Anosov vector fields and Fried sections

Abstract

The purpose of this paper is to prove that if $Y$ is a compact manifold, if $Z$ is an Anosov vector field on $Y$, and if $F$ is a flat vector bundle, there is a corresponding canonical nonzero section $\tau_{\nu}\left(i_{Z}\right)$ of the determinant line $\nu=\det H\left(Y,F\right)$. In families, this section is $C^{1}$ with respect to the canonical smooth structure on $\nu$. When $F$ is flat on the total space of the corresponding fibration, our section is flat with respect to the Gauss-Manin connection on $\nu$.