Showing papers on "Equivariant map published in 1988"

Structurally stable heteroclinic cycles

Abstract: This paper describes a previously undocumented phenomenon in dynamical systems theory; namely, the occurrence of heteroclinic cycles that are structurally stable within the space of Cr vector fields equivariant with respect to a symmetry group. In the space X(M) of Cr vector fields on a manifold M, there is a residual set of vector fields having no trajectories joining saddle points with stable manifolds of the same dimension. Such heteroclinic connections are a structurally unstable phenomenon [4]. However, in the space XG(M) ⊂ X(M) of vector fields equivariant with respect to a symmetry group G, the situation can be quite different. We give an example of an open set U of topologically equivalent vector fields in the space of vector fields on ℝ3 equivariant with respect to a particular finite subgroup G ⊂ O(3) such that each X ∈ U has a heteroclinic cycle that is an attractor. The heteroclinic cycles consist of three equilibrium points and three trajectories joining them.

Periodic solutions near equilibria of symmetric Hamiltonian systems

Abstract: We consider the effects of symmetry on the dynamics of a nonlinear hamiltonian system invariant under the action of a compact Lie group T, in the vicinity of an isolated equilibrium: in particular, the local existence and stability of periodic trajectories. The main existence result, an equivariant version of the Weinstein—Moser theorem, asserts the existence of periodic trajectories with certain prescribed symmetries Z c T x S 1 , independently of the precise nonlinearities. We then describe the constraints put on the Floquet operators of these periodic trajectories by the action of T. This description has three ingredients: an analysis of the linear symplectic maps that commute with a symplectic representation, a study of the momentum mapping and its relation to Floquet multipliers, and Krein Theory. We find that for some 2, which we call cylospetral , all eigenvalues of the Floquet operator are forced by the group action to lie on the unit circle; that is, the periodic trajectory is spectrally stable. Similar results for equilibria are described briefly. The results are applied to a number of simple examples such as T = SO(2), 0(2 ), Z n , D n , SU (2) ; and also to the irreducible symplectic actions of O(3) on spaces of complex spherical harmonics, modelling oscillations of a liquid drop.

On a notion of simplicial depth

Abstract: For a distribution F on Rp and a point x in Rp the simplicial depth D(x), which is the probability that x be inside a random simplex whose vertices are p + 1 independent observations from F, is introduced. D(x) can be viewed as a measure of depth of the point x relative to F, and its empirical version gives rise to a natural ordering of the data points from the center outward. This ordering provides an approach for detecting outliers in a multivariate data cloud and leads to the introduction of affine equivariant multivariate generalizations of the univariate sample median and L-statistics. This sample median is shown to be consistent for the center of any angularly symmetric distribution.

Convex bodies and algebraic geometry : an introduction to the theory of toric varieties

Abstract: 1. Fans and Toric Varieties.- 1.1 Strongly Convex Rational Polyhedral Cones and Fans.- 1.2 Toric Varieties.- 1.3 Orbit Decomposition, Manifolds with Corners and the Fundamental Group.- 1.4 Nonsingularity and Compactness.- 1.5 Equivariant Holomorphic Maps.- 1.6 Low Dimensional Toric Singularities and Finite Continued Fractions.- 1.7 Birational Geometry of Toric Varieties.- 2. Integral Convex Polytopes and Toric Projective Varieties.- 2.1 Equivariant Line Bundles, Invariant Cartier Divisors and Support Functions.- 2.2 Cohomology of Compact Toric Varieties.- 2.3 Equivariant Holomorphic Maps to Projective Spaces.- 2.4 Toric Projective Varieties.- 2.5 Mori's Theory and Toric Projective Varieties.- 3. Toric Varieties and Holomorphic Differential Forms.- 3.1 Differential Forms with Logarithmic Poles.- 3.2 Ishida's Complexes.- 3.3 Compact Toric Varieties and Holomorphic Differential Forms.- 3.4 Automorphism Groups of Toric Varieties and the Cremona Groups.- 4. Applications.- 4.1 Periodic Continued Fractions and Two-Dimensional Toric Varieties..- 4.2 Cusp Singularities.- 4.3 Compact Quotients of Toric Varieties.- Appendix. Geometry of Convex Sets.- A.1 Convex Polyhedral Cones.- A.2 Convex Polyhedra.- A.3 Support Functions.- A.4 The Mixed Volume of Compact Convex Sets.- A.5 Morphology for Convex Polytopes.- References.

