Vanishing of the Ruelle zeta function at the selected resonance crossings

Determine whether the Ruelle zeta function vanishes at zero for the selected metrics at the resonance-crossing parameters, equivalently whether the horizontal degree-two generalized zero-resonance space has dimension three rather than two.

Background

The paper constructs negatively curved real-analytic metrics and acyclic unitary characters for which the ordinary Fried identity fails. For each sufficiently small nonzero character parameter, a degree-one resonance crosses zero at a uniquely selected metric parameter. At such a crossing, the generalized zero-resonance space in horizontal degree two has dimension either two or three.

The dimension-two case yields a nonzero ordinary Ruelle zeta value at zero, whereas the dimension-three case yields a simple zero. The authors show that the torsion-normalized value tends to zero in either case, so resolving this multiplicity is not necessary for their counterexample.