## Periodic Reporting for period 3 - IPFLOW (Inverse Problems and Flows)

Reporting period: 2020-01-01 to 2021-06-30

The project IPFLOW is focused on the study of geometric flows, rigidity problems and inverse geometric problems, with a particular feature being the spectral approach of these problems. There are four aspects (Work Packages) that are studied thoroughly: 1) Inverse problems, 2) Rigidity phenomenon for Anosov flows, 3) Long time behaviour of hyperbolic flows and resonances, 4) Resonances of locally symmetric spaces.

WP1 is essentially focused on the boundary rigidity problem, where we try to determine a Riemannian metric on a manifold with boundary from the set of lengths of its geodesics with endpoints on the boundary, and/or from the tangent incoming/outgoing vectors of these geodesics (the so-called "scattering map"); these full data are called the "lens data". This question is important as it is a geometric model for X-ray tomography or for understanding the speed of propagation of the waves in the center of the earth by using the earthquake wave propagations. The problem asked by Michel has been solved in dimension 2 for metrics with no conjugate points on the unit disk by Pestov-Uhlmann, it has been solved recently in higher dimension in negative curvature for metrics on the unit ball by Stefanov-Uhlmann-Vasy.

In the closed case, there is an analog to the boundary rigidity problem: it asks if the marked length spectrum determines the isometry class of a negatively curved Riemannian metric This has been proved in dimension 2 by Croke and Otal in 1990 but the problem is largely open in higher dimension.

In the WP3, we propose to consider long time behaviour of the dynamic of hyperbolic flows, in particular understand the rate of mixing. Exponential mixing is known for contact Anosov flows since the works of Dolgopyat and Liverani in the early 2000's, but many other models of hyperbolic dynamics are still not fully understood. This type of problems is also almost equivalent to showing a gap of zeros for some dynamical zeta function (like Ruelle zeta function) that is constructed as an infinite product over the lengths of periodic orbits of the system, just like Riemann zeta function but where the prime numbers are replaced by the exponentials of the lengths of periodic orbits.

These zeta functions are some of the main aspects of our studies, in particular their meromorphic extension to the complex plane, the localization of its zeros/poles and what informations they carry on the flow or the geometry. An important parallel question on these zeta function is Fried conjecture which states that the value of Ruelle function at s=0 is given by some topological invariant; so far it has been solved for locally symmetric spaces (Fried, Moscovici-Stanton, Shen) but is open for general Anosov flows. In WP4, we propose to construct spectral approaches for studying flows appearing for locally symmetric spaces of all ranks.

WP1 is essentially focused on the boundary rigidity problem, where we try to determine a Riemannian metric on a manifold with boundary from the set of lengths of its geodesics with endpoints on the boundary, and/or from the tangent incoming/outgoing vectors of these geodesics (the so-called "scattering map"); these full data are called the "lens data". This question is important as it is a geometric model for X-ray tomography or for understanding the speed of propagation of the waves in the center of the earth by using the earthquake wave propagations. The problem asked by Michel has been solved in dimension 2 for metrics with no conjugate points on the unit disk by Pestov-Uhlmann, it has been solved recently in higher dimension in negative curvature for metrics on the unit ball by Stefanov-Uhlmann-Vasy.

In the closed case, there is an analog to the boundary rigidity problem: it asks if the marked length spectrum determines the isometry class of a negatively curved Riemannian metric This has been proved in dimension 2 by Croke and Otal in 1990 but the problem is largely open in higher dimension.

In the WP3, we propose to consider long time behaviour of the dynamic of hyperbolic flows, in particular understand the rate of mixing. Exponential mixing is known for contact Anosov flows since the works of Dolgopyat and Liverani in the early 2000's, but many other models of hyperbolic dynamics are still not fully understood. This type of problems is also almost equivalent to showing a gap of zeros for some dynamical zeta function (like Ruelle zeta function) that is constructed as an infinite product over the lengths of periodic orbits of the system, just like Riemann zeta function but where the prime numbers are replaced by the exponentials of the lengths of periodic orbits.

These zeta functions are some of the main aspects of our studies, in particular their meromorphic extension to the complex plane, the localization of its zeros/poles and what informations they carry on the flow or the geometry. An important parallel question on these zeta function is Fried conjecture which states that the value of Ruelle function at s=0 is given by some topological invariant; so far it has been solved for locally symmetric spaces (Fried, Moscovici-Stanton, Shen) but is open for general Anosov flows. In WP4, we propose to construct spectral approaches for studying flows appearing for locally symmetric spaces of all ranks.

Concerning WP1, we proved with M. Mazzucchelli and L. Tzou that the boundary distance of metrics with no conjugate points on simply connected surfaces with boundary determine the metric up to isometry, the novelty is that we do not assume that the boundary is strictly convex as it is the case in most known results. With R. Graham, P. Stefanov and G. Uhlmann, we also considered the boundary rigidity problem for asymptotically hyperbolic manifolds while with M.Lassas and L.Tzou we considered the case of asymptotically Euclidean metrics . Using this analysis, we show with M. Mazzucchelli and l. Tzou that asymptotically Euclidean metrics on R^n with no conjugate points are necessarily isometric to the the Euclidean metric.

The main achievement, since the starting of my IPFLOW project, is certainly the proof of the local rigidity of the marked length spectrum of manifolds with Anosov flow (including negatively curved closed manifolds) written last year with my PhD student T. Lefeuvre. This solved a local version of the long standing Burns-Katok conjecture, the paper was published by the Annals of Math. We show that two negatively curved metrics that are close enough and have the same marked length spectrum must be isometric. This approach uses the recent techniques of anisotropic Sobolev spaces and microlocal methods.

