The Liv\v{s}ic equation on differential forms over Anosov flows and applications

TL;DR

This paper investigates the solvability of the Livšic equation on differential forms over Anosov flows, revealing conditions under which solutions exist and linking flow properties to topological conjugacy.

Contribution

It establishes new solvability results for the Livšic equation on differential forms and connects flow asymmetry with topological conjugacy to suspensions.

Findings

Unique solutions in certain degrees for asymmetric flows

Characterization of flow conjugacy via $L^2$-closure properties

Link between flow asymmetry and geometric properties

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 Liv\v{s}ic equation $L_X \xi = \eta$ on the space of differential forms and show, for instance, that if the Anosov flow is \emph{asymmetric}, then the equation has a unique solution in the continuous category in degrees $2 \leq k \leq n-2$, where $n = \dim M$. Intuitively, an Anosov flow is asymmetric if in negative time it shrinks the volume of any $(n-2)$-dimensional parallelepiped exponentially fast when at least one side of it is in the strong unstable direction. As an application, we show that for volume-preserving asymmetric Anosov flows, the following result holds: the $L^2$-closure of the image of $L_X$ restricted to differential forms of degree $(n-1)$ contains the space of $L^2$-exact $(n-1)$-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.