Random cyclic dynamical systems

TL;DR

This paper analyzes the behavior of a specific cyclic dynamical system on large random subsets of a circle, revealing how the proportion of periodic points varies with rationality of a parameter, with implications for computational topology.

Contribution

It provides explicit asymptotic results for the expected fraction of periodic points in the system, connecting dynamical systems with topological data analysis through Catalan numbers.

Findings

Expected fraction of periodic points tends to 0 for irrational r.

Expected fraction tends to 1/q for rational r=p/q.

Results inform simplification of Vietoris-Rips complexes in topological data analysis.

Abstract

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.