A natural generalization of the Hénon map of the plane is a quadratic diffeomorphism that has a quadratic inverse. We study the case when these maps are volume preserving, which generalizes the the family of symplectic quadratic maps studied by Moser. In this paper we obtain a characterization of tliese maps for dimension four and less. In addition, we use Moser's result to construct a subfamil...

We investigate under which dynamical conditions the Julia set of a quadratic rational map is a Sierpiński curve.

In this paper, we consider the Frobenius endomorphism on twisted Edwards curve and give the characteristic polynomial of the map. Applying the Frobenius endomorphism on twisted Edwards curve, we construct a skew-Frobenius map defined on the quadratic twist of an twisted Edwards curve. Our results show that the Frobenius endomorphism on twisted Edwards curve and the skew-Frobenius endomorphism o...

Consider the one-parameter family of real quadratic maps fμ : [0, 1] → [0, 1] defined by fμ(x) = μx(1−x), with μ ∈ [0, 4]. Using Milnor-Thurston’s kneading theory [14], J. Guckenheimer [5] has classified up to topological conjugacy, a certain class of maps which includes the quadratic family. The idea of kneading theory is to encode information about the orbits of a map in terms of infinite seq...

We introduce a natural definition of Riesz measures and Wishart laws associated to an Ω-positive (virtual) quadratic map, where Ω ⊂ Rn is a regular open convex cone. We give a general formula for moments of the Wishart laws. Moreover, if the quadratic map has an equivariance property under the action of a linear group acting on the cone Ω transitively, then the associated Riesz measure and Wish...

For R-vector spaces U and V we consider a symmetric bilinear map B : U×U → V . This then defines a quadratic map QB : U → V by QB(u) = B(u, u). Corresponding to each λ ∈ V ∗ is a R-valued quadratic form λQB on U defined by λQB(u) = λ · QB(u). B is definite if there exists λ ∈ V ∗ so that λQB is positive-definite. B is indefinite if for each λ ∈ V ∗, λQB is neither positive nor negative-semidefi...

Dynamical systems, whether continuous or discrete, are used by physicists in order to study non-linear phenomena. In the case of discrete dynamical systems, one of the most used is the quadratic map depending on a parameter. However, some phenomena can depend alternatively of two values of the same parameter. We use the quadratic map xn+1 = 1 − axn when the parameter alternates between two valu...

Stability properties of the problem of minimizing a (nonconvex) linear-quadratic function over an Euclidean ball, known as the trust-region subproblem, are studied in this paper. We investigate in detail the case where the linear part of the objective function is perturbed and obtain necessary and sufficient conditions for the upper/lower semicontinuity of the Karush-Kuhn-Tucker point set map a...

The Fibonacci recurrence of the critical orbit appeared in the work of Branner and Hubbard on complex cubic polynomials [BH, §12] and in Yoccoz's work [Yl, Y2] on quadratic ones, as the "worst" pattern of recurrence. On the other hand, a real quadratic Fibonacci map was suggested by Hofbauer and Keller [HK] as a possible candidate for a map having a "wild" attractor (that is, a set A which is t...