@unpublished{59860, abstract = {{A three-dimensional quasi-Fuchsian Lorentzian manifold $M$ is a globally hyperbolic spacetime diffeomorphic to $\Sigma\times (-1,1)$ for a closed orientable surface $\Sigma$ of genus $\geq 2$. It is the quotient $M=\Gamma\backslash \Omega_\Gamma$ of an open set $\Omega_\Gamma\subset {\rm AdS}_3$ by a discrete group $\Gamma$ of isometries of ${\rm AdS}_3$ which is a particular example of an Anosov representation of $\pi_1(\Sigma)$. We first show that the spacelike geodesic flow of $M$ is Axiom A, has a discrete Ruelle resonance spectrum with associated (co-)resonant states, and that the Poincar\'e series for $\Gamma$ extend meromorphically to $\mathbb{C}$. This is then used to prove that there is a natural notion of resolvent of the pseudo-Riemannian Laplacian $\Box$ of $M$, which is meromorphic on $\mathbb{C}$ with poles of finite rank, defining a notion of quantum resonances and quantum resonant states related to the Ruelle resonances and (co-)resonant states by a quantum-classical correspondence. This initiates the spectral study of convex co-compact pseudo-Riemannian locally symmetric spaces.}}, author = {{Delarue, Benjamin and Guillarmou, Colin and Monclair, Daniel}}, booktitle = {{arXiv:2504.21762}}, title = {{{Spectra of Lorentzian quasi-Fuchsian manifolds}}}, year = {{2025}}, } @unpublished{58872, abstract = {{Given a non-compact semisimple real Lie group $G$ and an Anosov subgroup $\Gamma$, we utilize the correspondence between $\mathbb R$-valued additive characters on Levi subgroups $L$ of $G$ and $\mathbb R$-affine homogeneous line bundles over $G/L$ to systematically construct families of non-empty domains of proper discontinuity for the $\Gamma$-action. If $\Gamma$ is torsion-free, the analytic dynamical systems on the quotients are Axiom A, and assemble into a single partially hyperbolic multiflow. Each Axiom A system admits global analytic stable/unstable foliations with non-wandering set a single basic set on which the flow is conjugate to Sambarino's refraction flow, establishing that all refraction flows arise in this fashion. Furthermore, the $\mathbb R$-valued additive character is regular if and only if the associated Axiom A system admits a compatible pseudo-Riemannian metric and contact structure, which we relate to the Poisson structure on the dual of the Lie algebra of $G$.}}, author = {{Delarue, Benjamin and Monclair, Daniel and Sanders, Andrew}}, booktitle = {{arXiv:2502.20195}}, title = {{{Locally homogeneous Axiom A flows II: geometric structures for Anosov subgroups}}}, year = {{2025}}, } @article{53414, abstract = {{By constructing a non-empty domain of discontinuity in a suitable homogeneous space, we prove that every torsion-free projective Anosov subgroup is the monodromy group of a locally homogeneous contact Axiom A dynamical system with a unique basic hyperbolic set on which the flow is conjugate to the refraction flow of Sambarino. Under the assumption of irreducibility, we utilize the work of Stoyanov to establish spectral estimates for the associated complex Ruelle transfer operators, and by way of corollary: exponential mixing, exponentially decaying error term in the prime orbit theorem, and a spectral gap for the Ruelle zeta function. With no irreducibility assumption, results of Dyatlov-Guillarmou imply the global meromorphic continuation of zeta functions with smooth weights, as well as the existence of a discrete spectrum of Ruelle-Pollicott resonances and (co)-resonant states. We apply our results to space-like geodesic flows for the convex cocompact pseudo-Riemannian manifolds of Danciger-Gu\'eritaud-Kassel, and the Benoist-Hilbert geodesic flow for strictly convex real projective manifolds.