Bounded deviations in higher genus I: closed geodesics

TL;DR

This paper investigates bounded deviations of homeomorphisms on higher genus surfaces, focusing on closed geodesics, and provides criteria for periodic orbits, extending known results from the torus case.

Contribution

It introduces new results on bounded deviations with respect to closed geodesics on higher genus surfaces and offers a criterion for the existence of periodic orbits based on deviations.

Findings

Established bounds for deviations with respect to closed geodesics.

Derived a criterion linking large deviations to periodic orbits.

Extended bounded deviation results from the torus to higher genus surfaces.

Abstract

This is the first article of a series of two where we study the problem of bounded deviations for homeomorphisms of closed surfaces of genus $\ge 2$. This first part studies bounded deviations with respect to closed geodesics. As a byproduct of our proofs, we also get a criterion of existence of periodic orbits in terms of big deviation with respect to some closed geodesic. The combination with the second part [arXiv:2511.14226] generalises to the higher genus case most of the bounded deviations results already known for the torus.