Fractal Weyl law for the Ruelle spectrum of Anosov flows

On a closed manifold $M$, we consider a smooth vector field $X$ that generates an Anosov flow. Let $V\in C^{\infty}\left(M;\mathbb{R}\right)$ be a smooth potential function. It is known that for any $C>0$, there exists some anisotropic Sobolev space $\mathcal{H}_{C}$ such that the operator $A=-X+V$ has intrinsic discrete spectrum on $\mathrm{Re}\left(z\right)>-C$ called Ruelle-Pollicott resonances. In this paper, we show that the density of resonances is bounded by $O\left(\left\langle \omega\right\rangle ^{\frac{n}{1+\beta_{0}}}\right)$ where $\omega=\mathrm{Im}\left(z\right)$, $n=\mathrm{dim}M-1$ and $0<\beta_{0}\leq1$ is the H\"older exponent of the distribution $E_{u}\oplus E_{s}$ (strong stable and unstable). We also obtain some more precise results concerning the wave front set of the resonances states and the group property of the transfer operator. We use some semiclassical analysis based on wave packet transform associated to an adapted metric on $T^{*}M$ and construct some specific anisotropic Sobolev spaces

Spectra of differentiable hyperbolic maps

This note is about the spectral properties of transfer operators associated to smooth hyperbolic dynamics. In the first two sections (written in 2006), we state our new results relating such spectra with dynamical determinants, first announced at the conference ``Traces in Geometry, Number Theory and Quantum Fields" at the Max Planck Institute, Bonn, October 2005. In the last two sections, we give a reader-friendly presentation of some key ideas in our work in the simplest possible settings, including a new proof of a result of Ruelle on expanding endomorphisms. (These last two sections are a revised version of the lecture notes given during the workshop ``Resonances and Periodic Orbits: Spectrum and Zeta functions in Quantum and Classical Chaos" at Institut Henri Poincar\'e, Paris, July 2005.) (Revised version, submitted for publication)Comment: LaTe

Band structure of the Ruelle spectrum of contact Anosov flows

If X is a contact Anosov vector field on a smooth compact manifold M and V is a smooth function on M, it is known that the differential operator A=-X+V has some discrete spectrum called Ruelle-Pollicott resonances in specific Sobolev spaces. We show that for |Im(z)| large the eigenvalues of A are restricted to vertical bands and in the gaps between the bands, the resolvent of A is bounded uniformly with respect to |Im(z)|. In each isolated band the density of eigenvalues is given by the Weyl law. In the first band, most of the eigenvalues concentrate of the vertical line Re(z)=, the space average of the function D(x)=V(x)-1/2 div(X)/E_u where Eu is the unstable distribution. This band spectrum gives an asymptotic expansion for dynamical correlation functions.Comment: 12 page

The semiclassical zeta function for geodesic flows on negatively curved manifolds

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.Comment: 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 $\mathcal{L}^t_{k,\ell}$ 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 part