}}, author = {{Delarue, Benjamin and Monclair, Daniel and Sanders, Andrew}}, journal = {{Geometric and Functional Analysis (GAFA)}}, pages = {{673–735}}, title = {{{Locally homogeneous Axiom A flows I: projective Anosov subgroups and exponential mixing}}}, doi = {{10.1007/s00039-025-00712-2}}, volume = {{35}}, year = {{2025}}, } @article{53412, abstract = {{Let $M$ be a symplectic manifold carrying a Hamiltonian $S^1$-action with momentum map $J:M \rightarrow \mathbb{R}$ and consider the corresponding symplectic quotient $\mathcal{M}_0:=J^{-1}(0)/S^1$. We extend Sjamaar's complex of differential forms on $\mathcal{M}_0$, whose cohomology is isomorphic to the singular cohomology $H(\mathcal{M}_0;\mathbb{R})$ of $\mathcal{M}_0$ with real coefficients, to a complex of differential forms on $\mathcal{M}_0$ associated with a partial desingularization $\widetilde{\mathcal{M}}_0$, which we call resolution differential forms. The cohomology of that complex turns out to be isomorphic to the de Rham cohomology $H(\widetilde{ \mathcal{M}}_0)$ of $\widetilde{\mathcal{M}}_0$. Based on this, we derive a long exact sequence involving both $H(\mathcal{M}_0;\mathbb{R})$ and $H(\widetilde{ \mathcal{M}}_0)$ and give conditions for its splitting. We then define a Kirwan map $\mathcal{K}:H_{S^1}(M) \rightarrow H(\widetilde{\mathcal{M}}_0)$ from the equivariant cohomology $H_{S^1}(M)$ of $M$ to $H(\widetilde{\mathcal{M}}_0)$ and show that its image contains the image of $H(\mathcal{M}_0;\mathbb{R})$ in $H(\widetilde{\mathcal{M}}_0)$ under the natural inclusion. Combining both results in the case that all fixed point components of $M$ have vanishing odd cohomology we obtain a surjection $\check \kappa:H^\textrm{ev}_{S^1}(M) \rightarrow H^\textrm{ev}(\mathcal{M}_0;\mathbb{R})$ in even degrees, while already simple examples show that a similar surjection in odd degrees does not exist in general. As an interesting class of examples we study abelian polygon spaces.}}, author = {{Delarue, Benjamin and Ramacher, Pablo and Schmitt, Maximilian}}, journal = {{Transformation Groups}}, title = {{{Singular cohomology of symplectic quotients by circle actions and Kirwan surjectivity}}}, doi = {{10.1007/s00031-025-09924-0}}, year = {{2025}}, } @article{53413, abstract = {{For negatively curved symmetric spaces it is known that the poles of the scattering matrices defined via the standard intertwining operators for the spherical principal representations of the isometry group are either given as poles of the intertwining operators or as quantum resonances, i.e. poles of the meromorphically continued resolvents of the Laplace-Beltrami operator. We extend this result to classical locally symmetric spaces of negative curvature with convex-cocompact fundamental group using results of Bunke and Olbrich. The method of proof forces us to exclude the spectral parameters corresponding to singular Poisson transforms.}}, author = {{Delarue, Benjamin and Hilgert, Joachim}}, issn = {{0949-5932}}, journal = {{Journal of Lie Theory}}, number = {{(4)}}, pages = {{787----804}}, title = {{{Quantum resonances and scattering poles of classical rank one locally symmetric spaces}}}, volume = {{35}}, year = {{2025}}, } @unpublished{58873, abstract = {{We prove that the Patterson-Sullivan and Wigner distributions on the unit sphere bundle of a convex-cocompact hyperbolic surface are asymptotically identical. This generalizes results in the compact case by Anantharaman-Zelditch and Hansen-Hilgert-Schr\"oder.