On WP3, we studied with F. Faure the resonant states of the vector field generating an Anosov flow in dimension 3 that are associated to the Ruelle resonances in a strip close to the leading eigenvalue and showed that they enjoy invariance under unstable horocyclic flow. In a note with Dyatlov, we also showed that our previous result on Axiom A flows allows to prove meromorphic extension of the Ruelle zeta function of smooth Axiom A flows, a fact that was conjectured long time ago by S. Smale. With my ERC postdoc Benjamin Küster we study the resonance spectrum of the frame flow of hyperbolic 3 manifolds.

Another important contribution on the topic of dynamical zeta functions is my recent paper with Dang-Riviere-Shen, where we prove Fried conjecture for Anosov flows in small dimensions.

On WP4, we have extended my work with Dyatlov and Faure to the geodesic flow associated to general compact rank 1 locally symmetric spaces in joint work with J. Hilgert and T.Weich: we give a description of the first band of Ruelle resonances in terms of spectrum of Laplacian, and we show a quantum ergodicity result for the Ruelle resonant projectors associated to the 1st band of resonances.

Another important work with Y. Bonthonneau, J. Hilgert and T. Weich is the introduction of a spectral approach of the Weyl chamber flows of higher rank compact locally symmetric spaces. In that case there we define a notion of Ruelle-Taylor resonances by combining microlocal approaches and a notion of joint spectrum. This provides a completely new way to understand dynamics for locally symmetric spaces of higher rank.

The main achievement, since the starting of my IPFLOW project, is certainly the proof of the local rigidity of the marked length spectrum of manifolds with Anosov flow (including negatively curved closed manifolds) written last year with my PhD student T. Lefeuvre. This solved a local version of the long standing Burns-Katok conjecture, the paper was published by the Annals of Math. We show that two negatively curved metrics that are close enough and have the same marked length spectrum must be isometric. This approach uses the recent techniques of anisotropic Sobolev spaces and microlocal methods.

On WP3, we studied with F. Faure the resonant states of the vector field generating an Anosov flow in dimension 3 that are associated to the Ruelle resonances in a strip close to the leading eigenvalue and showed that they enjoy invariance under unstable horocyclic flow. In a note with Dyatlov, we also showed that our previous result on Axiom A flows allows to prove meromorphic extension of the Ruelle zeta function of smooth Axiom A flows, a fact that was conjectured long time ago by S. Smale. With my ERC postdoc Benjamin Küster we study the resonance spectrum of the frame flow of hyperbolic 3 manifolds.

Another important contribution on the topic of dynamical zeta functions is my recent paper with Dang-Riviere-Shen, where we prove Fried conjecture for Anosov flows in small dimensions.

On WP4, we have extended my work with Dyatlov and Faure to the geodesic flow associated to general compact rank 1 locally symmetric spaces in joint work with J. Hilgert and T.Weich: we give a description of the first band of Ruelle resonances in terms of spectrum of Laplacian, and we show a quantum ergodicity result for the Ruelle resonant projectors associated to the 1st band of resonances.

Another important work with Y. Bonthonneau, J. Hilgert and T. Weich is the introduction of a spectral approach of the Weyl chamber flows of higher rank compact locally symmetric spaces. In that case there we define a notion of Ruelle-Taylor resonances by combining microlocal approaches and a notion of joint spectrum. This provides a completely new way to understand dynamics for locally symmetric spaces of higher rank.

The most important progress performed in the first 30 months is the understanding of the marked length spectrum of Riemannian manifold with Anosov geodesic flow as a function on the space of metrics. With T. Lefeuvre, we were able to obtain completely new results in dimension higher than 2, while there were only very few results on the Burns-Katok conjecture excet in dimension 2. We plan to pursue this study and hope to show more global uniqueness results and maybe solve Burns Katok conjecture completely.

An important new progress in the field is the work with N.V. Dang, G. Riviere and S. Shen on Fried conjecture where weprove that the twisted Ruelle zeta function (twisted by a representation of the fundamental group) is locally constant with respect to the flow. My PhD student, Y. Chaubet, is pursuing this direction with NV. Dang.

Another important new work is the analysis, via anisotropic spaces and spectral approach, of the Weyl chamber flows of locally symmetric spaces of rank greater than 1 or more generally Anosov abelian group actions. This approach should allow us to understand the long-time dynamics. We expect some possible new results on the long time behaviour of the flow (exponential mixing) for a certain class of Anosov abelian actions, and maybe to show new rigidity results in that direction: such manifolds that possess Anosov abelian actions are conjectured to be very rare.

An important new progress in the field is the work with N.V. Dang, G. Riviere and S. Shen on Fried conjecture where weprove that the twisted Ruelle zeta function (twisted by a representation of the fundamental group) is locally constant with respect to the flow. My PhD student, Y. Chaubet, is pursuing this direction with NV. Dang.

Another important new work is the analysis, via anisotropic spaces and spectral approach, of the Weyl chamber flows of locally symmetric spaces of rank greater than 1 or more generally Anosov abelian group actions. This approach should allow us to understand the long-time dynamics. We expect some possible new results on the long time behaviour of the flow (exponential mixing) for a certain class of Anosov abelian actions, and maybe to show new rigidity results in that direction: such manifolds that possess Anosov abelian actions are conjectured to be very rare.