  We introduce a geometric dynamical system where iteration is defined as a
cycling composition of a finite collection of geometric maps, which act on a
space composed of three or more lines in $\mathbb{R}^2$. This system is
motivated by the dynamics of iterated function systems, as well as billiards
with modified reflection laws. We provide conditions under which this dynamical
system generates periodic orbits, and use this result to prove the existence of
closed nonsmooth curves over $\mathbb{R}^2$ which satisfy particular structural
constraints with respect to a space of intersecting lines in the plane.
