Minimizing laminations in regular covers, horospherical orbit closures, and circle-valued Lipschitz maps

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.