Fractal Weyl law for the Ruelle spectrum of Anosov flows

Faure, Frédéric; Tsujii, Masato

doi:10.5802/ahl.167

Faure, Frédéric; Tsujii, Masato

Fractal Weyl law for the Ruelle spectrum of Anosov flows

Annales Henri Lebesgue, Volume 6 (2023), pp. 331-426.

Metadata

DOI10.5802/ahl.167

Keywords Transfer operator, Ruelle resonances, decay of correlations, Semi-classical analysis

Abstract

On a closed manifold $M$ , we consider a smooth vector field $X$ that generates an Anosov flow. Let $V \in C^{\infty} (M; ℝ)$ be a smooth function called potential. It is known that for any $C > 0$ , there exists some anisotropic Sobolev space $ℋ_{C}$ such that the operator $A = - X + V$ has intrinsic discrete spectrum on $Re (z) > - C$ called Ruelle resonances. In this paper, we show a “Fractal Weyl law”: the density of resonances is bounded by $O ({〈 ω 〉}^{\frac{n}{1 + β_{0}}})$ where $ω = Im (z)$ , $n = dim M - 1$ and $0 < β_{0} \leq 1$ 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 $T^{*} M$ and construct some specific anisotropic Sobolev spaces.

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