Finding Lie groups that reduce the order of discrete dynamical systems

Discrete dynamical systems of the form xm+1 = f (xm) are considered, where xm is an n-component vector. Equations X = f (x) define a mapping f from an n-dimensional projective space into itself. Each component of f is a rational function, i.e. a ratio of polynomials in n dynamical variables. Maeda showed that when f commutes with each transformation gα of a Lie group, a reduction in the order of the dynamical system results. Given a discrete dynamical system, the difficulty is to find its continuous symmetries. We present a way of using f -invariant sets to find these symmetries. The approach taken is to arrange groups in order of increasing complication and to characterize the set of dynamical systems admitting each group. Criteria are given for recognizing and reducing the order of systems admitting subgroups of the projective general linear group in n variables, PGL(n), or certain Lie subgroups of the Cremona group of birational transformations in n variables, Cn. Quispel et al demonstrated the use of canonical group variables for achieving this reduction. We develop canonical coordinates for several groups with elementary Lie algebras and demonstrate reduction of order in each case. Results are used to reduce the order of several examples of recursion formulae taken from the literature on renormalizable lattice models.