Quandles derived from dynamical systems and subsets which are closed under quandle operations

A quandle is a set with a binary operation satisfying certain conditions related to Reidemeister moves in knot theory. First we give an example of a quandle with subsets which are not subquandles but closed under the quandle operation. We introduce a method to produce a quandle from an invertible dynamical system. Our example is generalized to such dynamical quandles.