Symmetries of Difference Systems on Manifolds

Abstract

Here, we shall study symmetries of difference systems of the type (2.1) via the so-called Lie symmetry vectorfields. This line of investigation was initiated by Maeda [201], see also chapter 10 of the monograph [280], for the simpler systems with f _k = f(x(k)). The main theorem of Maeda is that in the presence of such Lie symmetry vectorfields, a change of variable is possible such that the function f takes a simpler form with respect to a decomposition of the components of the new variable, and that in the case n = 1, the single equation can be linearized. We shall show that there is a similar decomposition in the case of the system (2.1). However, in the corresponding case n = 1, our result does not lead to linearization. Nevertheless, we shall demonstrate by two examples that linearization can occur for systems of a special nature.