Smoothness of Invariant Manifolds for Nonautonomous Equations

For semiflows generated by ordinary differential equations v’=A(t)v admitting a nonuniform exponential dichotomy, we show that for any sufficiently small perturbation f there exist smooth stable and unstable manifolds for the perturbed equation v’=A(t)v+f(t,v). As an application, we establish the existence of invariant manifolds for the nonuniformly hyperbolic trajectories of a semiflow. In particular, we obtain smooth invariant manifolds for a class of vector fields that need not be C^1+α for any α ∈ (0,1). To the best of our knowledge no similar statement was obtained before in the nonuniformly hyperbolic setting. We emphasize that we do not need to assume the existence of an exponential dichotomy, but only the existence of a nonuniform exponential dichotomy, with sufficiently small nonuniformity when compared to the Lyapunov exponents of the original linear equation. Furthermore, for example in the case of stable manifolds, we only need to assume that there exist negative Lyapunov exponents, while we also allow zero exponents. Our proof of the smoothness of the invariant manifolds is based on the construction of an invariant family of cones.Supported by the Center for Mathematical Analysis, Geometry, and Dynamical Systems, and through Fundação para a Ciência e a Tecnologia by Program POCTI/FEDER, Program POSI, and the grant SFRH/BPD/14404/2003.     

Invariant Measures Satisfying an Equality Relating Entropy,Folding Entropy and Negative Lyapunov Exponents

In this paper we prove that, for a C ² (non-invertible but non-degenerate) map on a compact manifold, an invariant measure satisfies an equality relating entropy, folding entropy and negative Lyapunov exponents if and, under a condition on the Jacobian of the map, only if the measure has absolutely continuous conditional measures on the stable manifolds. This work is supported by National Basic Research Program of China (973 Program) (2007CB814800).     

Locally Conformal Dirac Structures and Infinitesimal Automorphisms

We study the main properties of locally conformal Dirac bundles, which include Dirac structures on a manifold and locally conformal symplectic manifolds. It is proven that certain locally conformal Dirac bundles induce Jacobi structures on quotient manifolds. Furthermore we show that, given a locally conformal Dirac bundle over a smooth manifold M, there is a Lie homomorphism between a subalgebra of the Lie algebra of infinitesimal automorphisms and the Lie algebra of admissible functions. We also show that Dirac manifolds can be obtained from locally conformal Dirac bundles by using an appropriate covering map. Finally, we extend locally conformal Dirac bundles to the context of Lie algebroids.     

Poisson manifolds and their associated stacks

We associate to any integrable Poisson manifold a stack, i.e., a category fibered in groupoids over a site. The site in question has objects Dirac manifolds and morphisms pairs consisting of a smooth map and a closed 2-form. We show that two Poisson manifolds are symplectically Morita equivalent if and only if their associated stacks are isomorphic. We also discuss the non-integrable case.     

Renormalization-Group Method for Reduction of Evolution Equations; Invariant Manifolds and Envelopes

The renormalization group (RG) method as a powerful tool for reduction of evolution equations is formulated in terms of the notion of invariant manifolds. We start with derivation of an exact RG equation which is analogous to the Wilsonian RG equations in statistical physics and quantum field theory. It is clarified that the perturbative RG method constructs invariant manifolds successively as the initial value of evolution equations, thereby the meaning to set t₀=t is naturally understood where t₀ is the arbitrary initial time. We show that the integral constants in the unperturbative solution constitutes natural coordinates of the invariant manifold when the linear operator A in the evolution equation is semi-simple, i.e., diagonalizable; when A is not semi-simple and has a Jordan cell, a slight modification is necessary because the dimension of the invariant manifold is increased by the perturbation. The RG equation determines the slow motion of the would-be integral constants in the unperturbative solution on the invariant manifold. We present the mechanical procedure to construct the perturbative solutions hence the initial values with which the RG equation gives meaningful results. The underlying structure of the reduction by the RG method as formulated in the present work turns out to completely fit to the universal one elucidated by Kuramoto some years ago. We indicate that the reduction procedure of evolution equations has a good correspondence with the renormalization procedure in quantum field theory; the counter part of the universal structure of reduction elucidated by Kuramoto may be Polchinski's theorem for renormalizable field theories. We apply the method to interface dynamics such as kink–anti-kink and soliton–soliton interactions in the latter of which a linear operator having a Jordan-cell structure appears.     

