Quadratic Endomorphisms Research Articles

Sums and products of pairs of quadratic endomorphisms of a countable-dimensional vector space

Let V be a vector space with countable dimension over a field, and let u be an endomorphism of it which is locally finite, i.e. are linearly dependent for all x in V. We give several necessary and sufficient conditions for the decomposability of u into the sum of two square-zero endomorphisms. Moreover, if u is invertible, we give necessary and sufficient conditions for the decomposability of u into the product of two involutions, as well as for the decomposability of u into the product of two unipotent endomorphisms of index 2. Our results essentially extend the ones that are known in the finite-dimensional setting. In particular, we obtain that every strictly upper-triangular infinite matrix with entries in a field is the sum of two square-zero infinite matrices (potentially non-triangular, though) and that every upper-triangular infinite matrix (with entries in a field) with only on the diagonal is the product of two involutory infinite matrices.

Bifurcation of the Birth of a Closed Invariant Curve in a One-Parameter Family of Quadratic Mappings of the Plane

We give an example of a one-parameter family of quadratic endomorphisms of the plane, in which a closed invariant curve is born from an elliptic fixed point.

Remarks on Quadratic Mappings

We present several results concerning the geometry and topology of quadratic mappings. The main attention is given to certain basic properties, namely, properness, surjectivity, stability, and topology of fibers. The structure of singular sets, discriminants, bifurcation diagrams, and Pareto sets is also discussed. After describing the setting and algebraic methods for computing the topological degree and Euler characteristic that are crucial for our approach, we concentrate on the study of quadratic endomorphisms and quadratic mappings into the plane. We begin by considering homogeneous quadratic endomorphisms of the plane and give criteria of properness and possible values of topological degree, as well as some geometric information about the structure of singular sets and discriminants. Next, we deal with analogs of the above results for proper quadratic endomorphisms in arbitrary dimension. In particular, we obtain an explicit estimate for the topological degree of quadratic endomorphism in terms of dimension and present examples showing that this estimate is exact. After this we discuss homogeneous quadratic mappings from ℝn into the plane and obtain a number of results on the Euler characteristic and topology of fibers. Finally, we derive some corollaries in the case of numerical range mapping of a complex square matrix.

Almost Analytic Kähler Forms with Respect to a Quadratic Endomorphism with Applications in Riemann-Finsler Geometry

The almost analyticity with respect to a quadratic endomorphism T is introduced in an algebraic setting concerning a commutative and associative algebra \(\mathcal {A}\). Two main properties are proved: the first concerns the simultaneous closedness for an almost analytic 1-form \(\omega \) and \(T\omega \) while the second regards the vanishing of the interior product of such a form with the Nijenhuis tensor of T. Also, we introduce an extension of the Frolicher–Nijenhuis formalism to this framework as well as a hermitian type property. When \(\mathcal {A}\) is the algebra of smooth functions on a given (even dimensional) manifold we recover the classical notion of almost analytic 1-form. We study this analyticity and the hermitian type property for the Cartan 1-form of a Riemann-Finsler geometry. Also, we study the almost analytic functions on the tangent bundle of a Riemann-Finsler geometry with respect to the associated almost para-complex and almost complex structure of this geometry. We introduce two new types of Hessian and respectively Laplacian corresponding to these structures. Two types of gradient Ricci solitons are introduced in the tangent bundle.

On the quadratic Morse-Smale endomorphisms

For the quadratic map Fμ = (xy, (x − μ)2) we indicate an interval of parameter values, such that map Fμ with a parameter value in the indicated interval is a (nonsingular) Morse-Smale endomophism.

Sums of Quadratic Endomorphisms of an Infinite-Dimensional Vector Space

AbstractWe prove that every endomorphism of an infinite-dimensional vector space over a field splits into the sum of four idempotents and into the sum of four square-zero endomorphisms, a result that is optimal in general.

Endomorphisms of free modules as sums of four quadratic endomorphisms

We will prove that every endomorphism of a free right R-module of infinite rank can be decomposed as a sum of four endomorphisms which satisfy some fixed polynomial identities.

