No abstract

Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ àVb cb cb ed Tf "½ h Vi Tp rq ts ub wv yx ap V V t a Vv V rh ub q ap t G Db& rg a ¤l r g m #g k } ei ¹ ß ' §Ê p xk }g m 0× am # #g k } ei &p xm ¹ Ão r ei # 9 e g hg m r j a qi p xm q f r ep am ¬¼ F ¹ ß "¼ ݳ AE r e & rp a sp xãg m rg p x r & em j rg ¦ jg m r jg ¦ 9 Ù F À ¹ ß '¼ Øh ¾ ¹ ß y am hf am a } q # h p | ei jp a e h j ep xm p a qò Y i xÇ ¹ ß hÈ 3¼ Ú a ep am r`g £ p xk 3Í p f rf E ¢ j rg }g si p xm rf r ep am r 3 rm r ¥ q hg f rg jg k j # em hg Gp ak j s a P vp wo yg k § jg k j eg µg êé h r am r j p xm ¯ Gp ak ¹ ß h ¯ p f r ã Eg p x sp ak j r r j Al r h Gp xk ¦Í g z xg m |m rf Í Ø © y ¤ Gp xk } | E vp wo yg k Y jg k j g jp x h p xm (g zé } m ãg m rg k a p xm h jp sp ak jf rg k &Í Du x© g

No abstract

No abstract

No abstract

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].

73 pages, 5 figuresInternational audienceIn this article we prove that for a large class of operators, including Schroedinger operators, with hyperbolic classical flows, the smallness of dimension of the trapped set implies that there is a gap between the resonances and the real axis. In other words, the quantum decay rate is bounded from below if the classical repeller is sufficiently filamentary. The higher dimensional statement is given in terms of the topological pressure. Under the same assumptions we also prove a resolvent estimate with a logarithmic loss compared to nontrapping estimates

Any compact C ∞ manifold with boundary admits a Riemann metric on its interior taking the form x −4 dx 2 + x −2 h near the boundary, where x is a boundary defining function and h is a smooth symmetric 2-cotensor restricting to be positive-definite, and hence a metric, h, on the boundary. The scattering theory associated to the Laplacian for such a 'scattering metric' was discussed by the first author and here it is shown, as conjectured, that the scattering matrix is a Fourier integral operator which quantizes the geodesic flow on the boundary, for the metric h, at time π. To prove this the Poisson operator, of the associated generalized boundary problem, is constructed as a Fourier integral operator associated to a singular Legendre manifold.