TY - JOUR
T1 - The semiclassical zeta function for geodesic flows on negatively curved manifolds
AU - Faure, Frédéric
AU - Tsujii, Masato
N1 - Funding Information:
The authors thank Colin Guillarmou and Semyon Dyatlov for helpful discussions and useful comments during they are writing this paper. They also thank the anonymous referee for many (indeed more than 100!) comments to the previous version of this paper based on very precise reading, which was indispensable in correcting errors and making the paper more readable. F. Faure thanks the ANR agency support ANR-09-JCJC-0099-01. M. Tsujii thanks the support by JSPS KAKENHI Grant Number 22340035.
Publisher Copyright:
© 2016, Springer-Verlag Berlin Heidelberg.
PY - 2017/6/1
Y1 - 2017/6/1
N2 - We consider the semi-classical (or Gutzwiller–Voros) zeta functions for C∞ contact Anosov flows. Analyzing the spectra of the generators of some transfer operators associated to the flow, we prove that, for arbitrarily small τ> 0 , its zeros are contained in the union of the τ-neighborhood of the imaginary axis, | R(s) | < τ, and the half-plane R(s) < - χ0+ τ, up to finitely many exceptions, where χ0> 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.
AB - We consider the semi-classical (or Gutzwiller–Voros) zeta functions for C∞ contact Anosov flows. Analyzing the spectra of the generators of some transfer operators associated to the flow, we prove that, for arbitrarily small τ> 0 , its zeros are contained in the union of the τ-neighborhood of the imaginary axis, | R(s) | < τ, and the half-plane R(s) < - χ0+ τ, up to finitely many exceptions, where χ0> 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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U2 - 10.1007/s00222-016-0701-5
DO - 10.1007/s00222-016-0701-5
M3 - Article
AN - SCOPUS:84996968538
SN - 0020-9910
VL - 208
SP - 851
EP - 998
JO - Inventiones Mathematicae
JF - Inventiones Mathematicae
IS - 3
ER -