Chaos, order statistics and unstable periodic orbits

We present an interesting connection between order statistics and unstable periodic orbits of chaotic maps. This can be used to locale all the unstabl e periodic points (of a given other) of one-dimensional chaotic maps with c ontinuous invariant density. The densities of the ordered variates of the i terates are discontinuous exactly at unstable periodic points of the map. T his is illustrated using the logistic map, where densities corresponding to a small number of iterates have been obtained in closed form. This scheme can also be applied to a class of continuous-time systems where the success ive maxima of the time series behave as if they were generated from a unimo dal map. We demonstrate this by using the Lorenz model.