Local Diffeomorphisms Research Articles

The projective geometric theory of systems of second-order differential equations: straightening and symmetry theorems

In the framework of the projective geometric theory of systems of differential equations, which is being developed by the authors, conditions which ensure that a family of graphs of solutions of a system of second-order ordinary differential equations with unknown functions can be straightened (that is, transformed into a family of straight lines) by means of a local diffeomorphism of the variables of the system which takes it to the form (straightens the system) are investigated. It is shown that the system to be straightened must be cubic with respect to the derivatives of the unknown functions. Necessary and sufficient conditions for straightening the system are found, which have the form of differential equations for the coefficients of the system or are stated in terms of symmetries of the system. For the system consists of a single equation , and the tests obtained reduce to the conditions for straightening this equations which were derived by Lie in 1883.Bibliography: 34 titles.

Global inversion and covering maps on length spaces

In order to obtain global inversion theorems for mappings between length metric spaces, we investigate sufficient conditions for a local homeomorphism to be a covering map in this context. We also provide an estimate of the domain of invertibility of a local homeomorphism around a point, in terms of a kind of lower scalar derivative. As a consequence, we obtain an invertibility result using an analog of the Hadamard integral condition in the frame of length spaces. Some applications are given to the case of local diffeomorphisms between Banach–Finsler manifolds. Finally, we derive a global inversion theorem for mappings between stratified groups.

On the approximation of Lyapunov exponents and a question suggested by Anatole Katok

Let be a Hölder-continuous linear cocycle whose driving semiflow preserves a probability μ on a compact metric space X. Using the concept of two-sided quasi-Pesin orbits introduced in this paper, we show that the Lyapunov characteristic spectrum of μ can be approached arbitrarily by that of periodic points of f. So, if all periodic points of f have only positive Lyapunov exponents and such exponents are uniformly bounded away from zero, then μ also has only positive exponents. In our arguments, the exponential closing property of the driving semiflow is a basic condition, and it is easy to check that every C1-expanding map of a closed manifold obeys this closing property.Consequently, if f is a C1+Hölder local diffeomorphism of a closed manifold and if it is Hölder conjugated to some C1-expanding map, then f is itself expanding. This gives a positive answer to a question suggested by Anatole Katok.

EXISTENCE OF NATURAL AND CONFORMALLY INVARIANT QUANTIZATIONS OF ARBITRARY SYMBOLS

A quantization can be seen as a way to construct a differential operator with prescribed principal symbol. The map from the space of symbols to the space of differential operators is moreover required to be a linear bijection. In general, there is no natural quantization procedure, that is, spaces of symbols and of differential operators are not equivalent, if the action of local diffeomorphisms is taken into account. However, considering manifolds endowed with additional structures, one can seek for quantizations that depend on this additional structure and that are natural if the dependence with respect to the structure is taken into account. The existence of such a quantization was proved recently in a series of papers in the context of projective geometry. Here, we show that the construction of the quantization based on Cartan connections can be adapted from projective to pseudo-conformal geometry to yield the natural and conformally invariant quantization for arbitrary symbols, outside some critical situations.

Points and curves in the Monster tower

Cartan introduced the method of prolongation which can be applied either to manifolds with distributions (Pfaffian systems) or integral curves to these distributions. Repeated application of prolongation to the plane endowed with its tangent bundle yields the Monster tower, a sequence of manifolds, each a circle bundle over the previous one, each endowed with a rank 2 distribution. In an earlier paper (2001), the authors proved that the problem of classifying points in the Monster tower up to symmetry is the same as the problem of classifying Goursat distribution flags up to local diffeomorphism. The first level of the Monster tower is a three-dimensional contact manifold and its integral curves are Legendrian curves. The philosophy driving the current work is that all questions regarding the Monster tower (and hence regarding Goursat distribution germs) can be reduced to problems regarding Legendrian curve singularities.

Foliations and global inversion

We consider topological conditions under which a locally invertible map admits a global inverse. Our main theorem states that a local diffeomorphism f: M → ℝ^n is bijective if and only if H_{n − 1}(M) = 0 and the pre-image of every affine hyperplane is non-empty and acyclic. The proof is based on some geometric constructions involving foliations and tools from intersection theory. This topological result generalizes in finite dimensions the classical analytic theorem of Hadamard–Plastock, including its recent improvement by Nollet–Xavier. The main theorem also relates to a conjecture of the aforementioned authors, involving the well-known Jacobian conjecture in algebraic geometry.

