The Livšic equation on differential forms over Anosov flows and applications

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 LXξ=η 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 2≤k≤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 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 L2-closure of the image of LX restricted to differential forms of degree n−1 contains the space of L2-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.