A FAMILY OF COMPLETELY PERIODIC QUADRATIC DISCRETE DYNAMICAL SYSTEM

Abstract

There is a one-to-one correspondence between homogeneous quadratic dynamical systems and commutative (possibly nonassociative) algebras. The corresponding theory for continuous systems is well known (c.f. [Markus, 1960; Walcher, 1991; Kinyon & Sagle, 1995]). In this paper the dynamics on the boundary of the basin of attraction of the origin, ∂ B Att (0), in homogeneous quadratic discrete dynamical systems is considered. In particular, we consider the dynamical behavior in a family of systems corresponding to a family of algebras [Formula: see text] which admits nilpotents of rank 2 and idempotents. The complete periodicity of a system (and the corresponding algebra) is defined and it is proven that for every n > 2 there are some systems/algebras from [Formula: see text] which are on ∂ BAtt(0) completely periodic with period n. The dynamics on ∂ B Att (0) is considered via a special class of polynomials Pn, n ∈ ℕ ∪ {0, -1}, recursively defined by Pn(α) = 2αPn-2(α) + Pn-1(α); P-1(α) = 0, P0(α) = 1, n ∈ ℕ.