Robust ergodic properties in partially hyperbolic dynamics

We study ergodic properties of partially hyperbolic systems whose central direction is mostly contracting. Earlier work of Bonatti and Viana (2000) about existence and finitude of physical measures is extended to the case of local diffeomorphisms. Moreover, we prove that such systems constitute a C 2 C^2 -open set in which statistical stability is a dense property. In contrast, all mostly contracting systems are shown to be stable under small random perturbations.

Embedding smooth diffeomorphisms in flows

In this paper we study the problem on embedding germs of smooth diffeomorphisms in flows in higher dimensional spaces. First we prove the existence of embedding vector fields for a local diffeomorphism with its nonlinear term a resonant polynomial. Then using this result and the normal form theory, we obtain a class of local C k diffeomorphisms for k ∈ N ∪ { ∞ , ω } which admit embedding vector fields with some smoothness. Finally we prove that for any k ∈ N ∪ { ∞ } under the coefficient topology the subset of local C k diffeomorphisms having an embedding vector field with some smoothness is dense in the set of all local C k diffeomorphisms.

Embedding diffeomorphisms in flows in Banach spaces

AbstractThis paper concerns the problem of embedding, in the flow of an autonomous vector field, a local diffeomorphism near a hyperbolic fixed point in a Banach space. To solve the problem, we first extend Floquet theory to Banach spaces, and then prove that two C∞ hyperbolic diffeomorphisms are formally equivalent if and only if they are C∞-equivalent. The latter result is a version, in the Banach space context, of a classical theorem by Chen.

Local research of manifolds generated by families of self-adjoint operators

We consider V.I. Arnold’s manifold of self-adjoint operators with fixed multiplicity of eigenvalues and K. Uhlenbeck’s manifold of eigenvectors. Our aim is to consider the local analysis and the connection between these manifolds. We present the topological description of the spectrum perturbation problem, specifically the finite-multiple eigenvalue splitting problem. For investigation of manifolds, we use the local diffeomorphism introduced by D. Fujiwara, M. Tanikawa, and Sh. Yukita.

Birationality of étale maps via surgery

We use a counting argument and surgery theory to show that if $D$ is a sufficiently general algebraic hypersurface in $\Bbb C^n$, then any local diffeomorphism $F:X \to \Bbb C^n$ of simply connected manifolds which is a $d$-sheeted cover away from $D$ has degree $d=1$ or $d=\infty$ (however all degrees $d > 1$ are possible if $F$ fails to be a local diffeomorphism at even a single point). In particular, any \'etale morphism $F:X \to \Bbb C^n$ of algebraic varieties which covers away from such a hypersurface $D$ must be birational.

Regions Where the Exponential Map at Regular Points of Sub-Riemannian Manifolds is a Local Diffeomorphism

For a k-step sub-Riemannian manifold which admits a bracket generating vector at a point, we describe a region near the point where the exponential map is a local diffeomorphism. This is proved by taking the Taylor series of the exponential map and calculating the first nonzero term, which has order $ 2{\left( {{{\mathcal{D}}}_{{{\mathcal{H}}}} - n} \right)} $ , where n is the topological dimension and $ {{\mathcal{D}}}_{{{\mathcal{H}}}} $ is the Hausdorff dimension of the metric space associated to the sub-Riemannian manifold.

Optimal Extended Jacobian Inverse Kinematics Algorithms for Robotic Manipulators

Extended Jacobian inverse kinematics algorithms for redundant robotic manipulators are defined by combining the manipulator's kinematics with an augmenting kinematics map in such a way that the combination becomes a local diffeomorphism of the augmented taskspace. A specific choice of the augmentation relies on the optimal approximation by the extended Jacobian of the Jacobian pseudoinverse (the Moore-Penrose inverse of the Jacobian). In this paper, we propose a novel formulation of the approximation problem, rooted conceptually in the Riemannian geometry. The resulting optimality conditions assume the form of a Poisson equation involving the Laplace-Beltrami operator. Two computational examples illustrate the theory.

Hopf algebras of primitive Lie pseudogroups and Hopf cyclic cohomology

We associate to each infinite primitive Lie pseudogroup a Hopf algebra of ‘transverse symmetries,’ by refining a procedure due to Connes and the first author in the case of the general pseudogroup. The affiliated Hopf algebra can be viewed as a ‘quantum group’ counterpart of the infinite-dimensional primitive Lie algebra of the pseudogroup. It is first constructed via its action on the étale groupoid associated to the pseudogroup, and then realized as a bicrossed product of a universal enveloping algebra by a Hopf algebra of regular functions on a formal group. The bicrossed product structure allows to express its Hopf cyclic cohomology in terms of a bicocyclic bicomplex analogous to the Chevalley–Eilenberg complex. As an application, we compute the relative Hopf cyclic cohomology modulo the linear isotropy for the Hopf algebra of the general pseudogroup, and find explicit cocycle representatives for the universal Chern classes in Hopf cyclic cohomology. As another application, we determine all Hopf cyclic cohomology groups for the Hopf algebra associated to the pseudogroup of local diffeomorphisms of the line.

