The Fried conjecture in small dimensions

• 成果类型: Article
• 署名作者: Nguyen Viet Dang; Guillarmou, Colin; Riviere, Gabriel; Shen, Shu
• 署名单位: Centre National de la Recherche Scientifique (CNRS); Ecole Centrale de Lyon; Institut National des Sciences Appliquees de Lyon - INSA Lyon; Universite Claude Bernard Lyon 1; Universite Jean Monnet; Universite Paris Saclay; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universite de Lille; Universite Paris Cite; Sorbonne Universite
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-019-00935-9
• 发表日期: 2020
• 页码: 525-579
• 关键词: dynamical zeta-functions; anosov-flows; r-torsion; fredholm determinants; analytic-torsion; ruelle; spectrum; MAPS; resonances; systems

摘要

We study the twisted Ruelle zeta function.X (s) for smooth Anosov vector fields X acting on flat vector bundles over smooth compact manifolds. In dimension 3, we prove the Fried conjecture, relating Reidemeister torsion and.X (0). In higher dimensions, we show more generally that.X (0) is locally constant with respect to the vector field X under a spectral condition. As a consequence, we also show the 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 the Fried conjecture holds true.