The semiclassical zeta function for geodesic flows on negatively curved manifolds

• 成果类型: Article
• 署名作者: Faure, Frederic; Tsujii, Masato
• 署名单位: Communaute Universite Grenoble Alpes; Universite Grenoble Alpes (UGA); Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Kyushu University
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-016-0701-5
• 发表日期: 2017
• 页码: 851-998
• 关键词: transfer operators; sobolev spaces; anosov-flows; ruelle; determinants; spectrum

摘要

We consider the semi-classical (or Gutzwiller-Voros) zeta functions for contact Anosov flows. Analyzing the spectra of the generators of some transfer operators associated to the flow, we prove that, for arbitrarily small , its zeros are contained in the union of the -neighborhood of the imaginary axis, , and the half-plane , up to finitely many exceptions, where is the hyperbolicity exponent of the flow. Further we show that the density of the zeros along the imaginary axis satisfy an analogue of the Weyl law.