New resonance theory for Anosov flows connects spectral properties to mixing measures.
problem Defining and analyzing Ruelle-Taylor resonances for Anosov actions.
method Combining microlocal methods and J. Taylor's cohomological theory, defining Ruelle-Taylor resonances and proving Fredholm theory.
result Ruelle-Taylor resonances form a discrete subset of Cκ with λ=0 being a leading resonance. Study reveals a link between Ruelle-Pollicott resonances and cohomology eigenvalues for Anosov diffeomorphisms.
problem Understanding the speed of mixing in Anosov diffeomorphisms.
method Investigates Ruelle-Pollicott resonances on manifolds of any dimension, connecting them to cohomology eigenvalues of a quasi-compact transfer operator.
result Established a cohomological bound for the speed of mixing of Anosov diffeomorphisms.
Study SRB measures for Anosov actions on manifolds.
problem Characterize SRB measures for Anosov actions.
method Use Ruelle-Taylor resonances and properties of Sinai-Ruelle-Bowen measures.
result SRB measures have properties like smooth disintegrations, positive basins, and are unique under certain conditions.
Study of spectral properties of Lorentzian quasi-Fuchsian manifolds.
problem Understanding the spectral properties of Lorentzian quasi-Fuchsian manifolds.
method Analyzing the geodesic flow, Ruelle resonances, and pseudo-Riemannian Laplacian.
result Meromorphic extension of the resolvent of the pseudo-Riemannian Laplacian with poles of finite rank.
For compact and for convex co-compact oriented hyperbolic surfaces, we prove an explicit correspondence between classical Ruelle resonant states and quantum resonant states, except at negative integers where the correspondence involves holomorphic sections of line bundles.
For hyperbolic 3-manifolds, Ruelle zeta function vanishing order is 4 minus Betti number.
problem Analyzing the Ruelle zeta function at zero for perturbed hyperbolic 3-manifolds.
method Microlocal approach to dynamical zeta functions, first variation, new identity relating pushforwards of resonant and coresonant forms.
result The order of vanishing of the Ruelle zeta function at zero equals 4 minus Betti number for generic perturbations.
Study resonant forms for dissipative Anosov flows on 3-manifolds.
problem Determine resonant forms and their cohomology classes for dissipative Anosov flows.
method General theory including horocyclic invariance and local geometry analysis.
result Explicit computation of resonant forms and helicity for quasi-Fuchsian flows.
Study proves projective Anosov subgroups lead to mixing flows in specific spaces.
problem Understanding mixing properties of flows on specific geometric spaces.
method Constructing non-empty domain of discontinuity in homogeneous space, using spectral estimates for transfer operators.
result Exponential mixing, spectral gap, and meromorphic continuation of zeta functions established.
The study counts geodesics on curved surfaces with specific intersections.
problem Counting geodesics with exact intersection numbers on curved surfaces.
method Introduced a dynamical scattering operator and used Pollicott-Ruelle resonances.
result Asymptotic growth of geodesics with prescribed intersections.
We establish a direct classical-quantum correspondence on convex cocompact hyperbolic manifolds between the spectrums of the geodesic flow and the Laplacian acting on natural tensor bundles. This extends previous work detailing the correspondence for cocompact quotients.
The paper proves regularity of states on manifolds with unstable dynamics.
problem Propagation of regularity in dynamical systems with unstable manifolds.
method Leafwise semiclassical pseudodifferential calculus adapted to foliated spaces.
result Pollicott-Ruelle resonant states are smooth over entire manifolds if smooth on unstable leaves.
New dynamical torsion for contact Anosov flows connects to Reidemeister torsion.
problem Understanding contact Anosov flows and their properties.
method Introducing dynamical torsion and showing its properties.
result Locally constant ratio between dynamical and Turaev torsion.
The paper shows how to uniquely determine a connection up to gauge.
problem Determining a connection up to gauge from its holonomies.
method Using hyperbolic dynamical systems and transport operators.
result The primitive trace map is locally injective near generic points.
We give a new upper bound on the Selberg zeta function for a convex co-compact Schottky group acting on Hn+1: in strips parallel to the imaginary axis the zeta function is bounded by exp(C∣s∣δ) where δ is the dimension of the limit set of the group. This bound is more precise than the optimal…
Defines an equivariant Ruelle dynamical zeta function for flows on manifolds.
problem Defining a zeta function for equivariant flows on manifolds.
method Equivariant generalization of Guillemin's trace formula.
result Computes the equivariant Ruelle zeta function in various examples.
