Reversible linear and nonlinear discrete-time dynamics

It is shown that any causal discrete-time system can be realized with a reversible or invertible state variable representation. This bridges the gap with continuous time, where flows always satisfy this property, and settles an old debate on the relevance of invertibility in nonlinear discrete dynamics. As the result applies to the linear case, it is shown how the classic approach should be slightly modified, and several examples of the benefits that can be obtained are given. Difference algebra is employed for nonlinear systems. It leads to a most elegant characterization of causality, which enables reversibility to be demonstrated. Module theory suffices in the linear case, which is supplemented by elementary matrix manipulations