Iterating evolutes and involutes

We study iterations of two classical constructions, the evolutes and involutes of plane curves. We also study two kinds of their discretizations: the curves are replaced by polygons, and the evolutes are formed by the circumcenters of the triples of consecutive vertices, or by the incenters of the triples of consecutive sides. The dynamics of these two discrete systems are quite different. In addition to the theoretical study, we performed numerous computer experiments; some of the observations remain unexplained.