  For X a finite subset of the circle and for 0 < r <= 1 fixed, consider the
function f_r : X -> 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, as |X| -> infinity, the expected fraction of
periodic points of f_r tends to 0 if r is irrational and to 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.
