Dynamics of polynomials in finite and infinite Benz planes

  • Rafael Artzy


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 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 Fp (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 Pn+1(z)=PO(Pn(z)) is bounded if PO(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 PO that are polynomials of degree at least 2, with coefficients from the ground field.