A note on the theoretical existence of invariants: Part I

We propose a framework in which to study invariants of difference equations with the ultimate goal of establishing the (theoretical) existence of effective invariants. The paper is divided into two parts. Part I begins with several examples which illustrate some of the difficulties to be overcome and ends with a discussion of the connection between invariants and decompositions of the phase space. In Part II, which is more topological in nature, we will pursue the theme of invariant decompositions.