Fractal Weyl laws for asymptotically hyperbolic manifolds

Datchev, Kiril; Dyatlov, Semyon

doi:10.1007/s00039-013-0225-8

Cited by 31 publications

(45 citation statements)

References 55 publications

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“…We remark that a fractal upper bound for resonances should be valid for more general quotients \H n than the ones considered in [53]-see Guillarmou-Mazzeo Fig. 2) with real parts with | Re λ| ≤ 2 × 10 3 computed using a semiclassical zeta function [170].…”

Section: Theorem 13mentioning

confidence: 96%

Mathematical study of scattering resonances

2017

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Section: Theorem 13mentioning

confidence: 96%

Mathematical study of scattering resonances

2017

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“…For a canonical transformation κ with a fixed antiderivative F , we consider the class I comp h (κ) of compactly supported and microlocalized Fourier integral operators associated to κ -see for instance [GuSt77,Chapter 5], [GuSt13,Chapter 8], [DyGu14a,§3.2], [DaDy,§3.2], † [Dy,§3.2], and the references there. We adopt a convention that operators in I comp h (κ) act D (M 2 ) → C ∞ 0 (M 1 ).…”

Section: 2mentioning

confidence: 99%

Spectral gaps, additive energy, and a fractal uncertainty principle

2016

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Abstract. We obtain an essential spectral gap for n-dimensional convex co-compact hyperbolic manifolds with the dimension δ of the limit set close to n−1 2 . The size of the gap is expressed using the additive energy of stereographic projections of the limit set. This additive energy can in turn be estimated in terms of the constants in Ahlfors-David regularity of the limit set. Our proofs use new microlocal methods, in particular a notion of a fractal uncertainty principle.In this paper we study essential spectral gaps for convex co-compact hyperbolic quotients M = Γ\H n . To formulate our result in the simplest setting, consider n = 2 and take the Selberg zeta function [Bo07, (10.1)] ), where δ ∈ [0, 1] is the dimension of the limit set of the group (see (5.2)). Therefore Z M has finitely many singularities in {Im λ > − max(0,, Naud [Na] obtained the stronger statement that Z M has finitely many singularities in {Im λ > −β} for some β strictly greater than 1 2 − δ. Naud's result, generalized to higher dimensional quotients by Stoyanov [St11], is based on the method of Dolgopyat [Do] and does not specify the size of the improvement. Our first result in particular gives explicit estimates on the value of β when δ = 1 2 : Theorem 1. Let M = Γ\H 2 be a convex co-compact hyperbolic surface. Then for each ε > 0, the function Z M has finitely many singularities in {Im λ > −β + ε}, where β = 3 8 1 2 − δ + β E 16 , β E := δ exp − K(1 − δ) −28 (1 + log 14 C) .

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“…Note that this encodes the behaviour of the full symbol of A at h = 0 everywhere on T * X, as well as the behaviour at the fiber infinity ∂T * X for small, but positive, values of h -see [Va,§2.1]. We cannot use the more convenient space of classical operators, whose principal symbol is just a function on T * X (see [DaDy,§3.1]) because the symbol of the operator e sG(h) Pe −sG(h) (see §3) has the form p + ishH p G, with p ∈ S 1 (X) and H p G = O(log(2 + |ξ|)) narrowly missing the class S 0 (X). The (open) elliptic set ell h (A) ⊂ T * X is defined as follows:…”

mentioning

confidence: 99%

“…for an h-tempered family of distributions u = u(h), see for example [Zw,§8.4.2], [DaDy,§3.1]. Similarly to WF(u), the set WF h (u) can be characterized using the Fourier transform as follows: (x, ξ) ∈ WF h (u) if and only if there exists χ ∈ C ∞ c (X) supported in some coordinate neighbourhood, with χ(x) = 0, and a neighbourhood…”

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confidence: 99%

Dynamical zeta functions for Anosov flows via microlocal analysis

Dyatlov¹,

Zworski²

2016

Ann. Sci. École Norm. Sup.

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119

256

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Abstract. The purpose of this paper is to give a short microlocal proof of the meromorphic continuation of the Ruelle zeta function for C ∞ Anosov flows. More general results have been recently proved by Giulietti-Liverani-Pollicott [GiLiPo] but our approach is different and is based on the study of the generator of the flow as a semiclassical differential operator.The purpose of this article is to provide a short microlocal proof of the meromorphic continuation of the Ruelle zeta function for C ∞ Anosov flows on compact manifolds:Theorem. Suppose X is a compact manifold and ϕ t : X → X is a C ∞ Anosov flow with orientable stable and unstable bundles. Let {γ } denote the set of primitive orbits of ϕ t , with T γ their periods. Then the Ruelle zeta function,which converges for Im λ 1 has a meromorphic continuation to C.In fact the proof applies to any Anosov flow for which linearized Poincaré maps P γ for closed orbits γ satisfy | det(I − P γ )| = (−1) q det(I − P γ ), with q independent of γ. The meromorphic continuation of ζ R was conjectured by Smale [Sm] and in greater generality it was proved very recently by Giulietti, Liverani, and Pollicott [GiLiPo]. Another recent perspective on dynamical zeta functions in the contact case has been provided by Faure and Tsujii [FaTs1,FaTs2]. Our motivation and proof are however different from those of [GiLiPo]: we were investigating trace formulae for PollicottRuelle resonances [Po,Ru86] which give some lower bounds on their counting function. Sharp upper bounds were given recently in [DDZ, FaSj].

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Fractal Weyl laws for asymptotically hyperbolic manifolds

Cited by 31 publications

References 55 publications

Mathematical study of scattering resonances

Mathematical study of scattering resonances

Spectral gaps, additive energy, and a fractal uncertainty principle

Dynamical zeta functions for Anosov flows via microlocal analysis

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