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Thibault Lefeuvre

Publications and source records attributed to Thibault Lefeuvre.

At least 19 recordsLinked to original sources
arXiv2026

The spectrum of Anosov representations

Given a $P$-Anosov representation into a noncompact semisimple real algebraic group $G$, where $P < G$ is a parabolic subgroup, we construct a natural resonance spectrum for the classical dynamics associated with the representation. This spectrum is a complex analytic hypersurface in $(\mathfrak{a}_P^*)_{\mathbb{C}}$, the complexified dual of the Lie algebra of the split component of the associated Levi group $L < P$. We reinterpret several objects from the theory of Anosov representations within this spectral framework and investigate, in higher rank, questions that are classically related to Pollicott-Ruelle theory in the rank-one setting. In particular, the ``leading resonance''--which is now a hypersurface--is identified with the critical hypersurface of the representation and, as in the rank-one case, the resonant states lead to continuous families of invariant measures. We prove that the multivariate zeta functions and Poincar\'e series associated with Anosov representations admit a meromorphic extension to $(\mathfrak{a}_P^*)_{\mathbb{C}}$. We also establish an asymptotic expansion in inverse powers of time for the correlation function of the diagonal flow under a Diophantine condition on the representation. Most of our results concerning Anosov representations are obtained as a byproduct of a general theory of Axiom A actions of type $(k,1)$, where $k+1:=\dim \mathfrak{a}_P$, that we introduce in the article.

math.RT↗
arXiv2026

Zonal states and improved $L^\infty$ bounds for eigenfunctions of magnetic Laplacians on hyperbolic surfaces

We establish polynomially improved $L^\infty$ bounds for eigenfunctions of magnetic Laplacians on hyperbolic surfaces in the critical energy regime. We also show that, below the critical energy, the H\"ormander bound is saturated by explicit eigenstates, which we call magnetic zonal states. These states resemble zonal harmonics on the sphere and equidistribute on Lagrangian tori in phase space.

math.AP↗
arXiv2026

Propagation of regularity along unstable manifolds

Let $φ_t : M \to M$ be a flow on a smooth closed connected manifold $M$ that preserves and expands a foliation $F$. We establish a theorem of propagation of regularity along the leaves of $F$ for sections of vector bundles satisfying a transport equation involving the generator of a cocycle over $φ_t$. As a consequence, we prove a regularity result for Pollicott-Ruelle resonant states: if such state is smooth in restriction to a piece of an unstable leaf, then it is in fact smooth over the entire manifold. We also announce further applications related to joint integrability of extreme bundles of partially hyperbolic diffeomorphisms. The proofs rely on a leafwise semiclassical pseudodifferential calculus adapted to a foliated space, which may be of independent interest.

math.DS↗
arXiv2026

Semiclassical analysis of the magnetic Laplacian on hyperbolic surfaces

The magnetic Laplacian on hyperbolic surfaces provides a rich analytic framework in which a variety of quantum phenomena emerge. The present note, written for the \emph{Proceedings of the Journées EDP 2025}, is a concise overview of the main results obtained in [arXiv:2505.08584] and work in preparation by the author with L. Charles and A. Chabert.

math.AP↗
arXiv2025

Marked Poincaré rigidity near hyperbolic metrics and injectivity of the Lichnerowicz Laplacian in dimension 3

Let $M$ be a compact manifold without boundary equipped with a Riemannian metric $g$ of negative curvature. In this paper, we introduce the marked Poincaré determinant (MPD), a homothety invariant of $g$ depending on differentiable periodic data of its geodesic flow. The MPD associates to each free homotopy class of closed curves in $M$ a number which measures the unstable volume expansion of the geodesic flow along the associated closed geodesic. We prove a local MPD rigidity result in dimension 3: if $g$ is sufficiently close to a hyperbolic metric $g_0$ and both metrics have the same MPD, then they are homothetic. As a by-product of our proof, we show the Lichnerowicz Laplacian of $g_0$ is injective on the space of trace-free divergence-free symmetric 2-tensors, which, to our knowledge, is the first result of its kind in negative curvature.

math.DG↗
arXiv2025

Semiclassical defect measures of magnetic Laplacians on hyperbolic surfaces

On a closed hyperbolic surface, we investigate semiclassical defect measures associated with the magnetic Laplacian in the presence of a constant magnetic field. Depending on the energy level where the eigenfunctions concentrate, three distinct dynamical regimes emerge. In the low-energy regime, we show that any invariant measure of the magnetic flow in phase space can be obtained as a semiclassical measure. At the critical energy level, we establish Quantum Unique Ergodicity, together with a quantitative rate of convergence of eigenfunctions to the Liouville measure. In the high-energy regime, we prove a Shnirelman-type result: a density-one subsequence of eigenfunctions becomes equidistributed with respect to the Liouville measure.

