The semiclassical zeta function for geodesic flows on negatively curved manifolds - Archive ouverte HAL
Article Dans Une Revue Inventiones Mathematicae Année : 2016

The semiclassical zeta function for geodesic flows on negatively curved manifolds

Frédéric Faure
Masato Tsujii
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Résumé

We consider the semi-classical (or Gutzwiller-Voros) zeta functions for $C^\infty $ contact Anosov flows. Analyzing the spectra of the generators of some transfer operators associated to the flow, we prove that, for arbitrarily small $\tau >0$, its zeros are contained in the union of the $\tau $-neighborhood of the imaginary axis, $|\mathfrak {R}(s)|<\tau $, and the half-plane $\mathfrak {R}(s)<-\chi _0+\tau $, up to finitely many exceptions, where $\chi _0>0$ 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.
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Dates et versions

hal-01655862 , version 1 (01-11-2024)

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Frédéric Faure, Masato Tsujii. The semiclassical zeta function for geodesic flows on negatively curved manifolds. Inventiones Mathematicae, 2016, 208 (3), pp.851 - 998. ⟨10.1007/s00222-016-0701-5⟩. ⟨hal-01655862⟩
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