Fractal Weyl law for the Ruelle spectrum of Anosov flows

Abstract

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 function called potential. 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 resonances. In this paper, we show a fractal Weyl law: the density of resonances is bounded by $O\left({⟨\omega ⟩}^{\frac{n}{1+{\beta }_0}}\right)$ where $\omega =\mathrm{Im}\left(z\right)$, $n=\dim M-1$ and $0<{\beta }_0\le 1$ is the Hölder 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 and the invertibility of the transfer operator. Since the dynamical distributions $E_u$,$E_s$ are non smooth, we use some semi-classical analysis based on wave packet transform associated to an adapted metric $g$ on $T^{\ast }M$ and construct some specific anisotropic Sobolev spaces.

81 pages

References in corpus (4)

• Ergodic optimization in dynamical systems
• Anisotropic Sobolev spaces and dynamical transfer operators: C^infty foliations
• Flow-independent Anisotropic space, and perturbation of resonances
• Rational invariant tori, phase space tunneling, and spectra for non-selfadjoint operators in dimension 2

Cited by in corpus (4)

• Higher rank quantum-classical correspondence
• FBI Transform in Gevrey Classes and Anosov Flows
• Micro-local analysis of contact Anosov flows and band structure of the Ruelle spectrum
• Fractal Weyl laws and wave decay for general trapping