Second order homological obstructions on real algebraic manifolds

Let Y be a compact nonsingular real algebraic set whose homology classes (over Z/2) are represented by Zariski closed subsets. It is well known that every smooth map from a compact smooth manifold to Y is unoriented bordant to a regular map. In this paper, we show how to construct smooth maps from compact nonsingular real algebraic sets to Y not homotopic to any regular map starting from a nonzero homology class of Y of positive degree. We use these maps to obtain obstructions to the existence of local algebraic tubular neighborhoods of algebraic submanifolds of Rn and to study some algebro-homological properties of rational real algebraic manifolds.     

Hyperbolicity of the invariant sets for the real polynomial maps

In this paper, the conditions under which there exits a uniformly hyperbolic invariant set for the map f_a(x) = ag(x) are studied, where a is a real parameter, and g(x) is a monic real-coefficient polynomial. It is shown that for certain parameter regions, the map has a uniformly hyperbolic invariant set on which it is topologically conjugate to the one-sided subshift of finite type for A, where ∣a∣ is sufficiently large, A is an eventually positive transition matrix, and g has at least two different real zeros or only one real zero. Further, it is proved that there exists an invariant set on which the map is topologically semiconjugate to the one-sided subshift of finite type for a particular irreducible transition matrix under certain conditions, and one type of these maps is not hyperbolic on the invariant set.     

On the zeros of monotone operators of retarded type in a Banach space

The existence of zeros of operators which map into a Banach space E but whose domain is the space of continuous functions on some real interval into E are considered.     

Approximation of maps into spheres by regulous maps

Let X be a compact real algebraic set of dimension n. We prove that every Euclidean continuous map from X into the unit n-sphere can be approximated by a regulous map. This strengthens and generalizes previously known results.     

Regular versus continuous rational maps

We prove that a continuous map from a compact nonsingular real algebraic variety X into the unit 2-sphere can be approximated by regular maps if and only if it is homotopic to a continuous map which is regular in the complement of a Zariski closed subvariety A of X of codimension at least 3. The assumption on the codimension of A is essential.     

A generalized Harish-Chandra method of descent for reductive Lie algebras

Let G be a connected real reductive group and M a connected reductive subgroup of G with Lie algebras g and m, respectively. We assume that g and m have the same rank. We define a map from the space of orbital integrals of m into the space of orbital integrals of g which we call a transfer. We then consider the transpose of the transfer. This can be viewed as a map from the space of G-invariant distributions of g to the space of M-invariant distributions of m and can be considered as a restriction map from g to m. We prove that this map extends Harish-Chandra method of descent and we obtain a generalization of the radial component theorem. We give an application.     

CMV Matrices and Little and Big ?1 Jacobi Polynomials

We introduce a new map from polynomials orthogonal on the unit circle to polynomials orthogonal on the real axis. This map is closely related to the theory of CMV matrices. It contains an arbitrary parameter ?? which leads to a linear operator pencil. We show that the little and big ?1?Jacobi polynomials are naturally obtained under this map from the Jacobi polynomials on the unit circle.     

Ergodic transport theory and piecewise analytic subactions for analytic dynamics

We consider a piecewise analytic real expanding map f: [0, 1] ?? [0, 1] of degree d which preserves orientation, and a real analytic positive potential g: [0, 1] ?? ?. We assume the map and the potential have a complex analytic extension to a neighborhood of the interval in the complex plane. We also assume log g is well defined for this extension. It is known in Complex Dynamics that under the above hypothesis, for the given potential ?? log g, where ?? is a real constant, there exists a real analytic eigenfunction ? _?? defined on [0, 1] (with a complex analytic extension) for the Ruelle operator of ?? log g. Under some assumptions we show that $\frac{1} {\beta }\log \varphi _\beta$ converges and is a piecewise analytic calibrated subaction. Our theory can be applied when log g(x) = ?log f??(x). In that case we relate the involution kernel to the so called scaling function.     

Algebraic and real K-theory of Real varieties

An algebraic variety defined over the real numbers has an associated topological space with involution, and algebraic vector bundles give rise to Real vector bundles. We show that the associated map from algebraic K-theory to Atiyah's Real K-theory is, after completion at two, an isomorphism on homotopy groups above the dimension of the variety.     

On the dimension of some real spaces of bounded rank matrices

Given n∈N, let X be either the set of hermitian or real n×n matrices of rank at least n-1. If n is even, we give a sharp estimate on the maximal dimension of a real vector space V⊂X∪{0}. The results are obtained, via K-theory, by studying a bundle map induced by the adjunction of matrices.     

