Related papers: Real Polynomial Diffeomorphisms with Maximal Entro…

We consider the complexification of the Henon family of quadratic diffeomorphisms of the plane, and the region of parameter space corresponding to complex horseshoes. We discuss some conjectures about the global topology of the complex…

We consider the family of quadratic H\'enon diffeomorphisms of the plane ${\bf R}^2$. A map will be said to be a "horseshoe" if its restriction to the nonwandering set is hyperbolic and conjugate to the full 2-shift. We give a criterion for…

In this paper we use complex techniques to study the structure of real Henon diffeomorphisms of maximal topological entropy.

The purpose of this article is to investigate geometric properties of the parameter locus of the H\'enon family where the uniform hyperbolicity of a horseshoe breaks down. As an application, we obtain a variational characterization of…

A polynomial endomorphism $\sigma\in {\rm End}_K(P_n)$ is called a Jacobian map if its Jacobian is a nonzero scalar (the field has zero characteristic). Each Jacobian map $\sigma$ is extended to an endomorphism $\sigma$ of the Weyl algebra…

Zarhin has extensively studied restrictions placed on the endomorphism algebras of Jacobians $J$ for which the Galois group associated to their 2-torsion is insoluble and 'large' (relative to the dimension of $J$). In this paper we examine…

We define a family of diffeomorphism-invariant models of random connections on principal $G$-bundles over the plane, whose curvatures are concentrated on singular points. In a limit when the number of point grows whilst the singular…

In this article we analyze the global diffeomorphism property of polynomial maps $F:\mathbb{R}^n\rightarrow\mathbb{R}^n$ by studying the properties of the Newton polytopes at infinity corresponding to the sum of squares polynomials…

In his previous papers the author proved that in characteristic different from 2 the jacobian J(C) of a hyperelliptic curve C: y^2=f(x) has only trivial endomorphisms over an algebraic closure K_a of the ground field K if the Galois group…

We construct new examples of exceptional Hahn and Jacobi polynomials. Exceptional polynomials are orthogonal polynomials with respect to a measure which are also eigenfunctions of a second order difference or differential operator. The most…

The purpose of the current article is to investigate the dynamics of the H\'enon family $f_{a, b} : (x, y) \mapsto (x^2-a-by, x)$, where $(a, b)\in \mathbb{R}\times\mathbb{R}^{\times}$ is the parameter~\cite{H}. We are interested in certain…

The Jacobian conjecture over a field of characteristic zero is considered directly in view of the nonlinear partial differential equations it is associated with. Exploring the integrals of such partial differential equations, this work…

The Jacobian conjecture is a well-known open problem in affine algebraic geometry that asks if any polynomial endomorphism of the affine space $\mathbb{A}_{\mathbb{C}}^{n}$ ($n\geq2$) with jacobian $1$ is an automorphism. We present a…

In this study, a theory analogous to both the theories of polynomial-like mappings and Smale's real horseshoes is developed for the study of the dynamics of mappings of two complex variables. In partial analogy with polynomials in a single…

We construct an almost everywhere approximately differentiable, orientation and measure preserving homeomorphism of a unit $n$-dimensional cube onto itself, whose Jacobian is equal to $-1$ a.e. Moreover we prove that our homeomorphism can…

For K a field of characteristic 0 and d any integer number greater than or equal to 2, we prove the invertibility of polynomial endomorphisms of the affine space of dimension d over K of the form F=Id+H, where each coordinate of H is the…

The Jacobi polynomials $\hat{P}_n^{(\alpha,\beta)}(x)$ conform the canonical family of hypergeometric orthogonal polynomials (HOPs) with the two-parameter weight function $(1-x)^\alpha (1+x)^\beta, \alpha,\beta>-1,$ on the interval…

We describe an algorithm, based on the properties of the characteristic polynomials of Frobenius, to compute $\operatorname{End}_{\overline{K}}(A)$ when $A$ is the Jacobian of a nice genus-2 curve over a number field $K$. We use this…

Let H: C^2 -> C^2 be the Henon mapping given by (x,y) --> (p(x) - ay,x). The key invariant subsets are K_+/-, the sets of points with bounded forward images, J_+/- = the boundary of K_+/-, J = the union of J_+ and J_-, and K = the union of…

We call a partially hyperbolic diffeomorphism \emph{partially volume expanding} if the Jacobian restricted to any hyperplane that contains the unstable bundle $E^u$ is larger than $1$. This is a $C^1$ open property. We show that any…