Flow prolongation of some tangent valued forms

We study the prolongation of semibasic projectable tangent valued k-forms on fibered manifolds with respect to a bundle functor F on local isomorphisms that is based on the flow prolongation of vector fields and uses an auxiliary linear r-th order connection on the base manifold, where r is the base order of F. We find a general condition under which the Frolicher-Nijenhuis bracket is preserved. Special attention is paid to the curvature of connections. The first order jet functor and the tangent functor are discussed in detail. Next we clarify how this prolongation procedure can be extended to arbitrary projectable tangent valued k-forms in the case F is a fiber product preserving bundle functor on the category of fibered manifolds with m-dimensional bases and local diffeomorphisms as base maps.

Spontaneous Lorentz and diffeomorphism violation, massive modes, and gravity

Theories with spontaneous local Lorentz and diffeomorphism violation contain massless NambuGoldstone modes, which arise as field excitations in the minimum of the symmetry-breaking potential. If the shape of the potential also allows excitations above the minimum, then an alternative gravitational Higgs mechanism can occur in which massive modes involving the metric appear. The origin and basic properties of the massive modes are addressed in the general context involving an arbitrary tensor vacuum value. Special attention is given to the case of bumblebee models, which are gravitationally coupled vector theories with spontaneous local Lorentz and diffeomorphism violation. Mode expansions are presented in both local and spacetime frames, revealing the Nambu-Goldstone and massive modes via decomposition of the metric and bumblebee fields, and the associated symmetry properties and gauge fixing are discussed. The class of bumblebee models with kinetic terms of the Maxwell form is used as a focus for more detailed study. The nature of the associated conservation laws and the interpretation as a candidate alternative to Einstein-Maxwell theory are investigated. Explicit examples involving smooth and Lagrange-multiplier potentials are studied to illustrate features of the massive modes, including their origin, nature, dispersion laws, and effects on gravitational interactions. In the weak static limit, the massive mode and Lagrange-multiplier fields are found to modify the Newton and Coulomb potentials. The nature and implications of these modifications are examined.

On Local Integrability Conditions of Jet Groupoids

A jet groupoid ℛq over a manifold X is a special Lie groupoid consisting of q-jets of local diffeomorphisms X→X. As a subbundle of Jq(X,X), a jet groupoid can be considered as a system of nonlinear partial differential equations (PDE). This leads to the question if ℛq is formally integrable. On the other hand, each jet groupoid is the symmetry groupoid of a geometric object, which is a section ω of a natural bundle ℱ. Using the jet groupoids, we give a local characterisation of formal integrability for transitive jet groupoids in terms of their corresponding geometric objects.

Curves in a foliated plane

The paper is devoted to the classification of nonsingular and singular plane curve germs with respect to the group of local diffeomorphisms preserving the foliation of the plane by the phase curves of a fixed vector field, either nonsingular or singular. We define the multiplicity of a pair consisting of a plane curve and a vector field and prove an analog of the Tougeron theorem on finite determinacy. It leads, almost immediately, to a number of classification results; a part of them is contained in the work.

Market crashes, speculation and learning in financial markets

A natural conjecture is that if agents’ beliefs are almost correct then equilibrium prices should be close to rational expectations prices. Sandroni (J Econ Theory 82:1–18, 1998) gives a counterexample in an economy with sunspots and complete markets. We extend Sandroni’s result by showing that the conjecture is generically true for economies with complete markets. We consider a standard General Equilibrium model with large but finite horizon and complete markets. We show that, for almost every such economy, if conditional beliefs eventually become correct along a path of events then equilibrium prices of assets traded along this path converge to rational expectations equilibria in the sup-norm. Moreover, we establish that, generically, there exist along any such path local diffeomorphisms between individual beliefs and equilibrium prices.

Lefschetz fixed point theory on fibrations

For a fibre preserving map ϕ: E → E on a fibration (E, π, B), we construct a grading preserving map T(ϕ, π) between H*(E) and H*(B) that generalizes the Lefschetz number. If T(ϕ, π) is an isomorphism between H0(E) and H0(B), then π restricts to a surjective local diffeomorphism on each connected component of the fixed point set of ϕ under a transversality condition. This yields a characterization for the bundle H → G → G/H to be trivial when π1 (G/H) = 0.