Random cyclic dynamical systems

For X a finite subset of the circle and for 0 X which maps each point to the clockwise furthest element of X within angular distance less than 2 pi r. We study the discrete dynamical system on X generated by f_r, and especially its expected behavior when X is a large random set. We show that the expected fraction of periodic points of f_r is 0 if r is irrational and 1/q if r=p/q is rational with p and q coprime. These results are obtained via more refined statistics of f_r which we compute explicitly in terms of (generalized) Catalan numbers. The motivation for studying f_r comes from Vietoris-Rips complexes, a geometric construction used in computational topology. Our results determine how much one can expect to simplify the Vietoris--Rips complex of a random sample of the circle by removing dominated vertices.