Random cyclic dynamical systems

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• 作者：Michał Adamaszek^a ; ^{aszek@mimuw.edu.pl" class="auth_mail" title="E-mail the corresponding author} ; Henry Adams^b ; ^{adams@math.colostate.edu" class="auth_mail" title="E-mail the corresponding author} ; Francis Motta^c ; ^{motta@math.duke.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：60C05 ; 37H99 ; 05E45
• 刊名：Advances in Applied Mathematics
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：83
• 期：Complete
• 页码：1-23
• 全文大小：520 K

文摘

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