Asymptotic Invariants Research Articles

Asymptotic invariants of base loci

Le but de cet article est de definir et d'etudier systematiquement quelques invariants asymptotiques associes aux lieux de base des fibres en droites sur les varietes projectives lisses. Le comportement fonctionnel de ces invariants est lie au comportement ensembliste des lieux de base.

Zariski chambers, volumes, and stable base loci

The purpose of this paper is to investigate the behaviour of certain asymptotic invariants of line bundles on projective surfaces. In particular, we describe the volume of line bundles and their destabilizing numbers.

Asymptotic behaviour and the moduli space of doubly-periodic instantons

We study doubly-periodic instantons, i.e. instantons on the product of a 1-dimensional complex torus T with a complex line ³, with quadratic curvature decay. We determine the asymptotic behaviour of these instantons, constructing new asymptotic invariants. We show that the underlying holomorphic bundle extends to T2À1. The converse statement is also true, namely a holomorphic bundle on T2À1 which is flat on the torus at infinity, and satisfies a stability condition, comes from a doubly-periodic instanton. Finally, we study the hyperkähler geometry of the moduli space of doubly-periodic instantons, and prove that the Nahm transform previously defined by the second author is a hyperkähler isometry with the moduli space of certain meromorphic Higgs bundles on the dual torus.

Aubry-mather theory and the inverse spectral problem for planar convex domains

The inverse spectral problem is concerned with the question to what extent the spectrum of a domain determines its geometry. We find that, associated to a convex domain Ω in ℝ2, there is a convexfunction which is a length spectrum invariant under continuous deformations. It includes several geometric quantities, such as the lengths and Lazutkin parameters of caustics, as well as the asymptotic invariants discovered by Marvizi and Melrose. Via a Poisson relation, we also find invariants determined by the Laplace spectrum of Ω.

On Asymptotic Isoperimetric Constant of Tori

In this note we continue the study of asymptotic invariants of Riemannian tori (e.g. see [BuI] and references there). By asymptotic invariants we mean invariants which do not change under passing to nite covers. In [BuI] we show that the asymptotic volume growth of a Riemannian torus is at least as fast as that of a flat one. One may ask what are the possible values of \asymptotic isoperimetric constants for such metrics (see the denition below). We show that the asymptotic isoperimetric constant of a conformally flat torus is no less than that of a flat one, while for general metrics (in dimensions higher than 2) this constant may be arbitrarily small. Let (M;g) be the universal cover of a Riemannian n-torus. Definition. We dene the asymptotic isoperimetric constant (M;g )o f (M;g )b y

Geometry of growth: approximation theorems for $L^2$ invariants

In this paper we study the problem of approximation of the $L^2$-topological invariants by their finite dimensional analogues. We obtain generalizations of the theorem of Luck, dealing with towers of finitely sheeted normal coverings. We prove approximation theorems, establishing relations between the homological invariants, corresponding to infinite dimensional representations and sequences of finite dimensional representations, assuming that their normalized characters converge. Also, we find an approximation theorem for residually finite $p$-groups ($p$ is a prime), where we use the homology with coefficients in a finite field $\fp$. We view sequences of finite dimensional flat bundles of growing dimension as examples of growth processes. We study a von Neumann category with a Dixmier type trace, which allows to describe the asymptotic invariants of growth processes. We introduce a new invariant of torsion objects, the torsion dimension. We show that the torsion dimension appears in general as an additional correcting term in the approximation theorems; it vanishes under some arithmeticity assumptions. We also show that the torsion dimension allows to establish non-triviality of the Grothendieck group of torsion objects.

The normalized cyclomatic quotient associated with presentations of finitely generated groups

Given a presentation of ann-generated group, we define the normalized cyclomatic quotient (NCQ) of it, which gives a number between 1−n and 1. It is computed through an investigation of the asymptotic behavior of a kind of an “average rank”, or more precisely the quotient of the rank of the fundamental group of a finite subgraph of the corresponding Cayley graph by the size of the subgraph. In many ways (but not always) the NCQ behaves similarly to the behavior of the spectral radius of a symmetric random walk on the graph. In particular, it characterizes amenable groups. For some types of groups, like finite, amenable or free groups, its value equals that of the Euler characteristic of the group. We give bounds for the value of the NCQ for factor groups and subgroups, and formulas for its value on direct and free products. Some other asymptotic invariants are also discussed.

