Dynamics of polynomials in finite and infinite Benz planes

Abstract

The classical Benz planes, that is, Möbius, Minkowski, and Laguerre planes, can be coordinatized [cf. 1], respectively, by the field C of complex numbers, the ring of “double numbers” z=x+jy (x,y ∊ R) where an element j  not in R, with j²=1 is adjoined, and the ring of “dual numbers” z=x+ye where an element e not in R with e²=0 is adjoined to R. When the field R is replaced by another field, in our case finite prime fields F_p (p a prime), one also obtains coordinate structures for corresponding Benz planes. The dynamics of polynomials of degree at least 2 in the classical Möbius plane has attracted much attention recently because there fractal structures make their appearance. The question posed in this context has been for which values of z the sequence P_n+1(z)=P_O(P_n(z)) is bounded if P_O(z) is a function. This gave rise to the determination of Julia and Mandelbrot sets for such functions [cf. 2]. In this paper we will restrict ourselves to the case of Minkowski and Laguerre planes and to functions P_O that are polynomials of degree at least 2, with coefficients from the ground field.