Local Diffeomorphisms Research Articles

A Complete Classification for Pairs of Real Analytic Curves in the Complex Plane with Tangential Intersection

We study the group structure of centralizers of a holomorphic mapping that is tangent to the identity. We give a complete classification for pairs of real analytic curves intersecting tangentially via the Ecalle--Voronin theory. We also classify pairs of one-dimensional holomorphic involutions that have a high-order tangency, real analytic mappings on the real line, and exceptional non-Abelian subgroups of Cerveau and Moussu. Consequently, we prove the existence of holomorphic mappings z ? z + O(2) that are reversible by a formal antiholomorphic mapping, but not by any (convergent) antiholomorphic one. We also prove the existence of local real analytic mappings on the real line that are not equivalent with respect to any local real analytic diffeomorphism and yet whose complexifications are holomorphically equivalent.

Natural operators of smooth mappings of manifoldswith metric fields

In this paper we introduce the concepts of both a natural bundle and a natural operator generalized for the case of the category Mf m × Mf m of cartesian products of two manifolds and products of local diffeomorphisms. It is shown that any r-th order natural bundle over M × N has a structure of an associated bundle (P rM × P rN)Z G m r × G m r ]. We consider prolongations of such associated bundles and their reduction with respect to a chosen subgroup. The existence of a bijective correspondence between natural operators of order k and the equivariant mappings of the corresponding type fibers are proved. A basis of invariants of arbitrary order is constructed for natural operators of smooth mappings of manifolds endowed with metric fields or connections, with values in a natural bundle of order one.

On local diffeomorphisms ofR n that are injective

There are obtained conditions under which maps fromR n to itself are globally injective. In particular there are proved some partial results related to the Weak Markus-Yamabe Conjecture which states that if a vector field X:R n →R n has the property that, for allp ∈R n , all the eigenvalues ofD X (p) have negative real part, thenX has at most one singularity.

Dynamics of Local Automorphisms of Embedded CR-Manifolds

A classical theorem due to Wong [1] states that the ball is the unique strictly pseudoconvex bounded domain having a noncompact automorphism group. Rosay [2] extended this theorem to bounded domains such that an orbit accumulates at a strictly pseudoconvex domain at the boundary. Later, Efimov [3] got rid of the boundedness assumption. Schoen [4] and, independently, Spiro [5] proved the CR-version of this result, where the strict pseudoconvexity of the accumulation point of an orbit also plays a crucial role. In the present note, we suggest two local CR-versions of this result for hypersurfaces. These versions manifest different behavior depending on whether the Levi form is definite or not. Let M be a real-analytic Levi-nondegenerate hypersurface in some neighborhood U of the origin in C , n ≥ 1 . Denote by Aut0 M the group of local diffeomorphisms of M , that is, the group of germs of biholomorphisms taking M locally into itself and preserving the origin. A point q = 0 (treated as its own radius-vector −→ 0q) is said to be contractible if there is a sequence of local automorphisms Φn ∈ Aut0 M such that

The moduli space of torsion-free G2G2 structures

Let M be the moduli space of torsion-free G2 structures on a compact oriented G2 manifold M. The natural cohomology map π3:M→H3(M,ℝ) is known to be a local diffeomorphism [Compact Manifolds with Special Holonomy, Oxford University Press, 2000]. Let M1⊂M be the subset of G2 structures with volume (M)=1. We show every nonzero element of H4(M,ℝ)=H3(M,ℝ)* is a Morse function on M1 when composed with π3, and we compute its Hessian. The result implies a special case of Torelli’s theorem: if H1(M,ℝ)=0 and dimH3(M,ℝ)=2, the cohomology map π3:M→H3(M,ℝ) is one to one on each connected component of M. We formulate a compactness conjecture on the set of G2 structures of volume (M)=1 with bounded L2 norm of curvature. If this conjecture were true, it would imply that every connected component of M is contractible, and that every compact G2 manifold supports a G2 structure whose fundamental 4-form represents the negative of the (nonzero) first Pontryagin class of M. We also observe that when H1(M,ℝ)=0, and the volume of the torus H3(M,ℝ)/H3(M,ℤ) is constant along M1, the locus π3(M1)⊂H3(M,ℝ) is a hyperbolic affine sphere.

The moduli space of torsion-free [formula omitted] structures

Let M be the moduli space of torsion-free G 2 structures on a compact oriented G 2 manifold M . The natural cohomology map π 3 : M → H 3 ( M , ℝ ) is known to be a local diffeomorphism [Compact Manifolds with Special Holonomy, Oxford University Press, 2000]. Let M 1 ⊂ M be the subset of G 2 structures with volume ( M ) = 1 . We show every nonzero element of H 4 ( M , ℝ ) = H 3 ( M , ℝ ) * is a Morse function on M 1 when composed with π 3 , and we compute its Hessian. The result implies a special case of Torelli’s theorem: if H 1 ( M , ℝ ) = 0 and dim H 3 ( M , ℝ ) = 2 , the cohomology map π 3 : M → H 3 ( M , ℝ ) is one to one on each connected component of M . We formulate a compactness conjecture on the set of G 2 structures of volume ( M ) = 1 with bounded L 2 norm of curvature. If this conjecture were true, it would imply that every connected component of M is contractible, and that every compact G 2 manifold supports a G 2 structure whose fundamental 4-form represents the negative of the (nonzero) first Pontryagin class of M . We also observe that when H 1 ( M , ℝ ) = 0 , and the volume of the torus H 3 ( M , ℝ ) / H 3 ( M , ℤ ) is constant along M 1 , the locus π 3 ( M 1 ) ⊂ H 3 ( M , ℝ ) is a hyperbolic affine sphere.