Deformation quantization of Heisenberg manifolds

ForM a smooth manifold equipped with a Poisson bracket, we formulate aC*-algebra framework for deformation quantization, including the possibility of invariance under a Lie group of diffeomorphisms preserving the Poisson bracket. We then show that the much-studied non-commutative tori give examples of such deformation quantizations, invariant under the usual action of ordinary tori. Going beyond this, the main results of the paper provide a construction of invariant deformation quantizations for those Poisson brackets on Heisenberg manifolds which are invariant under the action of the Heisenberg Lie group, and for various generalizations suggested by this class of examples. Interesting examples are obtained of simpleC*-algebras on which the Heisenberg group acts ergodically.This work was supported in part by National Science Foundation grant DMS 8601900     

Theory of Tensor Invariants of Integrable Hamiltonian Systems. II. Theorem on Symmetries and Its Applications

The theorem on symmetries is proved that states that a Liouville-integrable Hamiltonian system is non-degene\-rate in Kolmogorov's sense and has compact invariant submanifolds if and only if the corresponding Lie algebra of symmetries is abelian. The theorem on symmetries has applications to the characterization problem, to the integrable hierarchies problem, to the necessary conditions for the strong dynamical compatibility problem, and to the problem on master symmetries. The invariant necessary conditions for the non-degenerate C-integrability in Kolmogorov's sense of a given dynamical system V are derived. It is proved that the C-integrable Hamiltonian system is non-degenerate in the iso-energetic sense if and only if the corresponding Lie algebra of the iso-energetic conformal symmetries is abelian. An extended concept of integrability of Hamiltonian systems on the symplectic manifolds M ⁿ , n= 2k, is introduced. The concept of integrability describes the Hamiltonian systems that have quasi-periodic dynamics on tori or on toroidal cylinders of an arbitrary dimension . This concept includes, as a particular case, all Hamiltonian systems that are integrable in Liouville's classical sense, for which . The A-B-C-cohomologies are introduced for dynamical systems on smooth manifolds. Received: 16 January 1996 / Accepted: 3 July 1996     

Energy landscapes, lowest gaps, and susceptibility of elastic manifolds at zero temperature

We study the effect of an external field on (1 + 1) and (2 + 1) dimensional elastic manifolds, at zero temperature and with random bond disorder. Due to the glassy energy landscape the configuration of a manifold changes often in abrupt, “first order”-type of large jumps when the field is applied. First the scaling behavior of the energy gap between the global energy minimum and the next lowest minimum of the manifold is considered, by employing exact ground state calculations and an extreme statistics argument. The scaling has a logarithmic prefactor originating from the number of the minima in the landscape, and reads ΔE ₁∼L ^θ[ln(L _z L ^{- ζ})]^-1/2, where ζ is the roughness exponent and θ is the energy fluctuation exponent of the manifold, L is the linear size of the manifold, and L_z is the system height. The gap scaling is extended to the case of a finite external field and yields for the susceptibility of the manifolds ∼L ^{2D + 1 - θ}[(1 - ζ)ln(L)]^1/2. We also present a mean field argument for the finite size scaling of the first jump field, h ₁∼L ^{d - θ}. The implications to wetting in random systems, to finite-temperature behavior and the relation to Kardar-Parisi-Zhang non-equilibrium surface growth are discussed. Received December 2000 and Received in final form April 2001     

Note on duality on polarized symplectic manifolds

In this note we use some of the results of [3] to derive a general duality theorem for the cohomologies of foliated structures on a manifold. The result is applied to the special case of a symplectic manifold M on which the foliation is given by a complex polarization F in the sense of geometric quantization. We obtain, for example, a rigorous proof of the fact that for a smooth function ƒ on M whose Hamiltonian vector field leaves F invariant, the spectrum of the corresponding prequantization operator v(ƒ) coincides with the spectrum of its transpose, under the above duality. This latter result was obtained by Simms in [12] under certain hypotheses. Proofs of the validity of those hypotheses are now available in the literature; cf. [3] and [7].     