Book Review: Geometric group theory, Vol. 2: Asymptotic invariants of infinite groups

Book Review: Geometric group theory, Vol. 2: Asymptotic invariants of infinite groups

Introduction to the modern theory of dynamical systems

Part I. Examples and Fundamental Concepts Introduction 1. First examples 2. Equivalence, classification, and invariants 3. Principle classes of asymptotic invariants 4. Statistical behavior of the orbits and introduction to ergodic theory 5. Smooth invariant measures and more examples Part II. Local Analysis and Orbit Growth 6. Local hyperbolic theory and its applications 7. Transversality and genericity 8. Orbit growth arising from topology 9. Variational aspects of dynamics Part III. Low-Dimensional Phenomena 10. Introduction: What is low dimensional dynamics 11. Homeomorphisms of the circle 12. Circle diffeomorphisms 13. Twist maps 14. Flows on surfaces and related dynamical systems 15. Continuous maps of the interval 16. Smooth maps of the interval Part IV. Hyperbolic Dynamical Systems 17. Survey of examples 18. Topological properties of hyperbolic sets 19. Metric structure of hyperbolic sets 20. Equilibrium states and smooth invariant measures Part V. Sopplement and Appendix 21. Dynamical systems with nonuniformly hyperbolic behavior Anatole Katok and Leonardo Mendoza.

Au bord de certains polyèdres hyperboliques

The framework of this article is that of the hyperbolic groups and spaces of M. Gromov. It is motivated by the following question: how to differentiate two hyperbolic groups up to quasi-isometry ? We illustrate this problem by detailing one of Gromov’s examples taken from Asymptotic invariants for infinite groups. We describe an infinite family of hyperbolic groups, two by two non quasi-isometric, for which the boundary is Menger’s curve. The method consists of the study of their quasi-conformal structure on the boundary, using a numeric invariant: P. Pansu’s conformal dimension.

Spectral geometry and SO(4) gravity in a Riemann-Cartan spacetime

The method of coincidence limits of DeWitt is extended to autoparallels of a U 4 manifold to determine the formulae for the first three coefficients ( b 0, b 2, b 4) in the asymptotic expansion of the trace of the heat kernel of a second order laplacian-type differential operator Δ in a Riemann-Cartan spacetime. The calculation is done in the context of SO(4) gravity, with Δ acting on arbitrary spin fields transforming as scalars under the action of the coordinate transformations. The coefficients in the expansion are polynomials in certain asymptotic invariants of the metric tensor, torsion tensor, curvature tensor of the group SO(4) under which the fields transform, and their derivatives.

Spectral geometry and the Kaehler condition for complex manifolds

Recently, the relationship between the geometry of a smooth manifold and the spectrum of its Laplacian has been explored extensively. It is known that the Euler number and signature of a manifold are determined by the spectrum. Furthermore, the arithmetic genus of a complex manifold is determined by the spectrum of its complex Laplacian. Patodi [7] proved that it is possible to determine whether a real manifold is flat, constant curvature, or Einstein from the spectrum of the Laplace operator. He showed that it is possible to determine ira space is isometric to the standard sphere from the spectral geometry. In this paper, we answer a problem proposed by M. Berger [2]: we will show that it is possible to determine whether a complex Hermitian manifold M is Kaehler from the spectrum of the complex Laplacian ([~w acting on forms of type (0,0), (1,0), and (0, 1). John Sacks and the author [9] have used this result to show that it is possible to determine whether a complex Hermitian manifold is isometric to complex projective space with the standard metric from the spectrum of the complex Laplacian. The proofs of these results depend on certain asymptotic invariants of the eigenvalues of these operators. These invariants are computed by integrating corresponding local invariants of the metric tensor over the manifold. In this paper, we compute the second term in the asymptotic expansion for the operators Dp, q= (~* (~+(~*) on forms of type (p, q). In Section 1 we derive a combinatorial formula for the second term in the asymptotic expansion of an arbitrary operator using invariance theory and the results of McKean-Singer [6]. In Section 2, we compute all the invariant polynomials which are of order 2 in the derivatives of a Hermitian metric. This computation also uses invariance theory. In Section 3, we combine these results to compute the second term in the asymptotic expansion of the operators Dp, q. In the final section we derive the result cited above.

