The semiclassical zeta function for geodesic flows on negatively curved manifolds

Abstract

We consider the semi-classical (or Gutzwiller-Voros) zeta function for $C^{\infty }$ contact Anosov flows. Analyzing the spectrum of transfer operators associated to the flow, we prove, for any $\tau >0$, that its zeros are contained in the union of the $\tau$-neighborhood of the imaginary axis, $|\Re (s)|<\tau$, and the region $\Re (s)<-{\chi }_0+\tau$, up to finitely many exceptions, where ${\chi }_0>0$ is the hyperbolicity exponent of the flow. Further we show that the zeros in the neighborhood of the imaginary axis satisfy an analogue of the Weyl law.

106 pages, 4 figures. We revised the previous version following comments by the anonymous referee. The main changes are A) the index $k$ in the transfer operators ${\mathcal{L}}_{k,\ell }^t$ is reversed, B) The content of Subsections 8.2 and 8.3 is exchanged, and C) We rewrote the proof of Lemma 9.4, 9.12 and Lemma 10.11 for clarify of the argument around estimates using integration by parts

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