Random cyclic dynamical systems
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.
- Publication:
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arXiv e-prints
- Pub Date:
- November 2015
- DOI:
- 10.48550/arXiv.1511.07832
- arXiv:
- arXiv:1511.07832
- Bibcode:
- 2015arXiv151107832A
- Keywords:
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- Mathematics - Combinatorics;
- Mathematics - Dynamical Systems;
- 60C05;
- 37H99;
- 05E45
- E-Print:
- Advances in Applied Mathematics, 2015