Study of Anosov flows using microlocal analysis for ergodicity and mixing properties.
problem Ergodicity and mixing properties of Anosov flows and their isometric extensions.
method Microlocal analysis of Pollicott-Ruelle resonances to study isometric extensions of Anosov flows.
result Ergodicity of frame flow on negatively-curved Riemannian manifolds under specific curvature assumptions.
Study of twisted Ruelle zeta function on hyperbolic manifolds and its relation to analytic torsion.
problem Analyzing the twisted Ruelle zeta function on hyperbolic manifolds.
method Investigating the twisted Ruelle zeta function associated with geodesic flow and acyclic representations.
result The twisted Ruelle zeta function equals the square of the refined analytic torsion multiplied by an exponential involving the eta invariant.
New formula connects surface singularity zeta function to Reidemeister-Turaev torsion.
problem Calculating Reidemeister-Turaev torsion for non-unitary representations.
method Ruelle zeta function and Reidemeister-Turaev torsion for compact hyperbolic orbisurfaces.
result Value of Ruelle zeta function at 0 equals Reidemeister-Turaev torsion.
Study geodesics on flat tori, focusing on convex bodies.
problem Analyze geodesics orthogonal to convex subsets on flat tori.
method Define anisotropic Sobolev spaces and study properties of geodesics.
result Compute residues of geometric Epstein function in terms of intrinsic volumes.
Survey on twisted dynamical zeta functions and Fried's conjecture.
problem Analyzing twisted dynamical zeta functions and their relation to Fried's conjecture.
method Review of existing literature and mini-course presentation.
result Discussion and validation of Fried's conjecture.
The paper connects geodesic flows and higher-dimensional Reidemeister torsion for hyperbolic orbifolds.
problem Understanding the relationship between geodesic flows and higher-dimensional Reidemeister torsion.
method Using the integral expression of the Ruelle zeta function and the Selberg zeta function.
result The absolute value at zero of the Ruelle zeta function equals the higher-dimensional Reidemeister torsion.
The space of convex projective structures has been well studied with respect to the topological entropy. But, to better understand the geometry of the structure, we study the entropy of the Sinai-Ruelle-Bowen measure and show that it is a continuous function.
We prove exponential decay of correlations for Hölder continuous observables with respect to any Gibbs measure for contact Anosov flows admitting Pesin sets with exponentially small tails. This is achieved by establishing strong spectral estimates for certain Ruelle transfer operators for such flows.
The study connects contact forms and Ruelle invariant in convex domains.
problem Understanding the relationship between contact forms and Ruelle invariant in convex domains.
method Using the extrinsic curvature and Ruelle invariant, the authors prove bounds and construct counterexamples.
result First examples of dynamically convex contact 3-spheres not strictly contactomorphic to convex boundaries.
We will show a theorem of a type of Cheeger and Muller for a noncompact complete hyperbolic threefold of finite vulume. As an application we will compute a special value of Ruelle L-function at the origin for a unitary local system which is cuspidal.
Zeta functions for non-unitary twists are shown to have analytic continuation.
problem Analytic continuation of zeta functions for non-unitary twists.
method Analytic continuation for compact locally-symmetric spaces with non-unitary twists.
result Zeta functions admit analytic continuation as meromorphic functions.
For a unitary local system of rank one on a complete hyperbolic threefold of finite volume which has only one cusp, the Ruelle L-function is defined. We will show that if the first cohomology group of the local system vanishes its value at s=0 is equal to the square of the Franz-Reidemeister torsion.
We define generalized currents associated with immersions of abstract solenoids with a transversal measure. We realize geometrically the full real homology of a compact manifold with these generalized currents, and more precisely with immersions of minimal uniquely ergodic solenoids. This makes precise and geometric De…
Proves functional equation for twisted Ruelle zeta function on hyperbolic surfaces.
problem Determines the functional equation for twisted Ruelle zeta functions.
method Analyzes scattering matrix and uses topological data of hyperbolic surfaces.
result Determines the order of the divisor of R(s,χ) at s=0 and computes its Laurent expansion.
Entropy study of geodesic flow on convex projective surfaces.
problem Entropy of Sinai-Ruelle-Bowen measure on convex projective surfaces.
method Analysis of Hilbert area and Blaschke metric.
result Entropy tends to zero if and only if the Hilbert area tends to infinity.
The paper proves a conjecture linking two metrics on manifold cohomology.
problem Proving a conjecture about metrics on manifold cohomology.
method Constructing complex structures, defining metrics, and proving the conjecture.
result Ray-Singer metric equals Milnor metric, linking analytic torsion to combinatorial data.
Analytic torsion equals dynamical zeta function for certain bundles.
problem Equalities between analytic torsion and dynamical zeta functions.
method Analytic torsion and Ruelle dynamical zeta function for admissible twists.
result Generalization of previous results to admissible twists.