Ergodicity of Lévy flows

We consider a stochastic differential equation (SDE) of jump type on a finite-dimensional connected smooth and oriented manifold M. The SDE is driven by a family ( ζ j , 1⩽ j⩽ n) of complete smooth vector fields on M and an n-dimensional Lévy process X with characteristics ( b, σ, ν), where b=( b j ) is a real vector, σ=( σ i j ) is a real matrix, 1⩽ j⩽ n, 1⩽ i⩽ m, m⩽ n and ν is a Lévy measure on R n−{0} . The induced flows of local diffeomorphisms ( γ t (., w), t⩾0) on M are assumed to be stochastically complete. We find a necessary and sufficient condition for irreducibility of the flows with respect to a volume measure. We apply this criterion to the Horizontal Lévy flows on the orthonormal frame bundle over a compact Riemannian manifold and prove that the spherical symmetric (isotropic) Lévy motion on M is ergodic with respect to the Riemannian measure on M.

Hyperbolic times: frequency versus integrability

We consider dynamical systems on compact manifolds, which are local diffeomorphisms outside an exceptional set (a compact submanifold). We are interested in analyzing the relation between the integrability (with respect to Lebesgue measure) of the first hyperbolic time map and the existence of positive frequency of hyperbolic times. We show that some (strong) integrability of the first hyperbolic time map implies the existence of a positive Lebesgue measure subset of points with positive frequency of hyperbolic times. We also present an example of a map with positive frequency of hyperbolic times at Lebesgue almost every point but whose first hyperbolic time map is not integrable with respect to the Lebesgue measure.

EXTENDED DIFFEOMORPHISM ALGEBRAS IN (QUANTUM) GRAVITATIONAL PHYSICS

We construct an explicit representation of the algebra of local diffeomorphisms of a manifold with realistic dimensions. This is achieved in the setting of a general approach to the (quantum) dynamics of a physical system which is characterized by the fundamental role assigned to a basic underlying symmetry. The mathematical formalism developed addresses the relevant gravitational notions by means of the addition of some extra structure. The specific manners in which this is accomplished, together with their corresponding physical interpretation, lead to different gravitational models. Distinct strategies are in fact briefly outlined, showing the versatility of the present conceptual framework.

Geodesic flow on the diffeomorphism group of the circle

We show that certain right-invariant metrics endow the infinite-dimensional Lie group of all smooth orientation-preserving diffeomorphisms of the circle with a Riemannian structure. The study of the Riemannian exponential map allows us to prove infinite-dimensional counterparts of results from classical Riemannian geometry: the Riemannian exponential map is a smooth local diffeomorphism and the length-minimizing property of the geodesics holds.

Two Characterizations of Diffeomorphisms of Euclidean Space

Two characterizations for a local diffeomorphism of ℝ n to be global one are given in terms of associated Wazewski equations. The two characterizations could be useful for the investigation of the Jacobian conjecture.

Noether currents for BF gravity

The conservation laws and currents for BF theory in four spacetime dimensions and BF gravity corresponding with their Lagrangian local gauge symmetries (local Lorentz transformations and diffeomorphisms) are computed using Noether's theorem. The physical meaning of the Noether charges that arise from the conservation laws is not explored.

Five-dimensional supergravity inN=1superspace

We give a formulation of linearized minimal five-dimensional supergravity in $\mathcal{N}=1$ superspace. Infinitesimal local 5D diffeomorphisms, local 5D Lorentz transformations, and local 5D supersymmetry are all realized as off-shell superfield transformations. Compactification on an ${S}^{1}{/Z}_{2}$ orbifold and couplings to brane-localized supermultiplets are very simple in this formalism. We use this to show that five-dimensional supergravity can naturally generate \ensuremath{\mu} and $B\ensuremath{\mu}$ terms of the correct size in gaugino- or radion-mediated supersymmetry breaking. We also include a self-contained review of linearized minimal 4D supergravity in superspace.

Non-zero Lyapunov exponents and uniform hyperbolicity

In this paper, we proved that if f is a C1 local diffeomorphism such that the Lyapunov exponents of every f invariant probability measure are positive, then f is uniformly expanding. We also present a version of this result for diffeomorphism with a non-uniformly hyperbolic set.

