Fried conjecture in small dimensions

Abstract : We study the twisted Ruelle zeta function ζX (s) for smooth Anosov vector fields X acting on flat vector bundles over smooth compact manifolds. In dimension 3, we prove Fried conjecture, relating Reidemeister torsion and ζX (0). In higher dimensions, we show more generally that ζX (0) is locally constant with respect to the vector field X under a spectral condition. As a consequence, we also show Fried conjecture for Anosov flows near the geodesic flow on the unit tangent bundle of hyperbolic 3-manifolds. This gives the first examples of non-analytic Anosov flows and geodesic flows in variable negative curvature where Fried conjecture holds true.
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Nguyen Viet Dang, Colin Guillarmou, Gabriel Riviere, Shu Shen. Fried conjecture in small dimensions. Inventiones Mathematicae, Springer Verlag, 2020. ⟨hal-01828967⟩

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