math.AP↗
arXiv2024

On the ergodicity of the frame flow on even-dimensional manifolds

It is known that the frame flow on a closed $n$-dimensional Riemannian manifold with negative sectional curvature is ergodic if $n$ is odd and $n \neq 7$. In this paper we study its ergodicity in the remaining cases. For $n$ even and $n \neq 8, 134$, we show that: if $n \equiv 2$ mod $4$ or $n=4$, the frame flow is ergodic if the manifold is $\sim 0.3$-pinched, if $n \equiv 0$ mod $4$, it is ergodic if the manifold is $\sim 0.6$-pinched. In the three dimensions $n=7,8,134$, the respective pinching bounds that we need in order to prove ergodicity are $0.4962...$, $0.6212...$, and $0.5788...$. This is a significant improvement over the previously known results and a step forward towards solving a long-standing conjecture of Brin asserting that $0.25$-pinched even-dimensional manifolds have an ergodic frame flow.

math.DS↗
arXiv2024

Semiclassical analysis on principal bundles

Let $G$ be a compact Lie group. We introduce a semiclassical framework, called Borel-Weil calculus, to investigate $G$-equivariant (pseudo)differential operators acting on $G$-principal bundles over closed manifolds. In this calculus, the semiclassical parameters correspond to the highest roots in the Weyl chamber of the group $G$ that parametrize irreducible representations, and operators are pseudodifferential in the base variable, with values in Toeplitz operators on the flag manifold associated to the group. This monograph unfolds two main applications of our calculus. Firstly, in the realm of dynamical systems, we obtain explicit sufficient conditions for rapid mixing of volume-preserving partially hyperbolic flows obtained as extensions of an Anosov flow to a $G$-principal bundle (for an arbitrary $G$). In particular, when $G = \mathrm{U}(1)$, we prove that the flow on the extension is rapid mixing whenever the Anosov flow is not jointly integrable, and the circle bundle is not torsion. When $G$ is semisimple, we prove that ergodicity of the extension is equivalent to rapid mixing. Secondly, we study the spectral theory of sub-elliptic Laplacians obtained as horizontal Laplacians of a $G$-equivariant connection on a principal bundle. When $G$ is semisimple, we prove that the horizontal Laplacian is globally hypoelliptic as soon as the connection has a dense holonomy group in $G$. Notably, this result encompasses all flat bundles with a dense monodromy group in $G$. We also prove a quantum ergodicity result for flat (and in some situations non-flat) principal bundles under a suitable ergodicity assumption. We believe that this monograph will serve as a cornerstone for future investigations applying the Borel-Weil calculus across different fields.

math.AP↗
arXiv2024

Marked length spectrum rigidity for Anosov surfaces

Let $Σ$ be a smooth closed oriented surface of genus $\geq 2$. We prove that two metrics on $Σ$ with the same marked length spectrum and Anosov geodesic flow are isometric via an isometry isotopic to the identity. The proof combines microlocal tools with the geometry of complex curves.

math.DG↗
arXiv2024

Stability estimates for the Holonomy Inverse Problem

On a Riemannian manifold $(M, g)$ with Anosov geodesic flow, the problem of recovering a connection from the knowledge of traces of its holonomies along primitive closed geodesics is known as the holonomy inverse problem. In this paper, we prove Hölder type stability estimates for this inverse problem: 1) locally, near generic connections; 2) globally, for line bundles, and for vector bundles satisfying a certain low-rank assumption over negatively curved base $(M, g)$. The proofs are based on a combination of microlocal analysis along with a new non-Abelian approximate Livsic Theorem in hyperbolic dynamics.

math.AP↗
arXiv2024

Isometric extensions of Anosov flows via microlocal analysis

We revisit the classical framework developed by Brin, Pesin and others to study ergodicity and mixing properties of isometric extensions of volume-preserving Anosov flows, using the microlocal framework developed in the theory of Pollicott-Ruelle resonances. The approach developed in the present note is reinvested in the companion paper [arXiv:2111.14811] in order to show ergodicity of the frame flow on negatively-curved Riemannian manifolds under nearly $1/4$-pinched curvature assumption (resp. nearly $1/2$-pinched) in dimension $4$ and $4\ell+2, \ell > 0$ (resp. dimension $4\ell, \ell > 0$).

math.DS↗
arXiv2024

Invariant distributions and the transport twistor space of closed surfaces

The purpose of this paper is to study transport equations on the unit tangent bundle of closed oriented Riemannian surfaces and to connect these to the transport twistor space of the surface (a complex surface naturally tailored to the geodesic vector field). We show that fibrewise holomorphic distributions invariant under the geodesic flow - which play an important role in tensor tomography on surfaces - form a unital algebra, that is, multiplication of such distributions is well-defined and continuous. We also exhibit a natural bijective correspondence between fibrewise holomorphic invariant distributions and genuine holomorphic functions on twistor space with polynomial blowup on the boundary of the twistor space. Eventually, when the surface is Anosov, we classify holomorphic line bundles over twistor space which are smooth up to the boundary. As a byproduct of our analysis, we obtain a quantitative version of a result of Flaminio, asserting that invariant distributions of the geodesic flow of a positively-curved metric on the 2-sphere are determined by their zeroth and first Fourier modes.

