Transitivity and an abelian Livsic theorem for covers

TL;DR

This paper extends the abelian Livsic theorem to certain regular covers of Anosov flows, demonstrating its validity for periodic orbits trivial in the cover, thus broadening its applicability in dynamical systems.

Contribution

It generalizes the abelian Livsic theorem to regular covers where the lifted flow is transitive, under conditions involving trivial periodic orbits.

Findings

The Livsic theorem holds for trivial periodic orbits in regular covers.

The result applies to transitive lifted flows on covers.

It broadens the scope of Livsic-type theorems in dynamical systems.

Abstract

We show that the abelian Liv\v{s}ic theorem recently obtained by A. Gogolev and F. Rodriguez Hertz for null-homologous periodic orbits of homologically full Anosov flows continues to hold when restricted to periodic orbits which are trivial with respect to any regular cover for which the lifted flow is transitive.