Constraint Nondegeneracy in Variational Analysis

This paper studies the sensitivity analysis of variational conditions defined over perturbed systems of finitely many nonlinear inequalities or equations, subject to additional fixed polyhedral constraints. If the system of constraints obeys a certain property called nondegeneracy, we show how to construct a local diffeomorphism of the feasible set to its tangent cone. Moreover, this diffeomorphism varies smoothly as the perturbation parameter changes. The original variational condition is then locally equivalent to a variational inequality defined over this (polyhedral convex) tangent cone. This extends stability results already known for variational inequalities over polyhedral convex sets to a substantially more general case. We also show that existence, local uniqueness, and Lipschitz continuity, as well as B-differentiability of the solution, can all be predicted from a single affine variational inequality that is easily computable in terms of the data of the unperturbed problem at the point in question.

Analogy between a 10D model for nonlinear wave–wave interaction in a plasma and the 3D Lorenz dynamics

This paper investigates nonlinear wave–wave interactions in a system that describes a modified decay instability and consists of three Langmuir and one ion-sound waves. As a means to establish that the underlying dynamics exists in a 3D space and that it is of the Lorenz-type, both continuous and discrete-time multivariable global models were obtained from data. These data were obtained from a 10D dynamical system that describes the modified decay instability obtained from Zakharov’s equations which characterise Langmuir turbulence. This 10D model is equivariant under a continuous rotation symmetry and a discrete order-2 rotation symmetry. When the continuous rotation symmetry is modded out, that is, when the dynamics are represented with the continuous rotation symmetry removed under a local diffeomorphism, it is shown that a 3D system may describe the underlying dynamics. For certain parameter values, the models, obtained using global modelling techniques from three time series from the 10D dynamics with the continuous rotation symmetry modded out, generate attractors which are topologically equivalent. These models can be simulated easily and, due to their simplicity, are amenable for analysis of the original dynamics after symmetries have been modded out. Moreover, it is shown that all of these attractors are topologically equivalent to an attractor generated by the well-known Lorenz system.

Monodromy of real isolated singularities

Complex conjugation on complex space permutes the level sets of a real polynomial function and induces involutions on level sets corresponding to real values. For isolated complex hypersurface singularities with real defining equation we show the existence of a monodromy vector field such that complex conjugation intertwines the local monodromy diffeomorphism with its inverse. In particular, it follows that the geometric monodromy is the composition of the involution induced by complex conjugation and another involution. This topological property holds for all isolated complex plane curve singularities. Using real morsifications, we compute the action of complex conjugation and of the other involution on the Milnor fiber of real plane curve singularities.

Markov structures for non-uniformly expanding maps on compact manifolds in arbitrary dimension

We consider non-uniformly expanding maps on compact Riemannian manifolds of arbitrary dimension, possibly having discontinuities and/or critical sets, and show that under some general conditions they admit an induced Markov tower structure for which the decay of the tail of the return time function can be controlled in terms of the time generic points needed to achieve some uniform expanding behavior. As a consequence we obtain some rates for the decay of correlations of those maps and conditions for the validity of the Central Limit Theorem. 1. Dynamical and geometrical assumptions Let M be a compact Riemannian manifold of dimension d ≥ 1 with a normalized Riemannian volume | · |, which we call Lebesgue measure. Let f : M → M be a C local diffeomorphism for all x ∈ M C, where C is some critical set, which may include points at which the derivative Dfx is degenerate, as well as points of discontinuity and points at which the derivative is infinite. We assume the following natural non-degeneracy condition on C, which generalizes the notion of non-flat critical points for smooth one-dimensional maps. Definition 1. The critical set C ⊂ M is non-degenerate if |C| = 0 and there is a constant β > 0 such that for every x ∈M C we have dist(x, C) . ‖Dfxv‖/‖v‖ . dist(x, C)−β for all v ∈ TxM , and the functions log detDf and log ‖Df−1‖ are locally Lipschitz with Lipschitz constant . dist(x, C)−β . We now state our two dynamical assumptions: the first is on the growth of the derivative and the second is on the approach rate of orbit to the critical set. Notice that for a linear map A, the condition ‖A‖ > 1 only provides information about the existence of some expanded direction, whereas the condition ‖A−1‖ 0) implies that every direction is expanded. Received by the editors November 5, 2002. 2000 Mathematics Subject Classification. Primary 37D20, 37D50, 37C40. Work carried out at the Federal University of Bahia, University of Porto and Imperial College, London. Partially supported by CMUP, PRODYN, SAPIENS and UFBA. c ©2003 American Mathematical Society

On the fiber product preserving gauge bundle functors on vector bundles

We present a complete description of all fiber product preserving gauge bundle functors $F$ on the category ${\mathcal V}{\mathcal B}_m$ of vector bundles with $m$-dimensional bases and vector bundle maps with local diffeomorphisms as base maps. Some co

Structural identifiability of non-linear systems using linear/non-linear splitting

In this paper, the local state space isomorphism theorem for non-linear systems is used to analyse structural identifiability. In particular it is shown that, under certain restrictions, it is possible to perform a linear/non-linear splitting of the analysis. The relatively straightforward linear analysis then restricts the class of local diffeomorphic transformations as given by the non-linear state space isomorphism theorem. This, in turn, leads to possible simplifications to the subsequent non-linear analysis by providing an efficient means for calculating the local state diffeomorphism.