on the first implication, I wanted to prove the converse. But I'm stuck. Any suggestions or ideas are gladly welcomed. and don't worry this isn't an unsolved problem or something where I would be stealing credit. It's a little research project in dynamical systems.