Fried conjecture in small dimensions

Abstract

We study the twisted Ruelle zeta function ${\zeta }_X(s)$ 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 ${\zeta }_X(0)$. In higher dimensions, we show more generally that ${\zeta }_X(0)$ 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.

arXiv comment42 pages. minor modifications, exposition improved

References in corpus (9)

• Ruelle-Perron-Frobenius spectrum for Anosov maps
• Upper bound on the density of Ruelle resonances for Anosov flows
• Pollicott-Ruelle resonances for open systems
• Killing and Conformal Killing tensors
• Ruelle zeta function at zero for surfaces
• Power spectrum of the geodesic flow on hyperbolic manifolds
• Analytic torsion, dynamical zeta functions, and the Fried conjecture
• Dynamical zeta functions for Axiom A flows
• Resonances for Symmetric Tensors on Asymptotically Hyperbolic Spaces

Cited by in corpus (12)

• Pollicott-Ruelle resonant states and Betti numbers
• Geodesic stretch, pressure metric and marked length spectrum rigidity
• The Ruelle zeta function at zero for nearly hyperbolic 3-manifolds
• Twisted Ruelle zeta function at zero for compact hyperbolic surfaces
• Dynamical torsion for contact Anosov flows
• Analytic torsion, dynamical zeta function, and the Fried conjecture for admissible twists
• Ruelle zeta function from field theory
• Wick squares of the Gaussian Free Field and Riemannian rigidity
• The twisted Ruelle zeta function on compact hyperbolic orbisurfaces and Reidemeister-Turaev torsion
• Anosov vector fields and Fried sections
• Perturbative BF theory in axial, Anosov, gauge
• Resolvent of vector fields and Lefschetz numbers