Sums of three quadratic endomorphisms of an infinite-dimensional vector space

Sums of three quadratic endomorphisms of an infinite-dimensional vector space

Conjugate connections with respect to a quadratic endomorphism and duality

The goal of this paper is to consider the notion of conjugate connection in a unifying setting for both almost complex and almost product geometries, having as model the works of Mileva Prvanovic. A main interest is in finding classes of conjugate connections in duality with the initial linear connection; for example in the exponential case of almost complex geometry we arrive at a rule of quantization.

Linearly triangularizable quadratic endomorphisms

In the article, we prove that any polynomial endomorphism of , where is a field of characteristic zero, of the form ‘identity + homogeneous component of degree two’ is linearly triangularizable when the Jacobian matrix of the quadratic term is nilpotent of index at most three. In particular, this proves the well-known Meisters and Olech Conjecture. In our approach, besides methods of the classical matrix theory, some elements of the theory of commutative non-associative algebras appear in the context of the so-called polarization algebras associated with quadratic homogeneous endomorphisms. A powerful role of polarization algebras in the investigation of polynomial endomorphisms is revealed in a recent paper of Umirbaev. His results and comments were the main inspiration for us.

Moduli Spaces of Quadratic Rational Maps with a Marked Periodic Point of Small Order

The surface corresponding to the moduli space of quadratic endomorphisms of $\mathbb{P}^1$ with a marked periodic point of order $n$ is studied. It is shown that the surface is rational over $\mathbb{Q}$ when $n\le 5$ and is of general type for $n=6$. An explicit description of the $n=6$ surface lets us find several infinite families of quadratic endomorphisms $f: \mathbb{P}^1 \to \mathbb{P}^1$ defined over $\mathbb{Q}$ with a rational periodic point of order $6$. In one of these families, $f$ also has a rational fixed point, for a total of at least $7$ periodic and $7$ preperiodic points. This is in contrast with the polynomial case, where it is conjectured that no polynomial endomorphism defined over $\mathbb{Q}$ admits rational periodic points of order $n>3$.

Almost analytic forms with respect to a quadratic endomorphism and their cohomology

The goal of this paper is to consider the notion of almost analytic form in a unifying setting for both almost complex and almost paracomplex geometries. We use a global formalism, which yields, in addition to generalizations of the main results of the previously known almost complex case, a relationship with the Frolicher-Nijenhuis theory. A cohomology of almost analytic forms is also introduced and studied as well as deformations of almost analytic forms with pairs of almost analytic functions.

ON THE NUMBER OF RATIONAL ITERATED PREIMAGES OF THE ORIGIN UNDER QUADRATIC DYNAMICAL SYSTEMS

For a quadratic endomorphism of the affine line defined over the rationals, we consider the problem of bounding the number of rational points that eventually land at the origin after iteration. In the article "Uniform bounds on pre-images under quadratic dynamical systems," by two of the present authors and five others, it was shown that the number of rational iterated preimages of the origin is bounded as one varies the morphism in a certain one-dimensional family. Subject to the validity of the Birch and Swinnerton-Dyer conjecture and some other related conjectures for the L-series of a specific abelian variety and using a number of modern tools for locating rational points on high genus curves, we show that the maximum number of rational iterated preimages is six. We also provide further insight into the geometry of the "preimage curves."

Bounded solutions of quadratic circulant difference equations

In this paper we develop some techniques to obtain global hyperbolicity for a certain class of endomorphisms of with This kind of endomorphisms are obtained from vectorial difference equations where the mapping defining these equations satisfy a circulant condition. In particular, we show that one-parameter families of these quadratic endomorphisms are hyperbolic for large values of the parameter.

On the Quadratic Endomorphisms of the Plane

Abstract In this article we establish the following result: if a nondegenerate quadratic endomorphism of the plane has no fixed points, then every point has empty omega-limit set and alpha-limit set. It is also shown that there exists a six parameter family open and dense in the space of all quadratic mappings of the plane (even those having fixed points). The degenerate case (when the quadratic forms of both components are linearly dependent), for which the theorem fails, is considered in the last section.

On the Dynamics of $n$-Dimensional Quadratic Endomorphisms

Considering a convex endomorphism F (its n coordinates are convex functions) and the one parameter family Fμ=F−μν, where ν is any vector of ℝn, we find sufficient conditions in order that for large values of the parameter, the dynamical behavior of Fμ is completely described: either the nonwandering set Ω(Fμ) is empty or Fμ restricted to Ω(Fμ) is an expanding map. These conditions are shown to be generic in the space of quadratic endomorphisms.