  We expose a connection between distance minimizing laminations and
horospherical orbit closures in $\mathbb{Z}$-covers of compact hyperbolic
manifolds. For surfaces, we provide novel constructions of $\mathbb{Z}$-covers
with prescribed geometric and dynamical properties, in which an explicit
description of all horocycle orbit closures is given. We further show that even
the slightest of perturbations to the hyperbolic metric on a $\mathbb{Z}$-cover
can lead to drastic topological changes to horocycle orbit closures.
