Ruelle-Pollicott Resonances in dynamics and semiclassical analysis- Friday 10:00-12:00 - BigBlueButton
(organised by V. Baladi, C. Guillarmou, and M. Tsujii)
February-April 2021

Friday February 19, 10:00-10:45
Tobias Weich
Title: Ruelle-Taylor resonances for \R^r-Anosov actions
Abstract: Given a compact manifold M with an Anosov action of higher rank we introduce a discrete resonance spectrum which generalizes the notion of Ruelle resonances for Anosov flows to higher rank actions. The construction of this spectrum is based on the notion of Taylor spectra for commuting operators.
In the special case that the Anosov action is a Weyl chamber flow I will sketch how the Ruelle Taylor resonances can be related to the joint eigenvalues of the commuting (quantum) differential operators on the corresponding locally symmetric space and what can be learn from such a correspondance about the spectrum of Ruelle Taylor resonances.
This is based on joint work with Yannick Guedes-Bonthonneau, Colin Guillarmou, Joachim Hilgert and Lasse Wolf.

Friday February 19, 11:00-11:45
Yannick Bonthonneau
Title: Bowen formula and more on R^r-Anosov actions.
Abstract: Elaborating on the talk of Tobias Weich, I will explain how we obtain detailed information about the resonant state corresponding the dominant resonance. As in the case of Anosov flows, this is a SRB-like measure. Our investigations lead to a Bowen-type formula. We also construct a type of dynamical determinant for these actions.

Friday February 26, 10:00-10:45
Gabriel Paternain
Title: The Ruelle zeta function at zero for nearly hyperbolic 3-manifolds
Abstract: The Ruelle zeta function is a natural function associated to the lengths of the closed geodesics of a closed negatively curved manifold, and it is known to have a meromorphic extension to the whole complex plane. In this talk I will explain the main ideas that go into the proof of the following result: for a generic conformal metric perturbation of a hyperbolic 3-manifold, the order of vanishing of the Ruelle zeta function at zero equals 4-b_1, contrary to the hyperbolic case where it is equal to 4-2b_1 (b_1 is the first Betti number). This is joint work with Mihajlo Cekic, Semyon Dyatlov and Benjamin Küster.

Friday February 26, 11:00-11:45
Frédéric Faure
Title: Micro-local analysis of contact Anosov flows and band structure of the Ruelle spectrum
Abstract: This is a work with Masato Tsujii. We develop a geometrical micro-local analysis of contact Anosov flow, such as geodesic flow on negatively curved manifold. We use the method of wave-packet transform presented in a previous paper and observe that the transfer operator is well approximated (in the high frequency limit) by a "quantization" of an induced transfer operator acting on sections of some vector bundle on the trapped set. This gives a few important consequences: The discrete eigenvalues of the generator of transfer operators, called Ruelle spectrum, are structured into vertical bands. If the right-most band is isolated from the others, most of the Ruelle spectrum in it concentrate along a line parallel to the imaginary axis and, further, the density satisfies a Weyl law as the imaginary part tend to infinity.

Friday March 5, 10:00-10:45
Mihajlo Cekic
Title: The holonomy inverse problem and the generic injectivity of the twisted X-ray transform
Abstract: On a Riemannian manifold with Anosov geodesic flow, the celebrated Burns-Katok conjecture asserts that the marked length spectrum, namely the length of all closed geodesics marked by the free homotopy of the manifold, should determine the metric (up to isometries). In a similar fashion, given a vector bundle equipped with a unitary connection, one can wonder if the data of the holonomy of the connection along closed geodesics (up to conjugacy) determines the connection (up to gauge-equivalence). This is called the holonomy inverse problem and turns out to be a very rich question, as it brings together different fields of mathematics: microlocal analysis, hyperbolic dynamical systems, theory of Pollicott-Ruelle resonances and Kähler geometry. In this talk I will discuss a linearization of this problem, namely the twisted X-ray transform on tensors, and explain how to obtain its generic injectivity with respect to the connection. This will be used to solve the problem locally (see the next talk). Joint work with Thibault Lefeuvre.

Friday March 5, 11:00-11:45
Thibault Lefeuvre
Title: The local holonomy inverse problem
Abstract: Following on the previous talk, I will explain how to solve locally the holonomy inverse problem in a neighborhood of a generic connection. The idea is to show a link between the moduli space of gauge-equivalent connections and the Pollicott-Ruelle resonances near 0 of a certain natural operator for this problem (the so-called "mixed connection").

