The Livšic Equation on Differential Forms Over Anosov Flows and Applications

Publication Date

1-1-2026

Document Type

Article

Publication Title

Dynamical Systems

DOI

10.1080/14689367.2026.2692522

Abstract

The goal of this paper is to explore the relationship between the geometric properties of an Anosov flow on a closed manifold M and the analytic properties of its infinitesimal generator X as a linear operator on the space of smooth differential forms of all degrees. In particular, we study the solvability of the Livšic equation (Formula presented.) on the space of differential forms and show, for instance, that if the Anosov flow is asymmetric, then the equation has a unique solution in the continuous category in degrees (Formula presented.), where (Formula presented.). Intuitively, an Anosov flow is asymmetric if in negative time it shrinks the volume of any (Formula presented.) -dimensional parallelepiped exponentially fast when at least one of its sides is in the strong unstable bundle. As an application, we show that for volume-preserving asymmetric Anosov flows, the following result holds: the (Formula presented.) -closure of the image of (Formula presented.) restricted to differential forms of degree n−1 contains the space of (Formula presented.) -exact (Formula presented.) -forms if and only if the sum of the strong bundles of the flow is uniquely integrable, in which case the flow is therefore topologically conjugate to a suspension of an Anosov diffeomorphism.

Keywords

Anosov flow, differential form, Livšic equation

Department

Mathematics and Statistics

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