EREDITARY HEMORRHAGIC telangiectasia (HHT) (Online Mendelian Inheritance in Man [OMIM] #187300) is a dominantly inherited genetic vascular disorder characterized by recurrent epistaxis; cutaneous telangiectasia; and visceral arteriovenous malformations (AVMs) that affect many organs, including the lungs, gastrointestinal tract, liver, and brain. Diagnosis is based on the Curaçao criteria and is considered definite if at least 3 of 4 criteria are fulfilled. 1 The criteria are spontaneous and recurrent epistaxis, Author Affiliations are listed at the end of this article.

Using a semiclassical approach we show that the spectrum of a smooth Anosov vector field V on a compact manifold is discrete (in suitable anisotropic Sobolev spaces) and then we provide an upper bound for the density of eigenvalues of the operator (−i) V , called Ruelle resonances, close to the real axis and for large real parts. RésuméPar une approche semiclassique on montre que le spectre d'un champ de vecteur d'Anosov V sur une variété compacte est discret (dans des espaces de Sobolev anisotropes adaptés). On montre ensuite une majoration de la densité de valeurs propres de l'opérateur (−i) V , appelées résonances de Ruelle, près de l'axe réel et pour les grandes parties réelles.

In this paper we construct a sequence of eigenfunctions of the "quantum Arnold's cat map" that, in the semiclassical limit, show a strong scarring phenomenon on the periodic orbits of the dynamics. More precisely, those states have a semiclassical limit measure that is the sum of 1/2 the normalized Lebesgue measure on the torus plus 1/2 the normalized Dirac measure concentrated on any a priori given periodic orbit of the dynamics. It is known (the Schnirelman theorem) that "most" sequences of eigenfunctions equidistribute on the torus. The sequences we construct therefore provide an example of an exception to this general rule. Our method of construction and proof exploits the existence of special values of for which the quantum period of the map is relatively "short", and a sharp control on the evolution of coherent states up to this time scale. We also provide a pointwise description of these states in phase space, which uncovers their "hyperbolic" structure in the vicinity of the fixed points and yields more precise localization estimates.

Abstract. 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.In this paper, we consider the characteristic frequencies of correlations,for the geodesic flow ϕ t on a compact hyperbolic manifold M of dimension n + 1 (that is, M has constant sectional curvature −1). Here ϕ t acts on SM , the unit tangent bundle of M , and µ is the natural smooth probability measure. Such ϕ t are classical examples of Anosov flows; for this family of examples, we are able to prove much more precise results than in the general Anosov case.An important question, expanding on the notion of mixing, is the behavior of ρ f,g (t) as t → +∞. Following [Ru], we take the power spectrum, which in our convention is the Laplace transformρ f,g (λ) of ρ f,g restricted to t > 0. The long time behavior of ρ f,g (t) is related to the properties of the meromorphic extension ofρ f,g (λ) to the entire complex plane. The poles of this extension, called Pollicott-Ruelle resonances (see [Po86a,Ru,FaSj] and (1.7) below), are the complex characteristic frequencies of ρ f,g , describing its decay and oscillation and not depending on f, g.For the case of dimension n + 1 = 2, the following connection between resonances and the spectrum of the Laplacian was announced in [FaTs13a, Section 4] (see [FlFo] for a related result and the remarks below regarding the zeta function techniques). Theorem 1. Assume that M is a compact hyperbolic surface (n = 1) and the spectrum of the positive Laplacian on M is (see Figure 1)