Equivariant minimax and minimal surfaces in geometric three-manifolds

Abstract: Minimal surfaces in Riemannian three-dimensional manifolds have played a major role in recent studies of the geometry and topology of 3-manifolds. In particular, stable and least area minimal surfaces have been used extensively [SY, MSY, HS]. On the other hand, explicit examples of unstable minimal surfaces have rarely been given. For the 3-sphere S with the standard metric of constant sectional curvature, there is the classical paper of Lawson [LH], showing that closed orientable surfaces of every genus occur as (unstable) minimal surfaces embedded in 5 3 . More recently, Karcher, Pinkall, and Sterling [KPS] have constructed several new examples in S. We have discovered new infinite families of minimal surfaces in S. More generally, we announce a number of new finite and infinite families of embedded minimal surfaces in geometric 3-manifolds, using the minimax procedure described below. Geometric structures on 3-manifolds were introduced by Thurston [TW]. (See also the excellent survey by Scott [SP1].) There are eight geometries: R , S , S x R, Nil, H 2 x R, S L ^ R ) , Solv, and H . A geometric structure on a 3-manifold S is a representation of E as a quotient of one of the above eight spaces divided out by a covering transformation group acting isometrically. Equivalently, E is locally isometric to one of these spaces, with its natural homogeneous space structure. The first six of these geometries give Seifert fiber spaces and we are mainly interested in such examples. The SO(2)-isometry actions associated with most Seifert fiber spaces yield infinite classes of embedded minimal surfaces. Note that interesting examples can also be obtained in the other two geometries; e.g., in hyperbolic geometry H 3 [PR3, §2]. The minimax procedure has proved to be a versatile and powerful means of constructing unstable minimal surfaces in 3-manifolds. For basic details of this technique, see [PJ, SS, P R l , and PR2]. Suppose that G is a finite group of isometries acting on a closed oriented Riemannian 3-manifold E. Assume that A is a Heegaard surface in E; i.e., the closures of the components of E ~ A are handlebodies K and K'. Assume furthermore that A is G-equivariant; i.e., gA = A for all g E G. We consider one-parameter smooth families At, t € [0,1], sweeping out E and having the following properties: Ao and Ai are graphs; At is isotopic to A for all 0 < t < 1; At is G-equivariant for all t; the handlebody Kt is chosen so that the orientation on At is induced from that on

The RO(G)-graded equivariant ordinary cohomology of complex projective spaces with linear ℤ/p actions

Abstract: INTRODUCTION. If X is a CW complex with cells only in even dimensions and R is a ring, then, by an elementary result in cellular cohomology theory, the ordinary eohomology H*(X;R) of X with R coefficients is a free, 7/-graded R-module. Since this result is quite useful in the study of well-behaved complex manifolds like projective spaces or Grassmannians, it would be nice to be able to generalize it to equivariant ordinary eohomology. The result does generalize in the following sense. Let G be a finite group, X be a G-CW complex (in the sense of [MAT, LMSM]), and R be a ring-valued eontravariant coefficient system JILL]. Then the G-equivariant ordinary Bredon cohomology H*(X; R) of X with R coefficients may be regarded as a coefficient system. If the cells of X are all even dimensional, then H*(X;R) is a free module over R in the sense appropriate to coefficient systems. Unfortunately, this theorem does not apply to complex projective spaces or complex Grassmannians with any reasonable nontrivial G-action because these spaces do not have the right kind of G-CW structure. In fact, if G is ~/p, for any prime p, and r / is a nontrivial irreducible complex G-representation, then the theorem does not apply to S ~, the one-point compactification of r 1. Moreover, the 2~-graded Bredon cohomology of S n with coefficients in the Burnside ring coefficient system is quite obviously not free over the coefficient system.

Intersection homology and torus actions

Abstract: in the Bialynicki-Birula decomposition of X (see 15, 6, 10, 13]). The same formula holds for appropriate integers my when C* is replaced by a torus T=(C*)r. In [9] it is shown that the formula (0.1) is valid even when X is singular, provided that the Bialynicki-Birula decomposition is "good". The aim of this paper is to generalize (0.1) to the case when X is singular in a different way, which involves replacing ordinary homology by intersection homology (with respect to the middle perversity). However only rational coefficients are considered. When X is nonsingular its intersection homology and ordinary homology coincide, but when X is singular its intersection homology behaves better in many respects than its ordinary cohomology. It is shown that when X is singular, just as when X is nonsingular, the rational intersection homology of X is determined by the action of T on an arbitrarily small neighborhood of the fixed point set XT . As might be expected, the formula (0.1) does not carry over directly when intersection homology replaces ordinary homology. The terms Hi_2m (F ; Z) appearing in the right-hand side are replaced by hypercohomology groups of certain complexes of sheaves over the F which depend upon how the F meet the singularities of X (see They y orem 2.3). ? 1 of this paper contains a review of a proof of (0.1) for rational coefficients which uses equivariant Morse theory. In ?2 it is shown how this proof can be extended to apply to singular varieties when intersection homology is used. The

