For X a finite subset of the circle and for 0<r≤1 fixed, consider the function fr:X→X which maps each point to the clockwise furthest element of X within angular distance less than 2πr. We study the discrete dynamical system on X generated by a4ad257df1bed471ace5a01c" title="Click to view the MathML source">fr, and especially its expected behavior when X is a large random set. We show that, as a44b6dc07dbca4dc9f6a0ae4d" title="Click to view the MathML source">|X|→∞, the expected fraction of periodic points of a4ad257df1bed471ace5a01c" title="Click to view the MathML source">fr tends to 0 if r is irrational and to if is rational with p and q coprime. These results are obtained via more refined statistics of a4ad257df1bed471ace5a01c" title="Click to view the MathML source">fr which we compute explicitly in terms of (generalized) Catalan numbers. The motivation for studying a4ad257df1bed471ace5a01c" title="Click to view the MathML source">fr 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.