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

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

Manifestation of the topological index formula in quantum waves and geophysical waves

Using semi-classical analysis in $\mathbb{R}^{n}$ we present a quite general model for which the topological index formula of Atiyah-Singer predicts a spectral flow with a transition of a finite number of eigenvalues transitions between clusters (energy bands). This model corresponds to physical phenomena that are well observed for quantum energy levels of small molecules [faure_zhilinskii_2000,2001] but also in geophysics for the oceanic or atmospheric equatorial waves [Matsuno_1966, Delplace_Marston_Venaille_2017]

Micro-local analysis of contact Anosov flows and band structure of the Ruelle spectrum

We develop a geometrical micro-local analysis of contact Anosov flow, such as geodesic flow on negatively curved manifold. We use the method of wave-packet transform discussed in arXiv:1706.09307 and observe that the transfer operator is well approximated (in the high frequency limit) by the quantization of an induced transfer operator acting on sections of some vector bundle on the trapped set. This gives a few important consequences: The discrete eigenvalues of the generator of transfer operators, called Ruelle spectrum, are structured into vertical bands. If the right-most band is isolated from the others, most of the Ruelle spectrum in it concentrate along a line parallel to the imaginary axis and, further, the density satisfies a Weyl law as the imaginary part tend to infinity. Some of these results were announced in arXiv:1301.5525.Comment: 78 pages, 9 figure

Power spectrum of the geodesic flow on hyperbolic manifolds

We describe the complex poles of the power spectrum of correlations for the geodesic flow on compact hyperbolic manifolds in terms of eigenvalues of the Laplacian acting on certain natural tensor bundles. These poles are a special case of Pollicott-Ruelle resonances, which can be defined for general Anosov flows. In our case, resonances are stratified into bands by decay rates. The proof also gives an explicit relation between resonant states and eigenstates of the Laplacian.MSRI (National Science Foundation (U.S.) (grant 0932078 000, Fall 2013)National Science Foundation (U.S.) (grant DMS-1201417)France. Agence nationale de la recherche ( grant ANR-13-BS01-0007-01