Friday March 12, 10:00-10:45
Gabriel Rivière
Title: Meromorphic continuation of Poincaré series
Abstract: On a compact negatively curved manifold, Poincaré series are zeta functions associated with the lengths of the geodesic arcs joining two given points of the manifold. I will explain how the recent progress made towards the the spectral analysis of Anosov flows allow to prove the meromorphic continuation of Poincaré series to the whole complex plane. I will also explain how this argument can be extended to prove the meromorphic continuation of generalized version of Poincaré series. In particular, this includes series associated with the lengths of the geodesic arcs orthogonal to a pair of closed geodesics on a negatively curved surface.
This is a joint work with Nguyen Viet Dang (Univ. Lyon 1).

Friday March 12, 11:00-11:45
Viet Dang
Title:Meromorphic continuation of Poincaré series
Abstract: This follows the talk of Gabriel Rivière. On a compact negatively curved surface, Poincaré series are zeta functions associated with the lengths of the orthogeodesic arcs joining two closed curves of the surface. When the closed curves are homologically trivial, I will explain how to prove the rationality of the value at 0 of the Poincaré series and also calculate it explicitely.

Friday March 19, 10:00-10:45
Malo Jézéquel
Title: FBI transform in Gevrey classes and Anosov flows
Abstract: I will explain how an analytic FBI transform may be used to study statistical properties for analytic and Gevrey Anosov flows, in the spirit of the work of Helffer and Sjöstrand in the context scattering theory. In deep contrast with the infinitely differentiable case, the generator of the flow acting on a suitable space is made (hypo-)elliptic, which gives a global control on the Ruelle spectrum. This is a joint work with Yannick Guedes Bonthonneau.

Friday March 19, 11:00-11:45
Stéphane Nonnenmacher
Title: Eigenmode delocalization on Anosov surfaces (joint with Semyon Dyatlov and Long Jin)
The geodesic flow on a compact Riemannian manifold of negative curvature (M,g) exhibits a chaotic (Anosov) dynamics. The corresponding quantum system, described by Schrödinger equation on (M,g), in particular its invariant states, embodied by the eigenmodes of the Laplace-Beltrami operator. The Quantum Unique Ergodicity conjecture states that, in the high frequency (=semiclassical) limit, the eigenstates asymptotically equidistribute on M. In the direction of QUE, Dyatlov and Jin have recently shown that, in the case of compact hyperbolic surfaces (namely, surfaces of constant negative curvature), the eigenstates are delocalized over M in the semiclassical limit. Their proof makes use of a new tool from harmonic analysis, a Fractal Uncertainty Principle proved by Bourgain and Dyatlov in 2016. In a joint work with Dyatlov and Jin, we extend the result to surfaces of (variable) negative curvature. One challenge of this extension is to control the regularity of the unstable and stable foliations, necessary to apply the Fractal Uncertainty Principle

Friday March 19, audio

Friday April 9, 10:00-10:45
Anke Pohl
Title: Selberg zeta functions with twists having non-expanding cusp monodromy
Abstract: The Selberg zeta function of a hyperbolic surface X is a dynamical zeta function that is well known to mediate between the geometric and the spectral properties of X. In particular, the observation that the zeros of the Selberg zeta function encode the spectral parameters (e.g., Laplace resonances) of X is a deep and highly influencial result. We report on the current status of a project to generalize these results to Selberg zeta function that are twisted by certain non-unitary representations.

Friday April 9, 11:00-11:45
Long Jin
Title: A local trace formula for Anosov flows
Abstract: In this talk, we present a local trace formula for Anosov flows on compact manifolds which relates Pollicott-Ruelle resonances to the periods of closed orbits. As an application, we show a weak lower bound for the counting function for resonances in a strip and thus the infinitude of the resonances. This is based on joint work with Maciej Zworski and Frederic Naud.

Friday April 16, 10:00-10:45
Yann Chaubet
Title: Closed geodesics with prescribed intersection numbers
Abstract: On a closed negatively curved surface, Margulis gave the asymptotic growth of the number of closed geodesics of bounded length, when the bound goes to infinity. In this talk, we will investigate such an asymptotic growth for closed geodesics of which certain intersection numbers (with a given family of pairwise disjoint simple closed geodesics) are prescribed.