On quadratic orbital networks

These are some informal remarks on quadratic orbital networks over finite fields Zp. We discuss connectivity, Euler characteristic, number of cliques, planarity, diameter and inductive dimension. We prove that for d = 1 generators, the Euler characteristic is always nonnegative and for d = 2 and large enough p the Euler characteristic is negative. While for d = 1, all networks are planar, we suspect that for d ≥ 2 and large enough p, all networks are non-planar. As a consequence on bounds for the number of complete subgraphs of a fixed dimension, the inductive dimension of all these networks goes 1 as p→∞. 1. Polynomial orbital networks Given a field R = Zp, we study orbital graphs G = (V,E) defined by polynomials Ti which generate a monoid T acting on R. We think of (R, T ) as a dynamical system where positive time T is given by the monoid of words w = w1w2 . . . wk using the generators wk ∈ A = {T1, . . . , Td } as alphabet and where {Tx | w ∈ R } is the orbit of x. The orbital network [1, 3, 4] is the finite simple graph G where V = R is the set of vertices and where two vertices x, y ∈ V are connected if there exists Ti such that Ti(x) = y or Ti(y) = x. The network generated by the system consists of the union of all orbits. As custom in dynamics, one is interested in invariant components of the system and especially forward attractors Ω(x) of a point x as well as the garden of eden, the set of points which are not in the image of any Ti. We are also interested in the inductive dimension of the network. This relates to the existence and number of cliques, which are complete subgraphs of G. Since most questions one can ask are already unsettled for quadratic polynomial maps, we restrict here to polynomial maps of the form Date: December 1, 2013. 1991 Mathematics Subject Classification. Primary: 05C82, 90B10,91D30,68R10 .