Finsler metrics on $1/n$-translation structures on surfaces

TL;DR

This paper introduces Finsler metrics on $1/n$-translation surfaces, analyzes their geodesics, and constructs a Liouville current that encodes lengths of closed curves, using multi-foliations.

Contribution

It defines compatible Finsler distances on $1/n$-translation surfaces and develops a new construction of Liouville currents based on multi-foliations.

Findings

Established a framework for Finsler metrics on $1/n$-translation surfaces.

Constructed Liouville currents encoding closed curve lengths.

Analyzed geodesic properties of these Finsler metrics.

Abstract

We define compatible Finsler distances on $1/n$-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.