A GEOMETRIC TREATMENT OF REDUCTION OF ORDER OF ORDINARY DIFFERENCE-EQUATIONS

We generalize the theory of Lie symmetries of ordinary difference equa tions to the non-autonomous case. A coordinate-invariant treatment in which solutions are sections of a fibre-bundle is employed. It is show n that Lie symmetries of difference equations must be projectable, whi ch is not the case for differential equations. In fact this result ext ends to partial difference equations. We also show that a time-like sy mmetry can be used to reduce a nonautonomous difference equation to th e autonomous case. Examples are given of this process and of the reduc tion of order of a non-autonomous system.