A Pairing Between Super Lie-Rinehart and Periodic Cyclic Homology

We consider a pairing producing various cyclic Hochschild cocycles, which led Alain Connes to cyclic cohomology. We are interested in geometrical meaning and homological properties of this pairing. We define a non-trivial pairing between the homology of a Lie-Rinehart (super-)algebra with coefficients in some partial traces and relative periodic cyclic homology. This pairing generalizes the index formula for summable Fredholm modules, the Connes-Kubo formula for the Hall conductivity and the formula computing the K⁰-group of a smooth noncommutative torus. It also produces new homological invariants of proper maps contracting each orbit contained in a closed invariant subset in a manifold acted on smoothly by a connected Lie group. Finally we compare it with the characteristic map for the Hopf-cyclic cohomology. The author was partially supported by the KBN grant 1P03A 036 26.     

Flows on Quaternionic-Kähler and Very Special Real Manifolds

BPS solutions of 5-dimensional supergravity correspond to certain gradient flows on the product M×N of a quaternionic-Kähler manifold M of negative scalar curvature and a very special real manifold N of dimension n0. Such gradient flows are generated by the ``energy function' f=P<sup>2</sup>, where P is a (bundle-valued) moment map associated to n+1 Killing vector fields on M. We calculate the Hessian of f at critical points and derive some properties of its spectrum for general quaternionic-Kähler manifolds. For the homogeneous quaternionic-Kähler manifolds we prove more specific results depending on the structure of the isotropy group. For example, we show that there always exists a Killing vector field vanishing at a point pM such that the Hessian of f at p has split signature. This generalizes results obtained recently for the complex hyperbolic plane (universal hypermultiplet) in the context of 5-dimensional supergravity. For symmetric quaternionic-Kähler manifolds we show the existence of non-degenerate local extrema of f, for appropriate Killing vector fields. On the other hand, for the non-symmetric homogeneous quaternionic-Kähler manifolds we find degenerate local minima. This work was supported by the priority programme ``String Theory'of the Deutsche Forschungsgemeinschaft.     

Existence and invariance of twisted products on a symplectic manifold

It is now well-known [1] that the twisted product on the functions defined on a symplectic manifold, play a fundamental role in an invariant approach of quantum mechanics. We prove here a general existence theorem of such twisted products. If a Lie group G acts by symplectomorphisms on a symplectic manifold and if there is a G-invariant symplectic connection, the manifold admits G-invariant Vey twisted products. In particular, if a homogeneous space G/H admits an invariant linear connection, T ^*(G/H) admits a G-invariant Vey twisted product. For the connected Lie group G, the group T ^*G admits a symplectic structure, a symplectic connection and a Vey twisted product which are bi-invariant under G.     

An adaptive method for computing invariant manifolds in non-autonomous, three-dimensional dynamical systems

We present a computational method for determining the geometry of a class of three-dimensional invariant manifolds in non-autonomous (aperiodically time-dependent) dynamical systems. The presented approach can be also applied to analyse the geometry of 3D invariant manifolds in three-dimensional, time-dependent fluid flows. The invariance property of such manifolds requires that, at any fixed time, they are given by surfaces in R³. We focus on a class of manifolds whose instantaneous geometry is given by orientable surfaces embedded in R³. The presented technique can be employed, in particular, to compute codimension one (invariant) stable and unstable manifolds of hyperbolic trajectories in 3D non-autonomous dynamical systems which are crucial in the Lagrangian transport analysis. The same approach can also be used to determine evolution of an orientable ‘material surface’ in a fluid flow. These developments represent the first step towards a non-trivial 3D extension of the so-called lobe dynamics — a geometric, invariant-manifold-based framework which has been very successful in the analysis of Lagrangian transport in unsteady, two-dimensional fluid flows. In the developed algorithm, the instantaneous geometry of an invariant manifold is represented by an adaptively evolving triangular mesh with piecewise C² interpolating functions. The method employs an automatic mesh refinement which is coupled with adaptive vertex redistribution. A variant of the advancing front technique is used for remeshing, whenever necessary. Such an approach allows for computationally efficient determination of highly convoluted, evolving geometry of codimension one invariant manifolds in unsteady three-dimensional flows. We show that the developed method is capable of providing detailed information on the evolving Lagrangian flow structure in three dimensions over long periods of time, which is crucial for a meaningful 3D transport analysis.     

