DYNAMICAL TORSION FOR CONTACT ANOSOV FLOWS

We introduce a new object, called dynamical torsion, which extends the potentially ill-defined value at 0 of the Ruelle zeta function ζ of a contact Anosov flow twisted by an acyclic representation of the fundamental group. The dynamical torsion depends analytically on the representation and is invariant under deformations among contact Anosov flows. Moreover, we show that the ratio between this torsion and the refined combinatorial torsion of Turaev, for an appropriate choice of Euler structure, is locally constant on the space of acyclic representations. In particular, for contact Anosov flows path connected to a geodesic flow of a hyperbolic manifold among contact Anosov flows, we relate the leading term of the Laurent expansion of ζ at the origin, the Reidemeister torsion and the torsions of the finite dimensional complexes of the generalized resonant states of both flows for the resonance 0. This extends previous work of [DGRS18] on the Fried conjecture near geodesic flows of hyperbolic 3-manifolds, to hyperbolic manifolds of any odd dimensions