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

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.