Finsler metrics on $1\mathrm{/}n$-translation structures on surfaces

We define compatible Finsler distances on $1\mathrm{/}n$-translation surfaces, we study their geodesics, and construct a Liouville current for each such metric, that is a geodesic current that encodes the information of the length of the closed curves. The construction is based on multi-foliations, a generalization of measured foliations of independent interest.