Reaction invariants in the control of continuous chemical reactors

For an arbitrary control law as applied to a continuous stirred tank reactor or a tubular reactor, it is shown that a subset of the thermodynamic states remains invariant to both the chemical reactions and the control actions. If in particular, the feed rate is used as a manipulator, it is shown that the asymptotic invariants resulting from the stoichiometric constraints remain unaffected by the control. The reduced reaction matrix and the mean residence time still determine the response modes entirely. It is shown how the reaction matrix and its eigenvalues change, as the condition change from fully isothermal through partly cooled to completely adiabatic modes of operation. The concept of reaction and control invariants is shown to carry over to a tubular reactor as well. For multicomponent mixtures, the residence time distribution of the reactor is shown to have direct and unique applications to the process dynamics when the reaction rates are linear in all the state variables.

Analytic Structure of the Triple-Regge Vertex

We study the triple-Regge, helicity-pole and related asymptotic behaviors of the six-line amplitude using the dual-resonance model (DRM) as a guide. We show that the triple-Regge vertex in the (DRM) can be expressed as a sum of four terms, which we interpret as exhibiting the allowed tree-like configurations of singularities in the asymptotic channel invariants. These four terms have counterparts in the helicity-pole limit where there exist several distinct terms with differing powers of the asymptotic variable, i.e., several different helicity poles. As an application we investigate the interesting property of the (DRM) that the triple-Regge or helicity-pole contribution to single-particle production at zero momentum transfer vanishes for trajectory intercepts near unity. We trace this effect to a nonsense zero. This suggests a mechanism for fulfilling one of the unitarity requirements for general six-line amplitudes. We discuss the possible generality of our results.

Asymptotic Invariants in Gravitational Radiation Fields

Asymptotic integral invariants are constructed for the electromagnetic and gravitational fields. The integrals are taken over closed two-dimensional surfaces embedded in a null hypersurface. In the absence of incoming radiation, the asymptotic behavior of the electromagnetic field ${F}^{\mathrm{ab}}$ and the Riemann tensor ${R}^{\mathrm{abcd}}$ is such that the integrals formed with these quantities are independent of the particular space-like surface of integration, as long as it lies in the same null hypersurface. Therefore, the integrals are related to the multipole structure of the charge distribution and the matter distribution, respectively. This relationship is shown explicitly for the electromagnetic field and for the linearized gravitational field. It follows that energy radiation as determined by the Einstein pseudotensor depends on the existence of a type III asymptotic behavior of the Riemann tensor. Finally, the asymptotic conditions are formulated under which the superpotential ${{U}_{m}}^{\mathrm{ns}}$ will also lead to asymptotically invariant integrals. It is pointed out that the linearized gravitational field with retarded potentials satisfies these conditions as do the asymptotic solutions for the Einstein field equations ${R}_{\mathrm{ab}}=0$, which have been constructed by Bondi and Newman. The significance of this result for the interpretation of the Bondi metric is discussed.

Quantization of Fields with Infinite-Dimensional Invariance Groups. II. Anticommuting Fields

The Green's function approach to the definition of commutators for fields possessing infinite-dimensional invariance groups is extended to the case of anticommuting fields. The discussion is restricted to fields which provide linear homogeneous or inhomogeneous representations of the group, a restriction which excludes no case of practical interest and facilitates setting up the formalism in a manifestly covariant way. Self-consistency of supplementary conditions, Huygens' principle, and reciprocity relations are established just as for commuting fields. Careful attention must be paid to the ordering of anticommuting factors, particularly in the demonstration of the Poisson-Jacobi identity. The invariance properties of the Poisson bracket are investigated in detail and the notion of conditional invariant is introduced. A special class of conditional invariants called asymptotic invariants, which give a complete physical characterization of initial and final states of the dynamical system, is studied in the final section.