math.DG↗
arXiv2024

Generic injectivity of the X-ray transform

In dimensions $\geq 3$, we prove that the X-ray transform of symmetric tensors of arbitrary degree is generically injective with respect to the metric on closed Anosov manifolds and on manifolds with spherical strictly convex boundary, no conjugate points and a hyperbolic trapped set. This has two immediate corollaries: local spectral rigidity, and local marked length spectrum rigidity (building on earlier work by Guillarmou, Knieper and the second author [arXiv:1806.04218], [arXiv:1909.08666]), in a neighbourhood of a generic Anosov metric. In both cases, this is the first work going beyond the negatively curved assumption or dimension $2$. Our method, initiated in [arXiv:2008.09191] and fully developed in the present paper, is based on a perturbative argument of the $0$-eigenvalue of elliptic operators via microlocal analysis which turn the analytic problem of injectivity into an algebraic problem of representation theory. When the manifold is equipped with a Hermitian vector bundle together with a unitary connection, we also show that the twisted X-ray transform of symmetric tensors (with values in that bundle) is generically injective with respect to the connection. This property turns out to be crucial when solving the $\textit{holonomy inverse problem}$, as studied in a subsequent article [arXiv:2105.06376].

math.AP↗
arXiv2023

The Holonomy Inverse Problem

Let $(M,g)$ be a smooth Anosov Riemannian manifold and $\mathcal{C}^\sharp$ the set of its primitive closed geodesics. Given a Hermitian vector bundle $\mathcal{E}$ equipped with a unitary connection $\nabla^{\mathcal{E}}$, we define $\mathcal{T}^\sharp(\mathcal{E}, \nabla^{\mathcal{E}})$ as the sequence of traces of holonomies of $\nabla^{\mathcal{E}}$ along elements of $\mathcal{C}^\sharp$. This descends to a homomorphism on the additive moduli space $\mathbb{A}$ of connections up to gauge $\mathcal{T}^\sharp: (\mathbb{A}, \oplus) \to \ell^\infty(\mathcal{C}^\sharp)$, which we call the $\textit{primitive trace map}$. It is the restriction of the well-known $\textit{Wilson loop}$ operator to primitive closed geodesics. The main theorem of this paper shows that the primitive trace map $\mathcal{T}^\sharp$ is locally injective near generic points of $\mathbb{A}$ when $\dim(M) \geq 3$. We obtain global results in some particular cases: flat bundles, direct sums of line bundles, and general bundles in negative curvature under a spectral assumption which is satisfied in particular for connections with small curvature. As a consequence of the main theorem, we also derive a spectral rigidity result for the connection Laplacian. The proofs are based on two new ingredients: a Livšic-type theorem in hyperbolic dynamical systems showing that the cohomology class of a unitary cocycle is determined by its trace along closed primitive orbits, and a theorem relating the local geometry of $\mathbb{A}$ with the Pollicott-Ruelle resonance near zero of a certain natural transport operator.

math.DS↗
arXiv2023

Marked boundary rigidity for surfaces of Anosov type

Let $Σ$ be a smooth compact connected oriented surface with boundary. A metric on $Σ$ is said to be of Anosov type if it has strictly convex boundary, no conjugate points, and a hyperbolic trapped set. We prove that two metrics of Anosov type with the same marked boundary distance are isometric (via a boundary-preserving isometry isotopic to the identity). As a corollary, we retrieve the boundary distance rigidity result for simple disks of Pestov and Uhlmann [arXiv:math/0305280]. The proof rests on a new transfer principle showing that, in any dimension, the marked length spectrum rigidity conjecture implies the marked boundary distance rigidity conjecture under the existence of a suitable isometric embedding into a closed Anosov manifold. Such an isometric embedding result for open surfaces of Anosov type was proved by the first author with Chen and Gogolev in [arXiv:2009.13665] while the marked length spectrum rigidity for closed Anosov surfaces was established by the second author with Guillarmou and Paternain in [arXiv:2303.12007].

math.DG↗
arXiv2023

Isospectral connections, ergodicity of frame flows, and polynomial maps between spheres

We show that on closed negatively curved Riemannian manifolds with simple length spectrum, the spectrum of the Bochner Laplacian determines both the isomorphism class of the vector bundle and the connection up to gauge under a low-rank assumption. We also show that flows of frames on low-rank frame bundles extending the geodesic flow in negative curvature are ergodic whenever the bundle admits no holonomy reduction. This is achieved by exhibiting a link between these problems and the classification of polynomial maps between spheres in real algebraic geometry.

math.DS↗