Non-almost periodic solutions of limit periodic and almost periodic homogeneous linear difference systems

We study limit periodic and almost periodic homogeneous linear difference systems. The coefficient matrices of the considered systems are taken from a given commutative group. We mention a condition on the group which ensures that, by arbitrarily small changes, the considered systems can be transformed to new systems, which do not possess any almost periodic solution other than the trivial one. The elements of the coefficient matrices are taken from an infinite field with an absolute value.