Livsic realization of 2D-behaviors with degree one autonomy

Early work on behaviors associated with N-D systems emphasized two extreme types of behaviors, namely, autonomous (no free variables) and controllable (existence of a global trajectory patching together the past part of some trajectory with the future part of some other trajectory). It was shown that any behavior can be decomposed as an autonomous part plus a controllable part. Later work gave a more refined classification of autonomous behaviors: nonautonomous behaviors are those with a free variable defined over the whole lattice and have autonomy degree 0 while autonomous behaviors are now further classified into those with autonomy degree k for k = 1; 2,...,N, infin, where k = infin for the zero behavior and otherwise N - k is the maximal dimension of a sublattice on which it is possible to assign a free variable. Independently of this work with motivation from multivariable operator theory, Livsic and coworkers introduced a certain type of overdetermined 2-D input/state/output linear system where inputs and outputs are required to satisfy their own compatibility difference equations. The purpose of this note is to identify the place of Livsic systems in the behavioral framework: the input-output behavior of a Livsic system roughly can be characterized as a 2-D behavior with autonomy degree equal to 1.