For X a finite subset of the circle and for class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0196885816300768&_mathId=si1.gif&_user=111111111&_pii=S0196885816300768&_rdoc=1&_issn=01968858&md5=2f428b6d8a94400b9d013d55c2883f26" title="Click to view the MathML source">0<r≤1class="mathContainer hidden">class="mathCode"> fixed, consider the function class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0196885816300768&_mathId=si2.gif&_user=111111111&_pii=S0196885816300768&_rdoc=1&_issn=01968858&md5=96a3f251f8c30e337c77563e165aa78d" title="Click to view the MathML source">fr:X→Xclass="mathContainer hidden">class="mathCode"> which maps each point to the clockwise furthest element of X within angular distance less than class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0196885816300768&_mathId=si3.gif&_user=111111111&_pii=S0196885816300768&_rdoc=1&_issn=01968858&md5=f57fe6198947357b6931b0ec72cefc65" title="Click to view the MathML source">2πrclass="mathContainer hidden">class="mathCode">. We study the discrete dynamical system on X generated by class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0196885816300768&_mathId=si365.gif&_user=111111111&_pii=S0196885816300768&_rdoc=1&_issn=01968858&md5=ac340b38a4ad257df1bed471ace5a01c" title="Click to view the MathML source">frclass="mathContainer hidden">class="mathCode">, and especially its expected behavior when X is a large random set. We show that, as class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0196885816300768&_mathId=si5.gif&_user=111111111&_pii=S0196885816300768&_rdoc=1&_issn=01968858&md5=28ef0b9a44b6dc07dbca4dc9f6a0ae4d" title="Click to view the MathML source">|X|→∞class="mathContainer hidden">class="mathCode">, the expected fraction of periodic points of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0196885816300768&_mathId=si365.gif&_user=111111111&_pii=S0196885816300768&_rdoc=1&_issn=01968858&md5=ac340b38a4ad257df1bed471ace5a01c" title="Click to view the MathML source">frclass="mathContainer hidden">class="mathCode"> tends to 0 if r is irrational and to class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0196885816300768&_mathId=si6.gif&_user=111111111&_pii=S0196885816300768&_rdoc=1&_issn=01968858&md5=31bfe08bb59f63aec83d9f14b0b63ab7">class="imgLazyJSB inlineImage" height="22" width="10" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0196885816300768-si6.gif">class="mathContainer hidden">class="mathCode"> if class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0196885816300768&_mathId=si7.gif&_user=111111111&_pii=S0196885816300768&_rdoc=1&_issn=01968858&md5=d7b4523d033975368fc7cc120677e360">class="imgLazyJSB inlineImage" height="19" width="40" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0196885816300768-si7.gif">class="mathContainer hidden">class="mathCode"> is rational with p and q coprime. These results are obtained via more refined statistics of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0196885816300768&_mathId=si365.gif&_user=111111111&_pii=S0196885816300768&_rdoc=1&_issn=01968858&md5=ac340b38a4ad257df1bed471ace5a01c" title="Click to view the MathML source">frclass="mathContainer hidden">class="mathCode"> which we compute explicitly in terms of (generalized) Catalan numbers. The motivation for studying class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0196885816300768&_mathId=si365.gif&_user=111111111&_pii=S0196885816300768&_rdoc=1&_issn=01968858&md5=ac340b38a4ad257df1bed471ace5a01c" title="Click to view the MathML source">frclass="mathContainer hidden">class="mathCode"> 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.