Torus-equivariant vector bundles on projective spaces

Abstract: Introduction For a free Z-module N of rank n, let T = TN be an ^-dimensional algebraic torus over an algebraically closed field k defined by N. Let X = TN emb (A) be a smooth complete toric variety defined by a fan Δ (cf. [6]). Then T acts algebraically on X A vector bundle E on X is said to be an equivariant vector bundle, if there exists an isomorphism ft: t*E-+E for each /^-rational point t in T, where t: Z > X i s the action of t Equivariant vector bundles have T-linearizations (see Definition 1.2 and [2], [4]), hence we consider T-linearizsd vector bundles. The 72-dimensional projective space P has a natural action of T and can be regarded as a toric variety. In [4], we classified indecomposable equivariant vector bundles of rank two on F. When n > 2, Hartshorne [3] constructed vector bundles of rank two from codimension two subschemes satisfying certain conditions. Bertin and Elencwajg [2] then used this method to construct equivariant vector bundles of rank two on P and showed that there exist no indecomposable equivariant vector bundles of rank two on P which are obtained in this way. In this paper, we generalize our method in [4] to show that there exist no indecomposable equivariant vector bundles of rank r (1 < r < ή) on P (Corollary 3.5) and that indecomposable equivariant vector bundles of rank n on P are isomorphic to E(d) or E*(d) for some integer d, where E is defined by an exact sequence

Fixed-Point Theory of Parametrized Equivariant Maps

Abstract: Preliminaries on group actions.- Equivariant vertical euclidean neighbourhood retracts.- The fixed point index of equivariant vertical maps.

Character, Character Cycle, Fixed Point Theorem and Group Representations

Abstract: Publisher Summary This chapter explores that among many methods to derive Weyl's character formula, there is an application of the fixed point theorem to a line bundle on the flag variety. Namely, any finite-dimensional irreducible representation of a reductive group G is obtained as the cohomology group of an equivariant line bundle on the flag variety. Therefore, the trace of the action of an element g of G is obtained as the sum of the contributions at each fixed point. On the other hand, Harish-Chandra defined the character of an infinite-dimensional representation of a real semisimple group G R as an invariant eigendistribution. The chapter presents a character formula in terms of the geometry of flag manifold as a conjecture and presents a proof of it for discrete series. The correspondence of Harish-Chandra modules and K -equivariant sheaves is completed by adding representations of G R and G R -equivariant sheaves. Thereafter, the character is calculated from G R -equivariant sheaves.

Equivariant theory of Yang-Mills connections over Riemannian manifolds of cohomogeneity one

Abstract: On etudie des connexions de Yang-Mills invariantes de groupe sur une variete de Riemann avec une grande action de groupe et on montre un analogue champ de jauge de la theorie invariante de la metrique d'Einstein et des sous-varietes minimales d'espaces symetriques

Stable maps into free g-spaces

Abstract: In this paper we introduce a systematic method for calculating the group of stable equivariant maps (X, Y\G (3, 18) into a G-free space or spectrum Y In fact the method applies without restriction on X whenever G is a p-group and Y is p-complete and satisfies standard finiteness assumptions The method is an Adams spectral sequence based on a new equivariant coho- mology theory c* (X) which we introduce in §1 This spectral sequence is quite calculable and provides a natural generalisation of the classical Adams spectral sequence based on ordinary modp cohomology It also geometrically realises certain inverse limits of nonequivariant Adams spectral sequences which have been useful in the study of the Segal conjecture (19, 5, 21, 9)

The recognition problem for equivariant singularities

Abstract: Singularity theory involves the classification of singularities up to some equivalence relation. The solution to a particular recognition problem is the characterisation of an equivalence class in terms of a finite number of polynomial equalities and inequalities to be satisfied by the Taylor coefficients of a singularity. The recognition problem can be simplified by decomposing the group of equivalences into a unipotent group and a group of matrices. Building upon results of Bruce and co-workers, (1985), the author shows for contact equivalence that in many cases the unipotent problem can be solved by just using linear algebra. He gives a necessary and sufficient condition for this, namely that the tangent space be invariant under unipotent equivalence. He then develops efficient methods for checking whether the tangent space is invariant, and gives several examples drawn from equivariant bifurcation theory.

The equivariant Conley index and bifurcations of periodic solutions of Hamiltonian systems

Abstract: An equivariant version of Conley's homotopy index theory for flows is described and used to find periodic solutions of a Hamiltonian system locally near an equilibrium point which is at resonance.

Minimum Risk Equivariant Estimators of Percentiles in Location­Scale Families of Distributions:

Abstract: The paper considers minimum risk equivariant estimation of percentiles in location­scale families of distributions. A general theory is developed, and it is then illustrated by the uniform, exponen...

Stability of periodic solutions in symmetric Hopf bifurcation

Abstract: The equivariant Hopf bifurcation theorem of Golubitsky and Stewart (1985) gives sufficient conditions, analogous to the classical Hopf bifurcation theorem, for the existence of periodic solutions to ordinary differential equations with symmetry. They show how to compute the asymptotic stability of these solutions on the assumption that the vector field is in Birkhoff normal form to infinite order. By studying perturbations of the Floquet operator we prove that it is sufficient for the vector to be in Birkhoff normal form to suitably high finite order, which can be achieved by an appropriate change of coordinates. Thus in principle the stability can be calculated without assuming Birkhoff normal form. A similar result has been obtained by Taliaferro (1987) using different methods.