Extension of Gaussian cylinder set measures

Let E be a one-to-one continuous map of the real and separable Hilbert space H into the real and separable Hilbert space K, with E having dense range. One considers Gaussian cylinder set measures on H defined by weak covariance operators. Such cylinder set measures may be used to induce, through E, Gaussian cylinder set measures on K. The result of this paper extends a result of Sato: it characterizes the norm of the spaces K for which the induced measure extends to a probability measure on the Borel sets of K. Such a result is of interest in the robustness study of signal detection.     

Continuous Maps on Aronszajn Trees

Assuming $\diamondsuit$ : Whenever B is a totally imperfect set of real numbers, there is special Aronszajn tree with no continuous order preserving map into B.     

Iterative sequence for asymptotically demicontractive maps in Banach spaces

Let E be a real Banach space and T:E→E an asymptotically demicontractive and uniformly L-Lipschitzian map with F(T):={x∈E:Tx=x}≠∅. We prove necessary and sufficient conditions for the strong convergence of the Mann iterative sequence to a fixed point of T.     

Wave propagation properties in oscillatory chains with cubic nonlinearities via nonlinear map approach

Free wave propagation properties in one-dimensional chains of nonlinear oscillators are investigated by means of nonlinear maps. In this realm, the governing difference equations are regarded as symplectic nonlinear transformations relating the amplitudes in adjacent chain sites (n, n + 1) thereby considering a dynamical system where the location index n plays the role of the discrete time. Thus, wave propagation becomes synonymous of stability: finding regions of propagating wave solutions is equivalent to finding regions of linearly stable map solutions. Mechanical models of chains of linearly coupled nonlinear oscillators are investigated. Pass- and stop-band regions of the mono-coupled periodic system are analytically determined for period-q orbits as they are governed by the eigenvalues of the linearized 2D map arising from linear stability analysis of periodic orbits. Then, equivalent chains of nonlinear oscillators in complex domain are tackled. Also in this case, where a 4D real map governs the wave transmission, the nonlinear pass- and stop-bands for periodic orbits are analytically determined by extending the 2D map analysis. The analytical findings concerning the propagation properties are then compared with numerical results obtained through nonlinear map iteration.     

Jost maps, ball-homogeneous and harmonic manifolds

Given a real number ε>0, small enough, an associated Jost map J_ε between two Riemannian manifolds is defined. Then we prove that connected Riemannian manifolds for which the center of mass of each small geodesic ball is the center of the ball (i.e. for which the identity is a J_ε map) are ball-homogeneous. In the analytic case we characterize such manifolds in terms of the Euclidean Laplacian and we show that they have constant scalar curvature. Under some restriction on the Ricci curvature we prove that Riemannian analytic manifolds for which the center of mass of each small geodesic ball is the center of the ball are locally and weakly harmonic.     

Semistable points with respect to real forms

We consider actions of real Lie subgroups G of complex reductive Lie groups on Kählerian spaces. Our main result is the openness of the set of semistable points with respect to a momentum map and the action of G.     

On amoebas of algebraic sets of higher codimension

The amoeba of a complex algebraic set is its image under the projection onto the real subspace in the logarithmic scale. We study the homological properties of the complements of amoebas for sets of codimension higher than 1. In particular, we refine A. Henriques’ result saying that the complement of the amoeba of a codimension k set is (k ? 1)-convex. We also describe the relationship between the critical points of the logarithmic projection and the logarithmic Gauss map of algebraic sets.     

Extension of maps and ordering

LetX be a topological space,Y a closed subspace and π:x→T, ψ:Y→T be two continuous maps. We shall say that ψ can be extended by π if there exists a continuous man η=ν(π, ψ):X→T such that: η| _x?y ?π, η| _Y =ψ. Clearly a similar definition can be given in the category of real or complex algebraic varietes. In this paper we give some sufficient conditions to ensure that map ψ can be extended by π. In particular we study the topological and the real algebraic case. It seems that the last setting is the more interesting.     

Mandelbrot-like sets in dynamical systems with no critical points

We report the discovery of an infinite quantity of Mandelbrot-like sets in the real parameter space of the Hénon map, a bidimensional diffeomorphism not obeying the Cauchy–Riemann conditions and having no critical points. For practical applications, this result shows to be possible to stabilize infinitely many complex phases by tuning real parameters only. To cite this article: A. Endler, J.A.C. Gallas, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Strongly stable real infinitesimally symplectic mappings

We prove that a mapA εsp(σ,R), the set of infinitesimally symplectic maps, is strongly stable if and only if its centralizerC(A) insp(σ,R) contains only semisimple elements. Using the theorem that everyB insp(σ,R) close toA is conjugate by a real symplectic map to an element ofC(A), we give a new proof of the openness of the set of strongly stable maps. Then we prove that the set of strongly stable maps is the interior of the set of all infinitesimally symplectic maps with purely imaginary or zero eigenvalues, and the connected components of this set are described. Finally, we give a new proof of the analytic conjugacy theorem for an analytic curve through a given strongly stable map.