Abelian Livsic theorems for Anosov flows

TL;DR

This paper presents two concise proofs of the abelian Livsic theorem for Anosov flows, extending the results to broader classes of orbits and covers, thus advancing the understanding of cohomological equations in dynamical systems.

Contribution

The authors provide simplified proofs of the abelian Livsic theorem and extend its applicability to null-homologous orbits and amenable covers.

Findings

Two short proofs of the abelian Livsic theorem

Extension to null-homologous orbits with positive density

Extension to amenable covers

Abstract

We give two short proofs of the abelian Livsi\v{c} theorem of Gogolev and Rodriguez Hertz. We show that these proofs may be extended to give new abelian Livsic theorems for positive density sets of null-homologous orbits and for amenable covers.