Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold

Anantharaman, Nalini; Nonnenmacher, Stéphane

doi:10.5802/aif.2340

Cited by 102 publications

(252 citation statements)

References 29 publications

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“…One can for example study semiclassical measures as in Gérard-Leichtnam [9], Zelditch [19], Zelditch-Zworski [20], Anantharaman [1], Anantharaman-Koch-Nonnenmacher [2], Anantharaman-Nonnenmacher [3]. The aim of these studies is generally to prove non-concentration theorems under geometric conditions on the geodesic flow (such as Anosov flow).…”

Section: Introductionmentioning

confidence: 99%

Semiclassical L p Estimates of Quasimodes on Curved Hypersurfaces

Hassell

Tacy

2010

J Geom Anal

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Abstract. Let M be a compact manifold of dimension n, P = P (h) a semiclassical pseudodifferential operator on M , andIn a previous article, the second-named author proved estimates on the L p norms, p ≥ 2, of u restricted to H, under the assumption that the u are semiclassically localised and under some natural structural assumptions about the principal symbol of P . These estimates are of the form Ch −δ(n,k,p) where k = dim H (except for a logarithmic divergence in the case k = n − 2, p = 2). When H is a hypersurface, i.e. k = n − 1, we have δ(n, n − 1, 2) = 1/4, which is sharp when M is the round n-sphere and H is an equator.In this article, we assume that H is a hypersurface, and make the additional geometric assumption that H is curved (in the sense of Definition 2.4 below) with respect to the bicharacteristic flow of P . Under this assumption we improve the estimate from δ = 1/4 to 1/6, generalising work of Burq-Gérard-Tzvetkov and Hu for Laplace eigenfunctions. To do this we apply the MelroseTaylor theorem, as adapted by Pan and Sogge, for Fourier integral operators with folding canonical relations.

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Section: Introductionmentioning

confidence: 99%

Semiclassical L p Estimates of Quasimodes on Curved Hypersurfaces

Hassell

Tacy

2010

J Geom Anal

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“…They also asked about the extension of this conjecture to manifolds without conjugate points [4]. In a recent work [16], we were able to prove that their conjecture holds for any surface with an Anosov geodesic flow (for instance surfaces of negative curvature).…”

Section: Kolmogorov-sinai Entropymentioning

confidence: 98%

“…Our strategy will be the same as in [16] (and also [4]) so it is probably better (and easier) for the reader to have a good understanding of the methods from these two references where the geometric situation is "simpler". We will focus on the main differences and refer the reader to [4,16] for the details of several lemmas. The crucial observation is that as in the Anosov case, surfaces of nonpositive curvature have continuous stable and unstable foliations and no conjugate points.…”

Section: Kolmogorov-sinai Entropymentioning

confidence: 99%

Entropy of Semiclassical Measures for Nonpositively Curved Surfaces

Rivière

2010

Ann. Henri Poincaré

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Abstract.We study the asymptotic properties of eigenfunctions of the Laplacian in the case of a compact Riemannian surface of nonpositive sectional curvature. To do this, we look at sequences of distributions associated to them and we study the entropic properties of their accumulation points, the so-called semiclassical measures. Precisely, we show that the Kolmogorov-Sinai entropy of a semiclassical measure μ for the geodesic flow g t is bounded from below by half of the Ruelle upper bound, i.e.where χ + (ρ) is the upper Lyapunov exponent at point ρ. The main strategy is the same as in Rivière (Duke Math J, arXiv:0809.0230, 2008) except that we have to deal with weakly chaotic behavior.

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“…Notez que Anantharaman [Ana08] a montré, pour toutes les variétés compactesà courbure négative, et sans aucune opérateurs de Hecke, que la limite quantique a entropie strictement positive, et cela aété encore accentué dans son travail conjoint avec Nonnenmacher [AN07] (voir aussi [AKN07]). Donc la contribution du Théorème 1 est qu'il donne des informations sur presque tous les composants ergodique d'une limite quantique.…”

Section: Version Française Abrégéeunclassified

“…Note that even without any Hecke operators, Anantharaman [Ana08] has shown (for general negatively curved compact manifolds) that any quantum limit has positive entropy, and this has been further sharpened in her joint work with Nonnenmacher [AN07] (see also [AKN07]). Hence the point of Theorem 1 is that it gives information on almost all ergodic components of a quantum limit.…”

Section: Introductionmentioning

confidence: 99%