Gravity in hyperspin manifolds

We seek the dynamics of a Bergmann manifold: a manifold of dimensionn=N ² supporting a bundle of spinor spaces of dimensionN, and a map from the tangent spaces to the Hermitian spinor forms. Even though the spin-vector is the fundamental variable of the theory, every invariant analytic function depending on and its firstm derivatives alone can be expressed in terms of the chronometric tensorg and its firstm derivatives. Bergmann manifolds of dimensionn > 4 do not have invariant second-order equations for. We find a family of invariant actions which lead tonth-order quasilinear equations of motion on Bergmann manifolds and reduce to the Einstein-Hilbert action forn=2. The resulting gauge particles have spin, 1/2,1, 3/2, and 2.     

Morita Invariant Properties of Twisted Poisson Manifolds

In this letter, we study some invariants of integrable twisted Poisson manifolds under Morita equivalence. We show that two twisted symplectic manifolds are Morita equivalent if and only if their fundamental groups are isomorphic. Moreover, the first cohomology group of twisted Poisson manifolds is invariant under Morita equivalence. Finally, we prove that if two twisted Poisson manifolds are Morita equivalent, then their representation categories are equivalent.      

On the generalized Jacobi equation

The standard text-book Jacobi equation (equation of geodesic deviation) arises by linearizing the geodesic equation around some chosen geodesic, where the linearization is done with respect to the coordinates and the velocities. The generalized Jacobi equation, introduced by Hodgkinson in 1972 and further developed by Mashhoon and others, arises if the linearization is done only with respect to the coordinates, but not with respect to the velocities. The resulting equation has been studied by several authors in some detail for timelike geodesics in a Lorentzian manifold. Here we begin by briefly considering the generalized Jacobi equation on affine manifolds, without a metric; then we specify to lightlike geodesics in a Lorentzian manifold. We illustrate the latter case by considering particular lightlike geodesics (a) in Schwarzschild spacetime and (b) in a plane-wave spacetime.     

L 2-Cohomology of Hyperkähler Quotients

Using an argument of Jost and Zuo, we give a criterion which implies that the L ² harmonic forms on a complete noncompact hyperk?hler manifold lie in the middle dimension and are invariant under the isometry group. This is applied to various examples, and in particular gives a verification of some of the predictions of Sen on monopole moduli spaces. Received: 20 September 1999 / Accepted: 2 November 1999     

Explicit form and convergence of 1-differential formal deformations of the poisson Lie algebra

Introducing the notion of an admissible graded Lie subalgebra A of the Nijenhui-Richardson algebra A(V) of the vector space V, it is shown that each cohomology class of a subcomplex C _A of the Chevalley-Eilenberg complex (C ⁰ M), extends in a cononical way as a graded cohomology class of weight — 1 of A. Applying this when V is the space N of smooth functions of a smooth manifold M, shows that the de Rham cohomology of M is induced by the graded cohomology of weight — 1 of the Schouten graded Lie algebra of M. This allows us to construct explicitly all 1-differential, nc formal deformations of the Poisson bracket of a symplectic manifold. The construction also applies for an arbitrary Poisson manifold but leads to only part of these deformations when the structure degenerates, as shown by an example.     

Intersection properties of invariant manifolds in certain twist maps

We consider the spaceN ofC ² twist maps that satisfy the following requirements. The action is the sum of a purely quadratic term and a periodic potential times a constantk (hereafter called the nonlinearity). The potential restricted to the unit circle is bimodal, i.e. has one local minimum and one local maximum. The following statements are proven for maps inN with nonlinearityk large enough. The intersection of the unstable and stable invariant manifolds to the hyperbolic minimizing periodic points contains minimizing homoclinic points. Consider two finite pieces of these manifolds that connect two adjacent homoclinic minimizing points (hereafter called fundamental domains). We prove that all such fundamental domains have precisely one point in their intersection (the Single Intersection theorem). In addition, we show that limit points of minimizing points are recurrent, which implies that Aubry Mather sets (with irrational rotation number) are contained in diamonds formed by local stable and unstable manifolds of nearby minimizing periodic orbits (the Diamond Configuration theorem). Another corollary concerns the intersection of the minimax orbits with certain symmetry lines of the map.     

Superstring and graded grassmannians

We introduce the infinite dimensional graded grassmannian manifolds in terms of free field operators, and study their properties. We show the embedding of the graded DiffS¹/S¹ manifold in the graded grassmannians and discuss some physical applications.