Dynamical zeta functions for Anosov flows via microlocal analysis

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].

Mathematical study of scattering resonances

“…2: meromorphic continuation for asymptotically hyperbolic spaces, fractal upper bounds in physical and geometric settings, resonance free strips in chaotic scattering and resonance expansions; we also provide some references to recent progress in some of the topics not covered in this survey; • Section 4 surveys the use of microlocal/scattering theory methods in the study of chaotic dynamical systems. Their introduction by Faure-Sjöstrand [84] and Tsujii [261] led to rapid progress which included a microlocal proof of Smale's conjecture about dynamical zeta functions [78], first proved shortly before by Giulietti-Liverani-Pollicott [109]. We review this and other results, again related to upper bounds, resonance free strips and resonances expansions.…”

“…The following theorem was first proved by Faure-Sjöstrand [84] for more specific weights G and by Dyatlov-Zworski [78]. The characterization of resonant states using a wave front set condition is implicit in [84] …”

“…We will now explain the proof [78] of the meromorphic continuation of ζ R . The first step [226] is a factorization of the zeta function valid in the case when the stable and unstable bundles, x → E s (x), x → E u (x), respectively, are orientable: 13 ζ R (λ) = where dim M = n, q = dim E s and the Poincaré map is defined by The relation of the zeta functions ζ j with the flow by the Atiyah-Bott-Guillemin trace formula (see [78, Appendix B] for a proof):…”

“…Thus, consider a compact oriented negatively curved Riemannian manifold ( , g). Following Fried [92,94] The proof of meromorphy of ζ R given in [78] and sketched above gives also the meromorphic continuation of ζ α : we only need to change the vector bundles on whose sections 1 i L X acts. The following statement was proved by Fried [92] in the case of manifolds of constant negative curvature and conjectured (in an even more precise form) in [94, p.181] for more general manifolds.…”

Commentary on “Differentiable dynamical systems” by Stephen Smale

“…A clear microlocal description of anisotropic spaces was given by Faure and Sjöstrand [FaSj11], and that was the starting point of my joint work with Dyatlov in which we gave a micorlocal proof of the theorem of Giulietti-Liverani-Pollicott [DyZw16]. As in previous works the connection with spectral methods is given by the Atiyah-Bott-Guillemin trace formula:…”