}}, author = {{Delarue, Benjamin and Palmirotta, Guendalina}}, booktitle = {{arXiv:2411.19782}}, title = {{{Patterson-Sullivan and Wigner distributions of convex-cocompact hyperbolic surfaces}}}, year = {{2024}}, } @article{53410, abstract = {{AbstractWe consider a geodesic billiard system consisting of a complete Riemannian manifold and an obstacle submanifold with boundary at which the trajectories of the geodesic flow experience specular reflections. We show that if the geodesic billiard system is hyperbolic on its trapped set and the latter is compact and non-grazing, the techniques for open hyperbolic systems developed by Dyatlov and Guillarmou (Ann Henri Poincaré 17(11):3089–3146, 2016) can be applied to a smooth model for the discontinuous flow defined by the non-grazing billiard trajectories. This allows us to obtain a meromorphic resolvent for the generator of the billiard flow. As an application we prove a meromorphic continuation of weighted zeta functions together with explicit residue formulae. In particular, our results apply to scattering by convex obstacles in the Euclidean plane.}}, author = {{Delarue, Benjamin and Schütte, Philipp and Weich, Tobias}}, issn = {{1424-0637}}, journal = {{Annales Henri Poincaré}}, keywords = {{Mathematical Physics, Nuclear and High Energy Physics, Statistical and Nonlinear Physics}}, number = {{2}}, pages = {{1607--1656}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{Resonances and Weighted Zeta Functions for Obstacle Scattering via Smooth Models}}}, doi = {{10.1007/s00023-023-01379-x}}, volume = {{25}}, year = {{2023}}, } @unpublished{53411, abstract = {{We compute a Riemann-Roch formula for the invariant Riemann-Roch number of a quantizable Hamiltonian $S^1$-manifold $(M,\omega,\mathcal{J})$ in terms of the geometry of its symplectic quotient, allowing $0$ to be a singular value of the moment map $\mathcal{J}:M\to\mathbb{R}$. The formula involves a new explicit local invariant of the singularities. Our approach relies on a complete singular stationary phase expansion of the associated Witten integral.}}, author = {{Delarue, Benjamin and Ioos, Louis and Ramacher, Pablo}}, booktitle = {{arXiv:2302.09894}}, title = {{{A Riemann-Roch formula for singular reductions by circle actions}}}, year = {{2023}}, } @article{31982, abstract = {{AbstractWe show that for a generic conformal metric perturbation of a compact hyperbolic 3-manifold $$\Sigma $$ Σ with Betti number $$b_1$$ b 1 , the order of vanishing of the Ruelle zeta function at zero equals $$4-b_1$$ 4 - b 1 , while in the hyperbolic case it is equal to $$4-2b_1$$ 4 - 2 b 1 . This is in contrast to the 2-dimensional case where the order of vanishing is a topological invariant. The proof uses the microlocal approach to dynamical zeta functions, giving a geometric description of generalized Pollicott–Ruelle resonant differential forms at 0 in the hyperbolic case and using first variation for the perturbation. To show that the first variation is generically nonzero we introduce a new identity relating pushforwards of products of resonant and coresonant 2-forms on the sphere bundle $$S\Sigma $$ S Σ with harmonic 1-forms on $$\Sigma $$ Σ .}}, author = {{Cekić, Mihajlo and Delarue, Benjamin and Dyatlov, Semyon and Paternain, Gabriel P.}}, issn = {{0020-9910}}, journal = {{Inventiones mathematicae}}, keywords = {{General Mathematics}}, number = {{1}}, pages = {{303--394}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{The Ruelle zeta function at zero for nearly hyperbolic 3-manifolds}}}, doi = {{10.1007/s00222-022-01108-x}}, volume = {{229}}, year = {{2022}}, } @article{32016, author = {{Delarue, Benjamin and Ramacher, Pablo}}, journal = {{Journal of Symplectic Geometry}}, number = {{6}}, pages = {{1281 -- 1337}}, title = {{{Asymptotic expansion of generalized Witten integrals for Hamiltonian circle actions}}}, doi = {{10.4310/JSG.2021.v19.n6.a1}}, volume = {{19}}, year = {{2021}}, }