The paper extends symplectomorphism quasi-morphisms to the entire disk group.
problem Extending quasi-morphisms to the symplectomorphism group of the disk.
method Showing homogeneous quasi-morphisms extend to the whole group.
result Second bounded cohomology of symplectomorphism group is infinite-dimensional.
We will prove that Ruelle L-function for a cuspidal local system on an odd dimensional hyperbolic manifold with finite volume satisfies a functional equation and an analog of the Riemann hypothesis. We will also compute its Laurent expansion at the origin and will prove that the second coefficient coincides with a rati…
Study reveals failure of uniqueness in dynamical invariants for 3D volume-preserving diffeomorphisms.
problem Uniqueness of dynamical invariants for 3D volume-preserving diffeomorphisms.
method Examined failure of uniqueness on integral homology spheres and arbitrary three-manifolds using local and global invariants.
result Failure of uniqueness is severe, with continuous and non-constant invariants appearing in C1-open sets of nonvanishing exact fields of fixed helicity. We extend Schwartzman theory beyond dimension 1 and provide a unified treatment of Ruelle-Sullivan and Schwartzman theories via Birkhoff's ergodic theorem for the class of immersions of solenoids with a trapping region.
Holomorphic vector bundles on Hopf manifolds admit flat connections.
problem Understanding flat connections on holomorphic vector bundles over Hopf manifolds.
method Defining resonant and non-resonant Mall bundles, proving the existence of flat connections on non-resonant bundles, and applying the Poincare-Dulac theorem.
result Non-resonant Hopf manifolds are linearizable, generalizing Kodaira's result.
We study the distribution of resonances for geometrically finite hyperbolic surfaces of infinite area by countting resonances numerically. The resonances are computed as zeros of the Selberg zeta function, using an algorithm for computation of the zeta function for Schottky groups. Our particular focus is on three aspe…
We develop theoretical foundations of Resonator Networks, a new type of recurrent neural network introduced in Frady et al. (2020) to solve a high-dimensional vector factorization problem arising in Vector Symbolic Architectures. Given a composite vector formed by the Hadamard product between a discrete set of high-dim…
We introduce the concept of solenoid as an abstract laminated space. We do a thorough study of solenoids, leading to the notion of ergodic and uniquely ergodic solenoids. We define generalized currents associated with immersions of oriented solenoids with a transversal measure into smooth manifolds, generalizing Ruelle…
We discuss inverse resonance scattering for the Laplacian on a rotationally symmetric manifold M=(0,∞)×Y whose rotation radius is constant outside some compact interval. The Laplacian on M is unitarily equivalent to a direct sum of one-dimensional Schrödinger operators with compactly supported potenti…
Complex valued analytic torsion and dynamical zeta function studied on locally symmetric spaces.
problem Analyzing the Ruelle dynamical zeta function on locally symmetric spaces with flat vector bundles.
method Meromorphic extension and regularisation of the dynamical zeta function, relating it to the complex valued analytic torsion.
result The leading term of the dynamical zeta function at zero is related to the regularised determinant of the flat Laplacian.
The study finds resonance points in polarised curves with polynomial conserved quantities.
problem Finding resonance points in polarised curves with polynomial conserved quantities.
method Using the non-orthogonality assumption on the conserved quantity, the study deduces the existence of resonance points.
result Every finite type polarised curve in the conformal 2-sphere with a polynomial conserved quantity admits a resonance point.
The resonant band is a useful notion for the computation of the nontrivial monodromy eigenspaces of the Milnor fiber of a real line arrangement. In this article, we develop the resonant band description for the cohomology of the Aomoto complex. As an application, we prove that real 4-nets do not exist.
We show that the resolvent of the Laplacian on SL(3,R)/SO(3) can be lifted to a meromorphic function on a Riemann surface which is a branched covering of C. The poles of this function are called the resonances of the Laplacian. We determine all resonances and show that the corresponding residue op…
Determinants remain constant along specific families of differential operators.
problem Local constancy of regularized determinants for differential operators.
method Analyzing families of operators Dτ=[δτ,d∇], showing flat-regularized determinant's constancy. result The flat-regularized determinant is constant in τ when restricted to im(δτ) under suitable assumptions. We prove the equality of the analytic torsion and the value at zero of a Ruelle dynamical zeta function associated with an acyclic unitarily flat vector bundle on a closed locally symmetric reductive manifold. This solves a conjecture of Fried. This article should be read in conjunction with an earlier paper by Moscovi…
We find a resonance free region polynomially close to the critical line on Conformally compact manifolds with polyhomogeneous metric.