Geometry of linear maps over finite fields

Abstract

The finite fields of degree two are reinterpreted as discrete phase spaces on the two-dimensional torus. The authors study dynamical systems obtained by iterating linear maps over these fields, from a geometrical viewpoint. These maps can be regarded as the two-dimensional discrete equivalent of a Bernoulli shift. They yield irregular motions, which may coexist with spatial order. They find that the dynamics of orbits of long period can be characterized as a percolation process. The question of randomness in dynamical systems over finite sets is discussed.